New preprints, ranked for me

Policy decisions often depend on whether the average treatment effect (ATE) exceeds a specific cutoff. In instrumental variable (IV) settings, point identification of the ATE requires strong assumptions that are often difficult to justify. Therefore, many researchers shift their attention to identification of the local (L) ATE or target bounds of the ATE. Both strategies have well-known drawbacks: the substantive relevance of the LATE is frequently questioned, and the bounds are often too wide to be informative. We propose a simple sensitivity analysis that allows researchers to maintain focus on the ATE, and then discuss its connection to related bounding procedures. Based on these results, researchers can decide, in a principled manner, whether the sign of the ATE is inferable under assumptions they deem credible. The proposed strategies are easy to apply, and we illustrate their value with two real-data examples. The R package ivate implements our procedures.

We extend the indirect inference principle to structural econometric models that are incomplete, in the sense that a single value of the structural parameter may be compatible with multiple distributions of the observed endogenous variables, so that the structural parameter is typically only partially identified. We define the binding function as the smallest norm of the auxiliary estimating equations over the set of data generating processes compatible with the structural model. Hence, we obtain a confidence region by inverting a test of the hypothesis that the binding function vanishes at the pseudo-true value. A dual characterization based on random set theory makes the binding function computationally tractable. We define the auxiliary model conditionally on the covariates to remove its dependence on the unknown covariate distribution and we show that the binding function is Lipschitz and Hadamard directionally differentiable, so that the numerical delta method delivers valid critical values. We show that the method provides significant computational advantages without relinquishing informativeness in a simulation experiment based on an entry game with 100 potential entrants.

We study time-varying quantile VAR models in which the conditional quantile does not propagate linearly through the recursion, and develop a toolkit for impulse response analysis, which to the best of our knowledge has not been established elsewhere. For example, we establish Bahadur representation for quantile VAR regression, infer model parameters via smoothed check function, and define and investigate the best linear projection of the time-varying tau-quantile impulse response via local projections. An application to U.S. industrial production growth and credit spreads from 1919-2025 uncovers structural shifts in tail relationships, demonstrating downside asymmetry diminishes at longer forecast horizons.

We develop a large-$n$, large-$T$ theory for the unsmoothed fixed-effects quantile regression estimator, paralleling Hahn and Newey (2004) for smooth nonlinear panel models. An exact rank-score identity for the profiled criterion, with a convexity argument, delivers the leading incidental parameter bias of the slope estimator, of order $T^{-1}$, in closed form. When $n$ and $T$ grow at the same rate, the estimator is asymptotically normal but in general not centred, and conventional intervals are invalid. Admissible growth of $n$ relative to $T$, up to $n=o(T^3)$, depends on the regularity of the conditional error densities. A split-panel jackknife removes the leading bias throughout this range and yields valid inference. In serially dependent panels whose dependence decays geometrically or polynomially, the bias acquires a term from the dependence between regressors and errors of different periods, the jackknife remains valid with contiguous but not with interleaved half-panels, and clustered standard errors are valid.

We consider multiplicity and uniqueness of equilibria in continuous-time dynamic discrete games where players move sequentially according to independent Poisson processes. With multiple equilibria, the structural model is incomplete: the reduced form and counterfactual predictions depend on how equilibria are selected. We provide closed-form sufficient conditions for uniqueness of Markov perfect equilibrium in this class of games. The conditions are semiparametric and can be verified from model primitives. To measure the coverage of the conditions and characterize multiplicity more broadly, we carry out a large-scale numerical search for equilibria of a dynamic duopoly model of entry and exit. Although uniqueness is the norm in our broad search, we find that multiplicity concentrates in economically relevant regions: where the market cannot profitably sustain both firms, where firms are very patient, and at intermediate decision frequencies that balance strategic interaction and commitment. When multiplicity is present, we typically find two stable asymmetric equilibria and one unstable symmetric equilibrium. Monte Carlo and counterfactual simulations show this structure matters for empirical work. A full-solution estimator that assumes uniqueness can converge to the unstable equilibrium, even when the data were generated by a stable one, while a two-step estimator recovers the correct equilibrium.

This note contains a story and a paper. The paper is about the admissibility of two-stage least squares with weak instruments, a topic we have worked on together since 2019. The story is about how we (or, rather, our LLMs) wrote the paper, and some thoughts on what this may mean for our field.

We review how residual confounding in large observational data sets can mistakenly lead researchers to declare a predictor variable statistically significant. As sample sizes become very large, standard errors decrease dramatically, causing unmeasured variables correlated with the predictor of interest to overwhelm the model and yield precise estimates of the wrong thing, the residual confounder. Three examples from environmental epidemiology are provided. Specification Curve Analysis, SCA, and Local Control Analysis, LCA, are offered to reveal model selection bias and true null effects.

Special-science explanations often concern distributions rather than individual outcomes. Since higher-level states depend on lower-level processes, what would make a higher-level explanation redundant? I argue that redundancy requires replacement without loss, not merely a reconstruction of individual outcomes. Distributional counterfactual dependence (DCD) identifies how changes in higher-level conditions alter an outcome distribution or a relevant feature of it. Against the background of stochastic supervenience, I distinguish this explanatory role from the dependence of higher-level states on lower-level bases. A replacement must be specified independently of the distribution to be explained and preserve the same counterfactual pattern at the grain and across the conditions relevant to the question. Applications to wage distributions, cognitive load, and selection clarify this requirement. A hypothetical comparison shows how a lower-level model can preserve the explanation and why matching selected outcome statistics may not suffice. DCD thus describes one form of non-reductive explanation that depends on counterfactual organization rather than additional higher-level productive powers.

We propose a test for essential heterogeneity in the marginal treatment effects model without imposing additive separability. On the observed support of the propensity score, no essential heterogeneity implies that the conditional mean of the outcome is affine in the propensity score, and this affine restriction is the sharp testable implication of the null. We implement the test using a projected Gaussian-process statistic that is Neyman-orthogonal to the first-step estimation error, so the multiplier bootstrap avoids re-estimating nuisance parameters. Monte Carlo experiments show accurate size control and good power, and, unlike the fixed-scale integrated conditional moment benchmark, the data-driven median rule retains power as the conditioning set grows in dimension. In an application to the Ugandan ELA programme, the omnibus statistic rejects the null of no essential heterogeneity for aspirations at the midline follow-up.

Under interference, the treatment of some units affects the outcomes of others, so that the effect on a treated unit consists of a direct and a spillover component. A non-treated comparison group then generally identifies neither component, but the difference between the total effect on the treated and the spillover effect on the comparison units. We show that both components are identified separately when multiple pre-treatment periods are available, without an uncontaminated comparison group. Under no anticipation, pre-treatment outcomes are free of both components for every unit. Extrapolating the pre-treatment trend within a group, as in interrupted time series approaches, therefore identifies the counterfactual outcome in the absence of any treatment. Applied to the treated it gives the total effect on them, applied to the non-treated at a given level of exposure the spillover effect on them. Disentangling the two components in addition requires effect homogeneity across treated and non-treated units with the same exposure and covariates, as well as common support in exposure. We derive doubly robust and Neyman-orthogonal expressions for the effects of interest, which permits controlling for covariates by machine learning, and establish asymptotic normality. A simulation study confirms the root-$n$ behaviour of the estimators and the reliability of the standard errors. We apply the framework to the compulsory licensing of enemy-owned patents in the United States in 1919. Licensing raised patenting in the licensed subclasses and in technologically close ones alike, so that a comparison-group design understates the total effect.

Regression adjustment is widely used to improve the precision of average treatment effect estimates from randomized controlled trials. Interacted regression weakly dominates simple regression in terms of its first-order behavior but, in 50-50 randomized trials, the asymptotic variance of both methods is the same and so relative performance is determined by higher-order behavior. Here, we conduct a higher order analysis to compare simple vs. interacted regression adjustments in the equal-probability case. The relative performance of the methods depends on a number of factors including feature skewness, treatment-effect heterogeneity, heteroskedasticity, and nonlinear misspecification, and can favor either estimator. However, in some practically important settings, including when features are skewed and outlying feature values are associated with higher outcome noise, simple regression can outperform interacted regression by a material amount.

Difference-in-differences (DiD) designs are typically used to identify and estimate average causal effects, even though treatment can affect the entire outcome distribution. This has led to a growing interest in distributional DiD methods. However, identifying the full counterfactual distribution requires a parallel trends assumption on the entire distributional dynamics, which can be too strong in practice. We address this problem by introducing a spectrum of identifying restrictions that lie between the average and fully distributional DiD by imposing parallel trends on $L$-moments. This corresponds to a basis expansion of the distributional dynamics. Importantly, each $L$-moment is interpretable as familiar distributional components: location, dispersion, skewness, kurtosis, etc. Researchers can pick and choose which features they want to target. For example, the second $L$-moment identifies the Gini mean difference, but does not require imposing mean-level or higher-order parallel trends. Taking a collection of $L$-moments can approximate distributional dynamics under weaker conditions since it leaves untargeted components unrestricted. We develop conditions under which higher-order $L$-moments decay, efficient estimators, and simultaneous inference across growing collections of moments. We demonstrate the benefits of our framework by exploring distributional changes to healthcare spending after the rollout of Medicare Part D.

Weakly dependent idiosyncratic errors are standard in mean factor models, but quantile factor models have been analyzed only under independence. The mean-case definition of weak dependence, a bound on the eigenvalues of the error covariance matrix, does not carry over: I show that it neither implies nor is implied by the property that controls quantile estimation, since uncorrelated errors can share a pervasive quantile component. I instead define approximate quantile factor models through a concentration condition on the idiosyncratic array. Under it, the quantile factor analysis (QFA) estimator is consistent at a rate that deteriorates by the square root of a dependence constant, which is minimax optimal, and the number of factors is consistently estimated. Dependence changes the asymptotic variance, for which I provide HAC and thresholded estimators. Monte Carlo experiments confirm these predictions. In FRED-QD and FRED-MD, standard errors ignoring dependence understate uncertainty, most strongly in the tails.

We study estimation of the average partial effect (APE) in an irregular correlated random coefficients panel data model in which the time dimension equals the number of random coefficients, $T=p$, and the density function is degenerate. We extend Graham and Powell (2012) in two directions. First, we allow the density of $d_i=\det(X_i)$ to vanish or diverge at the origin at the polynomial rate $|d|^{a}$ with $a>-1$, and show that the aggregate time effects and the APE remain identified for every such $a$, extending earlier work studying the case $a = 0$. Second, we propose a three-step estimator: an $m$th-order local polynomial regression over the stayers for the time effects; a local polynomial fit of the conditional mean coefficient just outside the trimming threshold; and an imputation step that extrapolates the latter into the stayer region rather than discarding stayers. We establish joint asymptotic normality in three regimes determined by the sign of $a$, and obtain a convergence rate approaching $n^{-(m+1)/(2m+3)}$ when $a=0$, against $n^{-1/3}$ for Graham and Powell (2012), whose estimator is the special case $m=0$, $a=0$. The results show that smoothness of the underlying conditional expectations can be traded for a faster convergence rate in irregularly identified models. Simulations confirm lower bias and root mean squared error across designs. We redo the overidentification analysis of a markup model in Raval (2023), allowing for correlated random elasticities.

In longitudinal studies, repeated treatments may be followed by different sequences of mediators and time-varying covariates. We develop semiparametric inference for path-specific effects in this setting, allowing the number, type, and ordering of intermediate variables to vary across treatment occasions. Under consistency, positivity, edge consistency, and the cross-world restrictions of a nonparametric structural equation model with independent errors, path-specific means are identified by the edge g-formula. We derive two parameterizations of the efficient influence function of the identifying functional. The first follows the observed-data factorization and uses conditional intermediate-variable densities. The second uses sequential regressions to replace the corresponding integrations and Bayes' rule to express density ratios through treatment-history regressions, thereby avoiding direct estimation of intermediate-variable densities. Each yields a one-step estimator that permits flexible nuisance estimation. We establish product-rate conditions for asymptotic linearity and semiparametric efficiency. The density-based estimator has blockwise multiple robustness, generalizing triply robust estimation, while the regression-based estimator has sequential multiple robustness. We further develop a sensitivity analysis for selected violations of cross-world independence assumptions used to identify these effects. Simulations examine finite-sample performance and robustness to model misspecifications. An application to the Framingham Heart Study examines sustained-smoking effects on systolic blood pressure along pathways through repeated body mass index measurements. The proposed methods are implemented in the R package flexPaths on Gihub at xxou/flexPaths.

Causal inference commonly defines an individual treatment effect as a quantity that varies across units. But we know such effects, even for a fixed unit, can also vary over time, across repeated studies, or over different methods of measurement, and they can even exhibit inherent stochasticity. In this work, we consider how such additional sources of variation create a need for more general notions of average treatment effects, including some estimands that have yet to be more formally explored by the causal inference community. This perspective also motivates a reconceptualization of our most common existing estimands, such as conditional average, sample average, and individual treatment effects (CATEs, SATEs, and ITEs), as subject to more axes of variation than are typically considered. We explore two main consequences of this reframing in the case where we have data from a single randomized controlled trial: (1) decision making in fixed samples and (2) uncertainty quantification during estimation. We find that different estimands, even for the same unit or the same sample, can qualitatively disagree on whether an effect is positive or negative. We demonstrate these issues empirically through a case study with real data and characterize through a simulation study how our assumptions about these sources of variation, even if we cannot observe the variation, can be consequential for estimation and proper coverage performance. Importantly, these phenomena arise even in the simplest of cases, where all structural and parametric assumptions are satisfied. More broadly, this paper aims to provide a preliminary investigation into an underexplored class of assumptions --- assumptions about unobserved non-confounders --- that should be considered when estimating treatment effects.

Bayesian comparison of nonlinear state space models requires integrating over a growing latent state path and a low-dimensional parameter vector. We propose a marginal likelihood approximation that integrates out the states with a numerical or simulation-based evaluator and applies a second-order corrected Laplace approximation to the integral over the static parameters. The correction is expressed through smoothing cumulants of the complete-data log density and, under derivative-growth and posterior-concentration conditions, reduces the log marginal likelihood error from $O(T^{-1})$ to $O(T^{-2})$. In stochastic volatility experiments, it removes 64\% to 99\% of the Laplace error at far lower cost than simulation-based estimators. Applications to bivariate exchange-rate volatility models and to a sample of $T = 19{,}266$ five-minute stock returns find the strongest support for heavy-tailed observations and for a transitory volatility factor.

Synthetic controls are used for one treated unit that lies outside its donors' range, with a pre-treatment window too short for residual-based standard errors to be trusted. We propose inference that separates the periods used to fit the synthetic control from the periods used to assess its error. Weights are fitted on an early block of pre-treatment periods, and the gap on a later block calibrates the bias and the noise. The resulting interval has finite-sample coverage under an explicit restriction on how far the counterfactual may depart from the extrapolated pre-treatment gap, for any fitting rule that uses only the early block. The leading restriction bounds the curvature of the gap and leaves its slope free, and the breakdown curvature at which the interval first includes zero states how much extrapolation a conclusion needs. We show when drift bounds calibrated from donor placebos are valid and how to allow for serial dependence. In simulations, conventional intervals undercover when the treated unit drifts from its donors, while the proposed interval covers whenever its restriction holds and the noise is serially independent. Re-analyses of three classic applications show that some well-known conclusions rest on far stronger extrapolation than others.

We study regression adjustment for ratios of treatment effects in completely randomized experiments. The existing literature adjusts the numerator and denominator separately, but this approach misses the efficiency target, loss of which matters more when richer or high-dimensional covariate information and flexible or regularized prediction methods are used to pursue further efficiency gains. We propose a two-step estimator that directly predicts the relevant transformed outcome. Using cross-fitting, our method accommodates high-dimensional first-stage prediction while keeping the final adjustment fixed-dimensional, thereby avoiding instability from a growing number of adjustment covariates. We further develop persistence-based LASSO theory that does not require sparsity, permits more covariates than observations, and allows cross-validated penalty selection under prediction-level conditions. For additional efficiency gains, our augmented two-step estimator nests the benchmark adjustment spaces and is therefore asymptotically no less efficient than the benchmark estimators, while its feasible conservative variance estimator satisfies the same efficiency ordering. Simulation studies and a real-data analysis demonstrate that our proposed estimator performs better than existing methods.

This paper develops a monthly framework for classifying Brazilian monetary policy that distinguishes the level of monetary restraint, changes in that stance, and statistical regimes associated with Selic adjustments. The framework combines a seven-state deterministic classifier, an empirical Markov chain, a Gaussian hidden Markov model, and Markov-switching regressions. Project outputs provide 145 classified observations from May 2014 to May 2026, with a partial final month. Under the implemented nominal neutral rate convention, restrictive states account for 88.3% of observations. A four-state HMM achieves a lower BIC, whereas the three-state model records a lower temporal validation loss. Switching regressions select three Selic adjustment regimes after solutions with degenerate variances are excluded. An additional sensitivity exercise reduces the restrictive share to 51.0% when the neutral benchmark is interpreted as real, demonstrating how strongly the results depend on benchmark measurement. The contribution is an auditable comparative application for monetary monitoring. The findings describe historical patterns conditional on the data and specifications; they do not establish predictive superiority or causal identification.

We propose a semiparametric real-time GARCH (SRT-GARCH) model that combines a smooth deterministic long-run volatility component with a stationary short-run process driven by contemporaneous return innovations. The model accommodates gradual changes in unconditional variance while allowing short-run volatility to respond directly to current information rather than relying solely on lagged observations. We derive the variance normalization, conditional moments, and closed-form conditional density of the model. For estimation, we develop a two-step procedure that combines kernel smoothing with conditional Gaussian maximum likelihood. Under Gaussian innovations and mild regularity conditions, we establish consistency and asymptotic normality of the short-run parameter estimator. The analysis exploits geometric moment contraction to control serial dependence and shows that estimating the long-run component adds a positive semidefinite rank-one adjustment to the oracle covariance matrix obtained when the long-run component is known. The resulting asymptotic covariance matrix admits a plug-in estimator that requires neither lag truncation nor additional tuning parameters. Monte Carlo simulations show good finite-sample performance of the estimator and the associated inference procedure. Applications to FTSE~100 returns and U.S. natural gas futures show that allowing for a time-varying long-run component improves forecasting relative to the parametric RT-GARCH benchmark, while the additional real-time feedback mechanism provides more pronounced gains in the natural gas application than in the FTSE application. The proposed model also delivers competitive one-step-ahead Value-at-Risk forecasts.

Bayesian nonparametric regression methods have wide applications in statistics and machine learning, but inference can often be expensive or require approximations. This paper introduces a class of recursive moment function updates which are closed form, interpretable, expedient, and fully quantify Bayesian uncertainty through the martingale posterior. The proposed recursive updates can be used off-the-shelf and encompass challenging scenarios such as nonparametric heteroscedasticity, heavy-tailed or skewed noise, and importantly generalize to the multi-output case with missing outcomes without any significant loss in tractability or speed. We show frequentist consistency under weak assumptions in the aforementioned scenarios, and demonstrate empirically that our proposed method is orders of magnitude faster than competing Bayesian nonparametric approaches, with competitive or even superior prediction and estimation accuracy.

We study the problem of predictive inference with outcomes missing not at random, when instrumental variables are available. While conformal prediction and weighted conformal prediction provide valid distribution-free inference when outcomes are missing completely at random and missing at random, respectively, inference is generally not possible without additional assumptions when outcomes are missing not at random. In this work, we consider a setting where we remain within the distribution-free framework but instead have access to an instrumental variable (IV). We explore the theoretical possibility of IV predictive inference and establish that it is characterized by a parameter that we refer to as the identification index. We derive an impossibility result for the case where the identification index is not sufficiently large, showing that any distribution-free approach fails to provide informative inference in this case. For the case where the identification index is sufficiently large, we introduce a practical method that achieves informative predictive inference. Our method leverages modern semiparametric efficiency theory which involves deriving the efficient influence function of the identifying functional for sharp IV bounds of the quantiles of the missing outcome, delivering a double robustness property in the coverage guarantee of the proposed framework for IV-based predictive inference. We illustrate the theoretical results and the performance of the proposed procedure through simulations and two real-data applications---FES wages and NHANES III self-reported heights.

Many causal panel estimators construct counterfactuals using weights or latent structure estimated directly from observed outcomes, making treatment-effect estimates sensitive to observation noise. Bayesian hierarchical models can account for this noise and borrow information across related observations, but existing Bayesian approaches to causal panel analysis generally combine outcome modeling and counterfactual imputation in estimator-specific formulations that can require bespoke sampling algorithms. We develop a framework that learns the latent untreated panel with a probabilistic model and applies established causal estimators to its posterior draws. The model accommodates non-Gaussian observations and structured pooling, while re-estimation across draws propagates uncertainty through counterfactual imputation. We establish conditions for posterior consistency of counterfactual imputation, characterize how pooling affects imputation risk, and develop fast cross-validation for model selection and calibration of predictive uncertainty for treatment effects. Our results show that probabilistic modeling can improve counterfactual imputation, particularly when observation noise is substantial and the pooling structure is informative.

Nonparametric inference often requires characterizing an unknown function, such as a cumulative distribution function (CDF), across its entire domain rather than a specific point. We develop confidence bands for distribution functions of a data stream that are valid simultaneously over all times and all points, even when each observation may depend arbitrarily on the past. We begin by deriving such bands for the time-averaged conditional CDF and show how their widths are determined by the depth of a binary search tree. We then show that this result is an instance of a framework through which one can construct valid confidence bands uniformly over time and over an arbitrary index set for any indexed family of time-averaged distribution functions. We illustrate the framework on importance-weighted and right-censored CDFs, as well as multivariate CDFs.

Proxy SVARs often rely on a single credible proxy, leaving proxy exogeneity untestable. Narrative arguments used to justify proxy validity imply a stronger form of exogeneity. We formalize the strong exogeneity assumption and propose a corresponding test based on internal synthetic proxies. The approach uses the external proxy to locate the target-shock direction in the reduced-form innovations and synthesize the candidate shock into an additional internal synthetic proxies, which allows to test the joint exogeneity of the original and internal synthetic proxy. We derive the large-sample theory for the ISP test and show that the identification strength is governed by a direct measure of the target shock's non-Gaussianity. We study weak identification under asymptotic Gaussianity and develop an identification-robust version of the test. Monte Carlo experiments demonstrate the favorable power of the test when the proxy has a low signal-to-noise ratio. Applications to monetary policy and oil supply news shocks find no evidence against the exogeneity of the benchmark proxies.

In this work, we develop a quantitative resolvent calculus for sample covariance matrices tailored to applications in high-dimensional statistics. We treat single resolvents, their derivatives, and products of two resolvents with different spectral parameters in the proportional-dimensional regime. Our bounds hold uniformly over compact subsets of $\mathbb C\setminus\mathbb R_{\geq0}$ with approximation errors of order $O_{\mathbb P}(\|\Theta_n\|_F/\sqrt n)$ for deterministic test matrices $\Theta_n$. These bounds require only uniformly bounded fourth moments and a uniformly bounded population covariance operator norm. Neither regular spectral edges nor convergence of the population spectral distribution is required. As a key technical ingredient, we develop an averaging argument for centered quadratic forms involving one or two leave-one-out resolvents. Applications of our results include precision-matrix shrinkage, prediction risk in ridge regression, and generalized cross-validation. For these problems, we obtain quantitative guarantees for estimation and data-driven parameter selection.

Structural equation modeling (SEM) is widely used to study relationships among unobserved constructs. Over recent years, the interest in computing predictions with SEM has grown. Different prediction procedures have been proposed for this purpose, including MLpredict, which is based on maximum likelihood (ML) estimation, and PLS(c)predict, which is based on partial least squares (PLS). The out-of-sample predictive performances of these procedures have not yet been compared. In our work, we conduct a comprehensive Monte Carlo study to compare out-of-sample predictions of MLpredict and PLS(c)predict for indicators associated with endogenous constructs, so-called operative predictions. Here, we vary key design factors regarding, for instance, the strengths of structural relations, the training sample size, and the construct types in the data generating processes, which determine the presence or absence of construct misspecification. Across the investigated conditions, MLpredict and PLS(c)predict exhibit similar predictive performance according to two assessment criteria, indicating that neither procedure is uniformly superior for operative prediction. Out-of-sample predictive performance is driven mainly by the strength of the relations between the variables, whereas training sample size and construct misspecification have comparatively little influence on it. An exception is the occasionally inflated prediction error of PLScpredict, which is in line with previous findings for ordinal indicators. Overall, our results suggest that, in the studied setting, fine-grained methodological differences between traditional SEM and PLS-based procedures matter less for out-of-sample predictive performance than properties of the data generating process, such as the coefficient of determination of endogenous constructs.

In this paper we propose a method for estimating conditional densities from an iid sample using a language-model agent. The agent writes the estimator as a short program that, for each covariate value, returns the estimated conditional distribution of the response. The program is built from a library of probability distributions, so the returned law can be a parametric distribution, a mixture or a multivariate factorization, with parameters given by explicit functions of the covariates. Since the number of operations and constants in the program is limited, the fitted conditional density is a formula that can be inspected and evaluated independently of the agent, while the choice of the model structure and of the fitting procedure is left to the agent. We call this approach agentic distributional regression (ADR). Treating the returned program as a random conditional density estimator, we derive a finite-sample error bound under explicit complexity and optimization assumptions, and an oracle inequality for selection using independent validation data. For a stably selected regular structure with sufficiently accurate fitting, we obtain asymptotic normality of the fitted constants. In simulation studies, on twelve real datasets and on a new dataset collected after the agents' training data, ADR with Claude Opus 5.5 has the best average performance among the methods considered.

For each time $t$, the regression function $m_t(x)=E[Y_t\mid X_t=x]$ of an outcome on contemporaneous covariates can be regarded as a field on covariate space. When the covariates of units move as a diffusion and each unit's outcome has its own dynamics, we show that $m_t$ satisfies a second-order evolution equation whose diffusion term is $\frac12\mathrm{tr}(\Sigma\nabla^2 m_t)$, the Hessian of the regression function weighted by the covariance $\Sigma$ of covariate increments, whose transport term is the drift of the time-reversed covariate process, and whose source term is the conditional mean rate of change of the outcome; this yields a Feynman-Kac representation along reversed covariate paths. We relate this Hessian to quadratic regression (under Gaussian design the best quadratic approximation has Hessian $E[\nabla^2 m(X)]$) and to kernel smoothing bias. For bistable replicator dynamics on covariate space, travelling fronts are exactly logistic in $(x,t)$, so one logistic regression identifies the diffusion-reaction ratio and the invading strategy. We then ask whether spread in covariate space reflects fluctuating unit characteristics or imitation among similar units. Imitation acts through exactly the same operator as covariate fluctuation, so the two are observationally equivalent in the regression field; panel data separate them because they identify $\Sigma$ directly. The two cannot be separated on a travelling front, and we propose a split-sample instrumental-variables test that works on seeded, non-front profiles, with size and power evaluated in agent-based experiments.

This study proposes and validates a methodology for reconstructing the Human Development Index (HDI) from individual-level survey microdata, enabling disaggregated estimates for demographic subpopulations while maintaining coherence with the official HDI. Data from 40,185 respondents in 21 countries from the European Social Survey (ESS7, 2014) were analysed, combined with World Bank indicators for life expectancy, gross national income per capita, and expected years of schooling. The health dimension was estimated from national life expectancy, adjusted by self-rated health; income, by a lognormal multiplier calibrated by the Gini coefficient; and education, by combining observed and expected years of schooling. The dimensions were aggregated by geometric mean, weighted by sampling weights, with confidence intervals estimated via the delta method. Among the 39,706 individuals with valid data (98.8%), the reconstructed HDI showed high agreement with the official 2014 HDI, with a mean absolute error of 0.0099 and a Pearson correlation of 0.917. Consistent differences across subgroups evidenced the methodology's convergent validity and its capacity to produce disaggregated estimates of human development, expanding the HDI's potential for analyses of socioeconomic and demographic inequalities.

Flexible modelling of the conditional distribution of a response given predictors is important in many applications where nonlinear effects, heteroskedasticity, and multimodality may be present. We introduce the SoftBART-CoRM, a predictor-dependent infinite mixture model that combines the interpretable component structure of Compound Random Measures (CoRMs) with SoftBART priors on both mixture weights and component-specific parameters. The model flexibly captures complex distributional features, while global--local shrinkage and predictor-selection priors mitigate overfitting and identify the covariates driving component allocation and kernel parameters. Posterior inference is performed using an efficient Markov chain Monte Carlo algorithm that leverages existing \texttt{R} implementations of SoftBART. Through simulation studies and real-data applications, we show that the proposed approach can adaptively shrink towards simpler mixture representations, provide interpretable inference on the predictors governing different aspects of the conditional distribution, and achieve superior out-of-sample predictive performance relative to benchmark models.

We study partial identification of conditional treatment effects with a continuous treatment under unobserved confounding. We target the conditional average potential outcome (CAPO) E[Y_x|z] and the conditional average treatment effect (CATE) between two treatment levels. We assume a bounded outcome and Lipschitz continuity of the response functions and of the propensity scores given the latent confounder, and show that these conditions imply the smooth treatment response and smooth treatment selection conditions of Kim et al. (2018). The combined bound of Kim et al. uses only two reference points on the identified diagonal and is generally loose. Bounding through every diagonal reference point instead gives explicit bounds whose average is the sharp identified set of the CAPO at every treatment, inside or outside the observational support. Capping the difference between the upper and lower bounds gives sharp causal bounds on the CATE. These bounds are informative without monotonicity or overlap assumptions.

Time-varying parameter (TVP) models are widely used to capture evolving relationships but inference is often computationally demanding due to high-dimensional latent states. The efficiency of iterative algorithms depends critically on the choice of latent state augmentation, with centering preferred for genuinely time-varying coefficients and noncentering favored when coefficients are nearly constant. We propose a partially noncentered parametrization that bridges these extremes via dimension-specific working parameters for both location and scale. Optimal working parameters are derived by analyzing EM convergence rates, and an alternating expectation-conditional maximization algorithm is developed to implement the approach. As the convergence rate of mean-field variational Bayes (MFVB) is closely related to EM, we extend the approach to develop partially noncentered MFVB algorithms for the TVP model, placing triple gamma priors on baseline coefficients and process variances to induce shrinkage. Simulations and empirical applications show that the resulting MFVB algorithms yield posterior approximations closer to Markov chain Monte Carlo than conventional centered or noncentered parametrizations. We also develop an efficient solver for the block tridiagonal inverse matrices arising in the proposed algorithms.

We establish asymptotic results for profile score statistics under sequences of parameters that may tend to the boundary of the parameter set. When the likelihood is profiled over a set with boundary points, the distribution of the statistic can depend, even asymptotically, on a distance to the boundary that cannot be estimated consistently. We address this in three ways. First, we give conditions under which the nuisance score has a root on an extended parameter set, and the profile score statistic formed with that root converges to a chi-squared distribution. Second, we show that the efficient score statistic converges to a chi-squared distribution under a separate condition that can hold even for estimators constrained to the original set. Both lead to confidence regions with asymptotically correct coverage probability, uniformly on sets that may contain boundary points, and the conditions allow the scores for the interest and nuisance parameters to grow at different rates or become asymptotically collinear. Third, we quantify the coverage error of applying the profile score statistic without an extended set. We approximate the distribution of the statistic and bound the error by a quantity that decays like a normal tail in the standardized distance to the boundary. The approximation also gives one-sided tests with least favorable critical values whose asymptotic size is at most the nominal level. We illustrate the results in a zero-inflated Poisson model and a one-way random effects model.

We study strongly adaptive regret for Online Convex Optimization (OCO) with time-varying movement costs, a setting that evaluates a learner's performance over arbitrary sub-intervals. This framework captures complex sequential decision-making problems where both the environment and the cost of switching decisions fluctuate over time. We propose a novel algorithm which, for any interval $[a,b] \subseteq [T]$, achieves a data-dependent regret bound of $ \widetilde{\mathcal{O}}\left( \sqrt{(D^2+DP_{[a,b]})\sum_{t \in [a,b]}(\|g_t\|^2+\lambda_t\|g_t\|)} \right) $, where $D$ is the diameter of the decision domain, $P_{[a,b]}$ is the path length of the comparator sequence over $[a,b]$, $g_t$ is the gradient of the loss, and $\lambda_t$ is the movement-cost coefficient at round $t$. Key to our approach is a novel reduction to sleeping experts with movement costs, a previously unexplored variant that is of independent interest. As a result of this development, we further establish the first strongly-adaptive guarantees under delayed feedback. Our approach also naturally extends to OCO with time-varying memory, where we obtain strongly-adaptive guarantees which improve the best-known bounds.

Modern interventions increasingly produce object-valued outcomes, such as distributions, covariance matrices, networks, and trees. Existing causal frameworks often exploit special geometry such as one-dimensional Wasserstein quantiles or geodesic Fr\'echet means, whereas metric statistics studies random objects without confounding adjustment. We propose the profile causal effect $\Delta(\omega,t)$, which describes where and at what scale a treatment shifts probability mass, including changes in dispersion, tails, or structure that leave mean-level summaries unchanged. Formally, $\Delta(\omega,t)=\Pr\{d(\omega,Y(1))\le t\}-\Pr\{d(\omega,Y(0))\le t\}$ contrasts the potential-outcome probabilities of the same anchor-centered metric ball and requires no structure beyond the metric. On grids fixed by a deterministic or auxiliary design, we combine cross-fitted doubly robust scores with high-dimensional Gaussian approximation and the multiplier bootstrap to obtain simultaneous confidence bands and a global test under regularity conditions. The bands bound treatment-induced changes in ball and shell probabilities simultaneously and, at fixed anchors, extend to all radii. A protocol guides the choice of anchors and radii. Across the experiments, the global test detects concentration effects that mean-quantile and Fr\'echet-mean summaries miss. Its empirical size is near or below the nominal level, even under confounding, where naive metric tests are substantially oversized.

Income inequality remains a central concern in economics and public policy, yet conclusions about its predictors often depend on both the factors considered and the measure of inequality used. This paper examines the predictive importance of a broad set of indicators covering structural factors, capital structure, financial-sector development, political institutions, regional classification, human capital, globalization, and technology. Using data for 88 countries from 1970 to 2017 and six inequality measures from the World Income Inequality Database (WIID), we apply a machine learning approach to assess the relative predictive importance of 35 candidate indicators across alternative income inequality measures. The analysis proceeds in three stages: a full specification including all candidate indicators, a category-representative specification with one common indicator from each conceptual category, and an inequality-specific specification allowing the most predictive indicator within each category to vary across inequality measures. Regional classification is the dominant predictor across all specifications and inequality measures, although its broad nature prevents identification of the specific regional characteristics underlying this result. Political and technology indicators provide the most consistent additional predictive information, whereas the importance of human capital, financial development, and globalization varies across inequality measures. Technology is particularly informative for top-income concentration, while returns to capital are more relevant for measures capturing gaps between the upper and lower parts of the income distribution.

Health researchers often study how a mixture of pollutants or nutrients relates to health. Quantile g-computation addresses this question by giving each exposure an ordered score and fitting a model for the outcome. For example, dividing an exposure into four quantile groups gives scores 0, 1, 2, and 3. The model can compare score 1 with score 2 for every exposure at once. These groups need not contain equal numbers of observations when measurements are tied. A one-score increase can represent different changes in the original units for different exposures. QuantileMix provides selected quantile g-computation methods in Python. It returns estimates, confidence intervals, predictions, and diagnostics for continuous, binary, count, and survival outcomes. Other models handle measurement limits, outcomes with several categories, and counts with many zeros. Fitting and prediction use NumPy and SciPy and do not require R. The package includes a reference manual, runnable examples, and numerical comparisons with R.

Regression analysis commonly features categorical variables and their interactions to model heterogeneous, group-specific effects. Such interactions are notorious for weak signals and cumbersome parameterizations, and thus urgently require shrinkage. However, the overwhelming default for linear models--using dummy coding, removing a reference group, and applying shrinkage--leads to unwelcome biases, asymmetries, and parameter incongruities across models. We resolve these limitations with a new parameterization-prior combination that couples abundance-based constraints for identifiability with customized shrinkage priors for categorical effects and interactions (BayesABCs). BayesABCs offers a surprising and unique invariance property: the joint posterior of all main effects is (nearly) unchanged by the addition of categorical interactions. We show how this, in conjunction with our shrinkage priors, effectively removes the principal drawbacks of including categorical interactions, thereby making heterogeneous effect estimation broadly feasible. We enforce the identifiability constraints via conditioning and provide a model representation that facilitates efficient posterior computing and direct analysis. Across simulations with a variety of categorical effects and interactions, BayesABCs consistently produces more accurate estimates and predictions, sharper inferences, and unmatched robustness to extraneous interactions relative to competing parameterizations and priors. We apply BayesABCs to study disparities in the error rates of automatic speech recognition systems.

We propose dynamic conditional densities with spline mixtures (DCD-SM) to detect whether the tail intensity of financial returns depends on past information after accounting for volatility and tail-order dynamics. The target is a weighted mixture of the conditional return densities of individual assets in a cross section. Individual distributions may differ, and returns within a date may be dependent. A normalized spline-mixture correction uses score-driven updates to change left and right tail intensities while preserving the reference tail order. We use Wald tests for persistence and assess empirical size, power, and forecast stability in simulations. With fitted references that have dynamic tail order, Wald tests at the nominal five-percent level detect dynamics that affect both tail intensities in the same direction in all 19 US and Japanese equity cross sections, for both raw returns and residuals after principal-component removal with training-fixed loadings. Evidence for dynamics in the imbalance between the two tail intensities is more limited. In the residuals, the correction improves out-of-sample one-percent lower-tail quantile loss relative to these references in 16 of 19 panels, with three significant positive differences at the five-percent level. Forecast gains are most evident in log scores and one-percent quantile losses and are less uniform for five-percent quantile loss.

Off-policy evaluation (OPE) is essential for assessing decision policies before deployment when online experimentation is costly or unsafe. Accurate estimation remains difficult when behavior-policy data are limited and the target policy favors states and actions rarely observed in those data. We propose InCOPE, a prior-data fitted network that learns a shared policy-value prediction rule across synthetic OPE tasks. We construct an OPE task prior over Markov decision processes and behavior-target policy pairs, generating behavior-policy datasets and exact target-policy values for pretraining. For a new tabular decision problem, the pretrained network uses the behavior-policy dataset, target policy, and known task settings as context to predict the policy's expected return without task-specific parameter updates. Experiments across four tabular domains, including a semi-synthetic clinical setting, demonstrate InCOPE's effectiveness for policy-value estimation.

Contextual decision making uses observed covariates to guide decisions under uncertainty. Two natural approaches are predict-then-optimize (PtO), which learns the conditional expected objective, and generate-then-optimize (GtO), which learns conditional uncertainty and optimizes using generated scenarios. We study when each approach is preferable by examining how the choice of learning target affects statistical accuracy and optimization tractability. Although PtO learns a narrower target, our regret bounds show how differences in smoothness can offset GtO's additional dimensional burden. Exploiting smoothness also raises an optimization tradeoff: within the class of nonnegative-weight PtO methods studied, a lower bound limits the gains from additional smoothness, while more flexible methods may introduce signed weights and lose the guarantee of preserving convexity. GtO preserves scenario-wise convexity, but requires learning a richer object and generating scenarios for optimization. Synthetic experiments and a real case study illustrate how these tradeoffs vary with data availability and problem structure. Together with the scope for reusing a model across tasks, our findings provide practical guidance for choosing between the approaches without identifying a universal winner.

In this paper we consider optimal stopping problems with full information and rank-dependent rewards. Our focus is on the following two problems: selecting one of the r best observations (Gusein-Zade's problem with full information) and minimizing the expected rank (Robbins' problem). In both settings we provide full characterization of the optimal stopping rules and corresponding optimal values. We also present numerical experiments in which the derived dynamic programming recursions are implemented, and approximations to optimal values are computed.

Papers in statistical theory rest on long proofs that few readers check line by line. We present KAPPA (Kit of Agents for Proof-checking Papers with a proof Assistant), open-source software that organizes language model agents, run by Claude Code, to translate the proofs of a LaTeX paper into the proof assistant Lean 4 with its library Mathlib and check them there, result by result. Each numbered result receives one of four verdicts: verified unconditionally; verified conditionally, with every cited or unproved input stated as a named hypothesis; issue recorded, with its place in the paper and the side at fault; or unfinished, with the reason. The kit follows the paper's own proof step by step, rather than finding a new one. Separate agents write and review, each statement is compared with the paper before it is proved and fingerprinted afterwards, and every verdict rests on evidence that can be rerun without a language model. The kit runs unattended for hours, waits out usage limits, re-checks only what a revision changed and reports its cost by role, model and result. We describe the design, walk through a complete run, and apply the kit to four example papers with answer keys and to three published statistical papers, checked blind in their versions from before public corrections. In all three, the kit found the published error at the step that the correction names, together with further problems that we confirmed by hand. Given the same papers directly, the same language model working with Lean classified fewer results correctly on the longest paper and reported some statements narrower than the paper's as proved; no verdict of the kit claimed more than its Lean evidence showed.

Causal identification under unobserved confounding typically relies on additional structure, such as a valid instrument or proxy variables. We show that heavy tailed variables offer an alternative route to identification by revealing treatment variation for which confounding becomes asymptotically negligible. Our key assumption is extreme propagation: when an unobserved confounder takes an extreme value, some observed variable in the system (called a safeguard) will become extreme with large probability. This way, selecting extreme treatment observations while safeguards remain non-extreme makes confounding asymptotically negligible relative to the treatment. We call this certification. We derive graphical conditions specifying which latent confounders must be monitored by safeguards and distributional conditions ensuring effective certification. The framework includes valid-instrument and double-proxy structures as special cases, as well as confounded instruments that are invalid in most approaches that do not exploit heavy tails. Geometrically, certification isolates the extremal ray whose slope equals the causal effect. This connects causal identification with multivariate extreme-value geometry and motivates a family of novel simple estimators. Simulations demonstrate performance gains under nonlinear confounding and heavy tails, while revealing limitations in light-tailed regimes (where cancellation can conceal extreme confounding from the safeguards). We validate our methodology in simulation studies and by recovering the ground-truth flow in a river discharge application.

We consider testing equality of mean functions for paired functional data, where one entire function is missing. To use all available information from the paired and unpaired observations, we propose globalized test statistics that are based on a weighted sum of point-wise paired-$t$ and Welch-$t$-type statistics. By the use of empirical process theory, we derive the asymptotic limit distributions of the resulting test statistics under the null hypothesis and show consistency against fixed alternatives. As they are non-pivotal and contain complex dependencies from the paired and incomplete observations, a permutation and parametric bootstrap procedures is proposed to compute critical values. The work includes a theoretical investigation of the asymptotic behaviour of the two resampling approaches and state regularity conditions under which the corresponding resampling tests are valid. Simulation studies show that the proposed methods provide accurate type I error control and competitive power for most settings under study. An illustrative analysis of functional emission data demonstrates the applicability of the methods.

We propose a neural exploratory framework for variational problems over graphons, i.e., symmetric measurable functions $[0,1]^2 \to [0,1]$, that arise as limits of dense graph sequences. Such problems are central to two parts of the mathematical literature: extremal graph theory, which studies graph parameter optimisation under given restrictions, often for those graphs with a large number of vertices, and the large deviation theory of dense random graphs, which characterises the structure of rare events through constrained graphon optimisation. We represent graphons by implicit neural networks and optimise graphon objectives by gradient descent. Our design combines three ingredients: a multi-scale sinusoidal residual architecture biased toward sharp, step-like graphons; an embedded solver that enforces a single empirical density constraint and is differentiated by implicit differentiation; and symmetry-aware Monte Carlo estimators. On generalised Tur\'an problems that previously required substantial human effort, our framework rediscovers known optimal graphons without human intervention. Applied to open instances, it produces candidate optima, including a previously unreported family of extremal structures. Applied to the variational problems from the large deviation theory of Erd\H{o}s--R\'enyi random graphs, the framework produces new candidate optimal graphons across both upper- and lower-tail regimes for multiple pattern graphs; for the case of the triangle graph and the upper-tail regime, these candidates improve upon the best-known reference construction of Lubetzky and Zhao. The structures found by our framework have also led to new theorems on the optimal graphons for both families of problems.

Statistical procedures often control expected loss at a prescribed level while leaving part of their risk budget unused. Building on the e-value representation of risk-controlled decisions and the e-closure principle for multiple testing, we formulate recertification, a general paradigm for improving procedures by exploiting computable slack in their validity certificates. A valid procedure's losses normalized by their risk budgets form a state-indexed e-value collection; conversely, decisions whose normalized losses are bounded by this collection inherit its risk guarantees. Recertification uses distributional structure to quantify slack, enlarge the certifying e-values while preserving validity, and construct less conservative decisions. We develop two applications. For conformal risk control (CRC) with a known loss family, recertification redistributes unused risk budget among the held-out points of each hypothetical augmented sample. The resulting controller is valid under every finite exchangeable law and no more conservative than CRC on any calibration sample. For convex losses on an interval of actions, repeated recertification converges geometrically to an admissible controller. For false discovery rate control with arbitrarily dependent e-values, recertification exploits unused conditional budgets in randomized e-BH. It yields a closed-form randomized procedure whose rejection set contains that of randomized e-BH for every realization of the shared randomization and whose expected number of rejections is maximal subject to the same conditional budget constraints. Recertifying with the larger e-collection of closed e-BH gives a second closed-form procedure whose rejection set contains that of the first. Experiments on CIFAR-100 hierarchical classification and multiple testing simulations demonstrate improved budget utilization, informativeness and power over the baselines.

Optimal designs for regression with asymmetric errors build on the second-order least squares estimator (SLSE) of Wang and Leblanc (2008), whose gain over ordinary least squares (OLS) comes from the error skewness. With i.i.d. errors the optimally weighted SLSE solves the same estimating equation as the degree-two Polynomial Maximization Method (PMM), a family whose degree-$S$ location estimator has asymptotic variance $g_S\le1$ times that of the sample mean. We derive optimal designs for this family when the errors are independent with constant variance but have standardised cumulants of order three and higher that vary with the regressor (cumulant heteroscedasticity). For a known reduction field $g_S(x)$, the $g_S$-weighted information matrix $M_{g_S}$ obeys the Kiefer-Wolfowitz equivalence theorem. A constant field leaves the OLS-optimal design unchanged; a non-constant field can change it but need not. For symmetric errors with varying kurtosis, the PMM$_2$ and conditional-SLSE designs cannot depart from the OLS-optimal one, whereas a degree-three design can. A stratify-and-pool PMM$_S$ estimator attains $N\mathrm{Cov}(\hat\theta)\to\sigma^2M_{g_S}^{-1}$ at every degree as all strata grow; its per-stratum reduction is below one only for symmetric errors with the symmetry imposed. For a quadratic model on $[0,1]$ with symmetric Beta errors of unit variance and dose-varying shape, the third-order design for the known field has $A$-efficiency $1.047$ relative to the OLS-optimal design, and $1.001$ under the $D$-criterion. A two-stage procedure that estimates the field from a one-third pilot builds designs of $A$-efficiency $1.014$-$1.020$ at $N=600$-$2400$; its realised mean-squared-error ratio against the OLS-optimal design ranges from $0.89$ (a loss) to $1.08$.

In this study, we introduce a new Bivariate Excess Wealth Transform (BEWT) to study excess wealth characteristics in bivariate settings. The proposed transform is constructed using conditional quantile functions and their corresponding conditional quantile densities. We establish fundamental properties of the proposed transform and derive some characteristic relationship for some well known bivariate lifetime models. Nonparametric weighted empirical estimation of BEWT has been proposed; theoretical and numerical validation of the method has also been considered in the study, along with a real data analysis based on the Eurosystem Household Finance and Consumption Survey.

While generalized Bayesian inference offers a principled approach to uncertainty quantification under general loss functions, its practical performance depends critically on the choice of the learning rate. Generalized posterior calibration, which selects the learning rate so that the resulting credible sets attain nominal frequentist coverage, is computationally intensive because it relies on repeated posterior sampling across bootstrap datasets and candidate learning rates. To overcome this practical difficulty, we propose a fast calibration method that combines Dirichlet-weighted bootstrap perturbations with importance sampling to estimate coverage while reducing the need for repeated posterior sampling. We further extend the framework to hierarchical models, where group-specific learning rates are calibrated and stabilized via shrinkage. Illustrative examples using quantile regression, support vector machines, and hierarchical Huber regression demonstrate that the proposed method substantially reduces computational cost while maintaining accurate calibration.

Estimating neural scaling laws involves an end-to-end pipeline: designing a scaling recipe, training models across scales, evaluating models on a target metric, and deploying an algorithm to fit a scaling law to the resulting data. This final fitting stage is often treated as routine: standard procedures minimize a non-convex fitting loss using local optimization from a grid of initializations, and return the best solution found. Yet such heuristic searches can fail in two distinct ways: they offer no guarantee of returning a nearly optimal solution, and they obscure whether fundamentally different parameter choices fit the data nearly as well. Indeed, we show that standard scaling law fitting is subject to a Rashomon effect (Breiman, 2001): the sublevel sets of near-minimizers correspond to broad parameter regions whose extrapolations diverge substantially at scale. We resolve these issues with a deterministic branch-and-bound algorithm that guarantees nearly optimal fitting loss and exhaustively maps the near-minimizers to the implied range of scaling curves.

Heterogeneity is a defining feature of modern data: individuals, subpopulations, sites, modalities, tasks, and time points may differ in distributions, associations, treatment effects, or decision objectives. This review develops a unified perspective in which heterogeneous analysis is a problem of structured information borrowing: deciding what should be shared, what should remain distinct, and how uncertainty about that structure should enter inference and decisions. We connect classical stratification, interaction models, finite mixtures, mixed-effects models, and empirical Bayes with subgroup discovery, individualized learning, synthetic data generation, large language models, federated learning, privacy, security, and mobile health. Precision medicine provides a central example: subgroup rules offer interpretable approximations to individualized response surfaces, whereas partially pooled models stabilize patient-specific estimation. Across applications, the same tensions recur: population averages can conceal consequential variation, unrestricted personalization is unstable, adaptive subgroup discovery creates selection bias, and borrowing across incompatible sources causes negative transfer. These difficulties are amplified in contemporary AI, where sparse user histories, entangled representations, heterogeneous preferences, feedback loops, distribution shift, and privacy constraints make reliable personalization and evaluation especially challenging. We conclude with an agenda for calibrated personalization that integrates multilevel representations, selective borrowing, subgroup-aware evaluation, and valid uncertainty quantification.

The Sinkhorn divergence, a debiased form of entropic optimal transport, has been widely used in various scientific disciplines due to its computational and statistical tractability. Unlike the Wasserstein distance, however, the Sinkhorn divergence is not a metric and it does not naturally admit a formal Riemannian structure. These properties are central to many applications of optimal transport methods that utilize interpolation, barycentric averaging, parallel transport or other geometric operations. Recent work addresses these shortcomings by constructing a formal Riemannian tangent space and a metric tensor on the space of probability measures that quadratically approximates the Sinkhorn divergence. In this work, we extend this framework with two major threads of contributions. For a suitable class of finite-smoothness cost functions, we prove Hadamard differentiability of the Sinkhorn metric tensor and we derive the Levi-Civita covariant derivative in this geometry; we then use the covariant derivative to obtain the geodesic and parallel transport equations, each of which amount to an ODE on a fixed Reproducing Kernel Hilbert Space (RKHS) with smooth coefficients. Our second contribution is to linearize the solutions to the geodesic and parallel transport equations using standard tools from Banach space ODE theory. We show that under standard regularity conditions these linearizations imply $\sqrt{n}$-limiting Gaussian processes centered at population values for the plugin estimators of the exponential map, the logarithmic map, parallel transport and the induced Sinkhorn distance. Moreover, we show that the nonparametric bootstrap is consistent for all objects mentioned. To the best of our knowledge, these are the first $\sqrt{n}$--limit theorems and bootstrap consistency results for canonical geometric operations in a transport-based geometry on the space of probability measures.

Consider the question: "Is it true that at least 80% of patients would benefit more from receiving a new treatment, $Rx$, than from receiving a standard treatment, $C$?" Conventional treatment efficacy analyses based on average treatment effects cannot directly answer this question. Yet, even if the true mean treatment effects were known exactly, the answer to this question could still be either "yes" or "no", depending on the variability in patients' treatment responses and the dependence between their potential outcomes. This tutorial introduces a proportion-based perspective on individualized treatment benefit through the ETZ-powered Counterfactual Individual Benefit Analysis framework. We present two metrics, the individual benefit probability and the individual benefit lower bound, which enable confidence-based assessments of whether a prespecified proportion of patients achieves a clinically meaningful treatment benefit. We develop these metrics in both the factual random-pair and counterfactual same-patient settings. In particular, the ETZ decomposition provides a transparent characterization of different sources of outcome variability and enables more precise uncertainty quantification for counterfactual treatment comparisons. We illustrate the framework and its practical implementation using patient-level data from a Phase II randomized clinical trial. By shifting the focus from average treatment effects to population proportions of individual benefit, this tutorial provides a complementary perspective for evaluating treatment efficacy and informing individualized treatment decisions while controlling the type I error rate at a prespecified level.

Estimating a population mean is fundamental across scientific applications, but observing every outcome can be prohibitively expensive. Often, covariates and machine learning (ML) predictions are available for the full population, even when only a small subset of outcomes can be observed. How can we leverage this information to proactively select observations to label, update ML predictions as labels arrive, and construct statistically valid confidence intervals? We develop a general, without-replacement formulation of Pro-Active Inference (PAI), extending our earlier with-replacement approach and building on Active Statistical Inference. Rather than making fixed-order label-skip decisions, PAI proactively selects which observation to label next from the remaining covariate pool while accommodating online ML updates. We establish PAI's finite-sample unbiasedness and asymptotic validity, develop an improved finite-sample variance estimator, and extend PAI to superpopulation inference without Bonferroni corrections or $\alpha$-splitting. We characterize PAI's potential efficiency gains over Batch Active Inference in a quasi-oracle setting, highlighting the role of prediction-residual imbalance. Across eight diverse datasets, PAI achieves greater sampling efficiency than Sequential Active Inference in the vast majority of tested settings with near-nominal statistical coverage, using the same ML predictor and uncertainty score. However, comparisons with simpler baselines such as Prediction-Powered Inference and uniform sampling, as well as the closely related Prediction-Powered Active Testing, reveal when additional adaptivity helps or hurts statistical efficiency. Through interpretable, theory-motivated diagnostics and frozen-ML ablations, we investigate when and why PAI gains or loses efficiency, highlighting the initial residual-estimation problem and subsequent online learning dynamics.

While the goal of classical statistical theory is to minimize the variance of unbiased estimators, an additional goal of some recent research is to minimize or constrain other notions of the instability of estimators. In particular, a subset of the authors recently introduced a notion of instability termed "sensitivity", which represents the expected amount that an estimator would change if its inputs were infinitesimally perturbed. In this work, we study the multiobjective problem of minimizing both the variance and sensitivity over unbiased estimators in a given statistical model, and we aim to determine Pareto optimal tradeoffs between these two objectives. Our main results show that the Pareto front for these tradeoffs is precisely determined by endowing the statistical model with the Hellinger-Kantorovich geometry, which interpolates between the Hellinger and Wasserstein geometries that are known to be associated, respectively, with the separate minimization of variance and sensitivity. We also develop a $Z$-estimation technique which constructs estimators that asymptotically achieve Pareto optimal pairs of variance and sensitivity. While computing the Pareto front and constructing the resulting estimators in practice requires solving a continuum of partial differential equations, we fully execute this program in some parametric examples of interest where the analysis is already non-trivial. Our results also shed light on some classical statistical problems, for example by showing that the maximum likelihood estimator achieves the smallest asymptotic sensitivity among all estimators achieving the optimal asymptotic variance.

The surrogate index method is the modern standard for estimation of long-term effects in a designed experiment using only short-term outcomes. This method relies on the surrogacy and comparability assumptions which may be violated in practice leading to estimation bias. We propose a new method called Experiment Reservoir Surrogate (ERS) which leverages a reservoir of historical long-term experiments that is common in large-scale experimentation platforms. ERS avoids the surrogacy assumption by predicting the long-term average treatment effect directly, and relaxes the comparability assumption by matching to similar historical experiments, without requiring every experiment in the reservoir to be comparable to the target experiment. Moreover, ERS is memory-efficient, as it only requires aggregate data for each time period, instead of raw individual-level data. We theoretically justify the improved efficiency of ERS and demonstrate the improvement empirically on a reservoir of experiments from the online platform DoorDash.

We establish a functional central limit theorem (CLT) for fluctuations of time-integrated functionals of empirical measures arising from interacting McKean--Vlasov particle systems. Our framework allows for coefficients with unbounded spatial growth and dependence on arbitrarily high-order moments of the measure variable, extending the scope of the existing fluctuation theory. The proof uses a novel approach combining linear functional differentiation with new well-posedness results and estimates for perturbed backward parabolic equations in polynomially weighted Schauder spaces, yielding a martingale-type representation of the fluctuations with an asymptotically negligible remainder. The limiting process is shown to be centred Gaussian with an explicit covariance structure. As a statistical application, we derive the asymptotic theory underlying hypothesis tests for both the temporal constancy of population characteristics and parametric restrictions on the drift coefficient.

Recently, Bachoc et al. (2026) proposed an axiomatic approach to fairness for repeated bilateral trade. Their resulting objective is a H\''older mean of seller and buyer utilities with non-positive order $\lambda$, interpolating between Rawlsian fairness at $\lambda = -\infty$ and Nash fairness at $\lambda = 0$. Here we study this family of H\''older means in the complementary setting of orders $\lambda \in (0,1]$, which constitutes a tradeoff between fairness and efficiency, with the weight of efficiency increasing with $\lambda$, and where $\lambda=1$ yields the existing gain from trade criterion, focusing solely on efficiency. We obtain an axiomatic characterization of these H\''older means, relying on a Pigou-Dalton principle of transferring from rich to poor. From an online learning perspective, we show that, contrary to the case $\lambda \le 0$, the problem is unlearnable in full generality, even under seller's and buyer's valuation independence. We focus on seller's and buyer's valuations with distributions having bounded densities, in which case we obtain matching lower and upper bounds on the cumulative regret, of order $\widetilde{\Theta}(T^{2/3})$ with respect to the time horizon $T$. The upper bounds rely on a reconstruction identity where the expected objective is written as a gain-from-trade term plus a two-dimensional integral involving the seller and buyer distribution and survival functions, and a kernel function. For $0 < \lambda < 1/2$, the strong singularity of the kernel is addressed by two main ingredients. First, we propose a new zooming algorithm that sequentially rejects or refines layers of the set of prices. Second, we propose a tailored decomposition of the two-dimensional integral, enabling us to truncate it with a controlled impact on the truncation error. For $1/2 \le \lambda \le 1$, the singularity of the kernel is mild enough, so that the existing scouting bandit algorithm, already proposed for the case $\lambda=1$, can be extended.

Nonlinear structural system identification remains challenging when governing physics is only partially known and measurements are sparse. Deep Bayesian filtering addresses these challenges by combining learnable dynamical representations with probabilistic state-space models, physical inductive biases, and uncertainty-aware inference. This paper presents, for the first time, a unified tutorial treatment of deep Bayesian filtering for nonlinear structural system identification, connecting model formulation, learning, posterior inference, and evaluation. Amortized variational inference and differentiable Kalman filtering are introduced as complementary strategies within this broader framework. The Deep Markov Model (DMM) provides a data-driven reference, while the Physics-guided Deep Markov Model (PgDMM) and Neural Extended Kalman Filter (Neural EKF) serve as illustrative instances through which methodological choices and their implications are examined. Dynamic state estimation and joint state and parameter estimation are demonstrated on a three-dimensional shear-frame benchmark with hysteretic Bouc-Wen connections and sparse acceleration measurements. The comparative analysis yields new observations concerning the relationships among reconstruction accuracy, predictive fidelity, physical interpretability, and estimation robustness. Physical structure improves latent-state interpretability without necessarily ensuring faithful predictive dynamics, while comparable typical reconstruction accuracy can conceal substantial differences in estimation robustness. These findings motivate practical guidelines for model construction, training, and evaluation that distinguish reconstruction, forecasting, state recovery, and parameter identification. The tutorial concludes by outlining challenges in uncertainty calibration, identifiability, and validation on real structures.

We study the asymptotic behaviour of the solutions to the unconstrained generalized variational inference problem regularized by an $\alpha$-divergence. Under mild conditions, the minimizer is explicit up to a scalar normalizing constant, and we show that it admits a rescaling under which it converges in $L^1$ to a fixed non-Gaussian law, at a concentration scale around the empirical loss minimizer that departs from the typical $n^{-1/2}$ in the classical Bernstein--von Mises theorem. For $\alpha\in(0,1)$ and $d<2/(1-\alpha)$ the limit is a Student-$t$ distribution and the posterior width shrinks faster than the sampling error of the empirical minimizer. Conversely, for $\alpha>1$ the limiting distribution is a compactly supported Pearson type-II distribution, and the posterior width shrinks more slowly than the sampling error. The KL divergence ($\alpha=1$) is the unique member of the $\alpha$-divergence family whose posterior width matches the sampling scale and whose limit is Gaussian. We discuss the implications of these qualitatively different regimes for the frequentist coverage of credible regions, and a toy example on intercept-only logistic regression illustrates the theoretical claims.

Meta-analysis aims to synthesize multiple studies to obtain a summary estimate of a population parameter of interest. However, existing meta-analysis methods require strong assumptions that render them not robust to bias and heterogeneity of the studies. In particular, fixed-effect meta-analysis assumes a common parameter across the studies, whereas random-effects meta-analysis models heterogeneity as independent and identically distributed random effects, which still implies a common parameter marginally over the random effects. Due to their limitations, we propose a bias-aware meta-analysis framework that allows for heterogeneous studies with fixed, unknown biases. With no bias, our bias-aware meta-analysis reduces to fixed-effect meta-analysis; with a given bound on the bias, we can still use the fixed-effect meta-analysis point estimator and standard error, but we must inflate the critical value for confidence intervals from the standard normal quantile to a folded normal quantile. We recommend reporting the maximum bias for significance, the maximum value of the bias such that the confidence intervals exclude the null value. We propose a formal model checking procedure and recommend reporting the minimum bias for compatibility, the minimum value of the bias such that the bias-aware meta-analysis model is compatible with the observed data. We discuss the practical implementation of our method and, in particular, the approaches to calibrating the maximum bias for significance. We also compare our method with traditional fixed-effect and random-effects meta-analyses.

Recent work established minimax rates for pointwise estimation of the conditional average treatment effect (CATE), but the fundamental limits under global losses remained unresolved. We derive matching minimax lower and upper bounds under the $L_2$ and $L_\infty$ risks on certain covariate-density and support classes. For the lower bounds, we develop a many-fuzzy-hypotheses method that replaces each fixed alternative in a multiple-hypothesis construction with a mixture over nuisance functions that leaves its CATE unchanged. A localized product-mixture $\chi^2$ bound is used to control the divergence of each resulting mixture from a common baseline, enabling multiple well-separated CATE hypotheses to be handled simultaneously. For the upper bounds, we analyze a modified local polynomial R-learner. The $L_2$ argument extends the pointwise bias--variance analysis, while the $L_\infty$ argument combines concentration inequalities for U-statistics with a continuity argument enabled by the use of a continuous kernel and additional regularity of the second-order basis. The resulting rates show that the pointwise elbow phenomenon persists globally: the $L_2$ transition threshold is unchanged, while the $L_\infty$ rate has an additional logarithmic factor and a transition threshold adjusted by $1-\log\log n/\log n$.

I analyze decision problems where a planner must choose, based on data from an experiment in a pilot population, how to simultaneously distribute a limited number of treatments that generate network spillovers across individuals in a target population. A global treatment rule maps the target population's observed covariates and network features into a jointly feasible allocation. Under local-response, stability, support, and policy-relevant overlap conditions, I identify the resulting welfare using a transported inverse propensity weighted criterion and construct an empirical welfare maximizer. I derive a finite-sample regret bound under restrictions on the complexity of network spillovers and the complexity of the global rule class, measured by its Natarajan dimension. The upper bound uses independent node-level variation, dyadic network formation without imposing a bounded maximum degree. A minimax lower bound establishes a corresponding rate comparison under additional welfare-relevant richness. Finally, I compare oracle optimal global and local rules (which allocate treatments on an individual-by-individual basis) and bound the welfare gap between them. An empirically calibrated oracle comparison illustrates the magnitude of this welfare gap.

We study the limiting spectral distribution and extreme eigenvalues of the sample correlation matrix $\hat{\boldsymbol{R}}$ from a high-dimensional vector autoregressive (VAR) processes. Depending on whether roots of the characteristic polynomial of given VAR lie on (nonstationary) or outside (super-nonstationary) the unit circle, $\hat{\boldsymbol{R}}$ presents different types of asymptotic spectral properties. Under both nonstationary and super-nonstationary regimes, the empirical spectral distribution of $\hat{\boldsymbol{R}}$ degenerates in probability to a point mass at zero. The rescaled extreme eigenvalues of $\hat{\boldsymbol{R}}$ are asymptotically equivalent to the eigenvalues of a finite-dimensional symmetric random matrix determined by the characteristic roots. Moreover, under super-nonstationarity, $\hat{\boldsymbol{R}}$ degenerates in probability to a matrix with finite rank.

We determine the optimal regret in agnostic multiclass online learning with arbitrary, possibly infinite, label spaces. For every hypothesis class of multiclass Littlestone dimension $d<\infty$, we prove a regret upper bound of $O(\min\{T,\sqrt{dT}\})$ over $T$ rounds, matching the known lower bound. This resolves a question of Hanneke et al. (2023) and extends the optimal binary regret bound of Alon et al. (2021). Our proof strengthens the one-sided expert cover of Hanneke et al. (2023) to a fractional cover. We combine this cover with a minimax reduction to a finite expert game, whose best-response rewards provide the disagreement bounds needed for chaining.

Model selection and hyperparameter tuning pose a central challenge in off-policy evaluation (OPE) because errors of fitted Q-functions and occupancy-ratio estimates cannot be directly evaluated from held-out transitions. Bellman-residual and minimax validation methods often require auxiliary regressions or optimization over separate critic classes, adding computational and tuning costs. We introduce pairwise Bellman validation (PBV), a tuning-free minimax procedure for selecting among finite classes of fitted Q-functions and occupancy-ratio estimators using held-out transitions. PBV constructs critics from pairwise candidate differences, requiring only finitely many empirical moment calculations without auxiliary validation models or separate critic optimization. We establish finite-sample oracle inequalities relative to the best available candidate, even under candidate-class misspecification. Our key insight is that Q-function errors and log ratios between candidate and true occupancy ratios provide oracle critics whose Bellman and adjoint Bellman moments control estimation error. Pairwise candidate differences approximate these unknown critics, while penalties account for comparator approximation error. For Q-function selection, we first develop unweighted PBV, which evaluates pairwise Bellman moments under the offline distribution and admits oracle guarantees under a pairwise coercivity condition relating the offline distribution to the target-policy dynamics. We then develop target-aligned PBV (TA-PBV), which uses marginalized importance weighting to provide error control without pairwise coercivity. For occupancy-ratio selection, we develop adjoint PBV (A-PBV), which evaluates adjoint Bellman moments against log-ratio critics. On Hopper and Deep OPE, unweighted PBV performs competitively, while target-occupancy weighting can improve performance in settings with controlled overlap.

Accurate forecasting often relies on both past target values and exogenous variables as inputs. In many applications, however, the time series of interest are non-stationary and the relationship between target and inputs evolves over time, requiring forecasters that adapt online. Moreover, inputs affect the target not only through their current values but also through their recent joint dynamics. We address these challenges in three steps. First, to encode the series dynamics, we rely on signatures of the input paths over sliding windows. Motivated by the universality of linear functionals of signatures, we fit a regularized linear regression on these features to capture nonlinear, path-dependent effects. Second, the simplicity of the linear model allows us to update it online as new data arrives, using a Kalman filter to track dynamic effects. Third, to capture dependencies at several time scales, we combine forecasters built on different window lengths through online aggregation. For French national electricity load forecasting, our method outperforms various state-of-the-art approaches. It also remains competitive across several benchmark datasets with exogenous variables.

In many applications, functional pairs are collected sequentially from a data stream. Existing function-on-function regression methods are primarily designed for batch data, and bootstrap inference has become one of the most popular approaches. However, this procedure may be infeasible when observations cannot be stored. In this paper, we propose a one-pass functional SGD estimator that updates upon each observation's arrival without data storage, and a weighted rolling validation rule is provided for step size selection for implementation. Further, an online multiplier bootstrap is developed for inference that updates perturbed stochastic gradient paths in parallel, bypassing estimating an infinite-dimensional covariance operator. Asymptotic behaviors are established for the proposed method, and conditional bootstrap validity is justified for the bias-centered pointwise estimation error. Simulation studies and two real data applications confirm the empirical effectiveness of the proposed procedures. To the best of our knowledge, this is the first theoretically justified one-pass framework for bootstrap inference on a bivariate coefficient surface with streaming functional data.

Randomized experiments involving complex, unstructured treatments such as text, images, or video, are increasingly common in social science research. The goal of many of these studies is to estimate the effect of a structured, interpretable attribute that generalizes beyond the treatments used in the experiment (e.g., the presence of a linguistic or visual feature). There are two key challenges when conducting such studies: observed effects of the attribute of interest (i) may be confounded by other correlated attributes, leading to poor internal validity; and (ii) may not generalize beyond the treatments used in the experiment, leading to poor external validity. We propose a framework for designing experiments with unstructured treatments that leverages generative AI or large databases to sample treatments in a way that explicitly navigates the tradeoff between internal and external validity. We show how to conduct conservative asymptotic inference with a difference-in-means estimator, and develop a calibrated AI-assisted estimator that incorporates zero-shot AI predictions to improve precision without introducing bias. Throughout, we work in a finite population framework, where treatment assignment is the only source of randomness, and treat AI generations and predictions as black-boxes with no guarantees on their performance. We illustrate and evaluate our approach by designing two hypothetical experiments, one with text and another with images.

Foundation models provide powerful representations for unstructured data, such as text and images, and parameter-efficient fine-tuning (PEFT) makes their adaptation to downstream tasks increasingly practical. For neural estimators that lack a tractable sampling distribution, resampling-based procedures offer a flexible route to statistical inference, but they often require hundreds of model fits, making them computationally prohibitive at foundation-model scale. We introduce HIFI, a two-stage fine-tuning framework for statistical inference on parameters such as conditional means. Stage 1 adapts a foundation model once on a larger sample, capturing the complex structure of the data and controlling estimation bias through a single fit. Stage 2 applies a resampling inference procedure to a smaller independent sample, concurrently fitting multiple lightweight PEFT estimators to the residual to quantify estimation variability without relearning the full task. HIFI accommodates a broad class of resampling procedures, including ensemble subsampling and the nonparametric bootstrap. Across vision and text experiments, HIFI scales to pretrained models ranging from tens of millions to more than ten billion parameters while providing accurate point estimates and well-calibrated confidence intervals across diverse architectures, and reduces training compute by up to 91.9% relative to direct resampling. An application to gene-expression inference illustrates the practical value of the framework for uncertainty-aware scientific inference.

We study squared-error estimation when a nuisance parameter is unidentified on a submodel but affects a target that vanishes there. For models with higher-order profile expansions, we give sufficient conditions under which the largest critical-path minimax risk is the exact asymptotic minimax risk over the full parameter space. An efficient observable feature, negligible calibration error and stable recovery yield one estimator attaining the leading constant uniformly, with the profile coordinates unknown. The estimator uses calibration estimates, a local feature mean and an empirical recovery feature. The constant is expressed through a bounded-normal decision problem. We verify the conditions in two models. In centered Gaussian logistic errors-in-variables regression with one surrogate and two unknown variances, we determine the fifth-order global coefficient risk and its exact constant. A sparse extension separates variance-decomposition uncertainty from direction estimation and specifies when the scalar constant is preserved. In a degree-balanced replicated-network model, we determine the sixth-order global edge-contrast risk when block proportions are unknown. Triangles identify a parameter combination; a bowtie supplies the additional information needed for the contrast.

Selecting adjustment covariates for a distribution-valued outcome must preserve dependence that scalar summaries can discard, and surveillance outcomes are also dependent across space and time. Under unconfoundedness and a relative-overlap condition we identify average treatment effects, stochastic and incremental intervention effects, and natural direct and indirect effects for such outcomes, with doubly and triply robust scores. For independent data, adjustment sets are selected exhaustively by distance-covariance sufficient dimension reduction: we give a root-n projector expansion, a basis-invariant criterion tensor, and a simultaneous outer confidence collection calibrated by fractional sampling with a finite-population correction. For dependence, units are functions of a latent Gaussian field, exponential in space and Ornstein-Uhlenbeck in time, plus independent noise. Such data are $\rho$-mixing at an exponential rate, and the covariance between any two square-integrable scores is a power series in the latent correlation with positive semidefinite coefficients. Consequences include a central limit theorem for all the causal estimators, the exact undercoverage of independence-based intervals, a consistent Mehler covariance estimator under finite chaos order, and an asymptotically conservative convex-chord estimator; block fractional sampling extends the collection under an assumed linearization. In simulations, fractional sampling raises weak-signal collection coverage from .69-.78 to at least .98 while retaining nearly all candidate sets, and the dependence-aware estimators substantially reduce undercoverage, although none is exact when few units fall within a correlation range. A descriptive analysis of U.S. syphilis surveillance and Medicaid expansion illustrates the estimands and compares dependence-aware with conventional inference.

Given a random probability distribution \(F\), whose realizations are defined on \(\mathbb R^d\), and a kernel \(k\), our aim is to analyze the properties of the kernel mean embedding, \(\mu_F\), of \(F\) as a random element of the corresponding reproducing kernel Hilbert space \(\mathcal H\). This establishes a rigorous framework in which kernel statistical techniques for distributional data can be stated both in the sample and population settings and, thus, the asymptotic properties of sample procedures can be properly established. As an illustration of this approach we focus on the principal components of \(F\). We obtain a general convergence result and the explicit expression of the spectral decomposition for the uncentered covariance operator of a normal probability distribution with a Gaussian distributed expectation. The performance of this kernel principal components procedure is illustrated with simulations and the analysis of the intensity distribution in synthetic aperture radar images.

Language model performance depends strongly on the mixture of domains in the pretraining data corpus. Popular data mixing methods select these domain weights by training many small proxy models on random mixes and then fitting a regression model to predict performance from the mix. We recognize this practice as a domain attribution problem, where the regression measures each domain's contribution to the target metric. Through this lens, the standard Dirichlet sampling of proxy data mixes is a poor regression design that requires tuning and cannot support a linear fit for hundreds of domains. Building on classical optimal experimental design, we introduce LinMix, which samples mixes at the vertices of the capped simplex so that each domain is at its cap or absent. This hyperparameter-free vertex sampling decorrelates domain effects, enabling an interpretable linear model whose coefficients are the domain attributions. The optimal mix given by LinMix provably keeps only domains with positive attribution values and requires no search over candidate mixes. We further establish the optimality of vertex sampling, and prove that Dirichlet sampling inflates the attribution error when its mixes concentrate around the prior data mix. Pretraining experiments with 1B-parameter OLMo2 and 4B Qwen3.5 language models show that LinMix outperforms Dirichlet-based data mixing across reasoning, code, and math benchmarks, with relative gains of 46% on MBPP and 14% on GSM8K. At 288 fine-grained domains, only vertex sampling supports a linear fit, and LinMix improves benchmark accuracy by 11% on average over the best nonlinear baseline.

Reporting the best of $M$ noisy, unbiased scores overstates the winner's value by up to $c\sqrt{V}$, where $V$ is the total noise variance and the sharp constant $c$ depends on how the noise is dependent. It was known only at the two extremes, arbitrary dependence ($\sqrt{(M-1)/M}$) and i.i.d. noise. We determine it for two classes in between: for independent, not identically distributed noise it is $\sqrt{(M-1)/(2M)}$, and for pairwise nonnegatively correlated noise it is $(M-1)/M$; for uncorrelated noise this answers the maximum version of an open question about the expected range. Explicit finite constructions attain both. As a corollary, the bias of the cross-validation estimator is bounded sharply within each class. All theorems are formally verified in Lean 4. Across 448 language-model and classifier selection pipelines, designs with per-candidate items realize a median 0.75 of the independent-class ceiling. At a fixed item budget, scoring all candidates on shared items is markedly less optimistic than giving each candidate its own items.

Scatter plots draw one mark per observation, so observations that share a location are hidden. In a two-item Likert survey of 2,757 respondents, 98.7% of the data lie under other points and the plot shows only 36 dots. The density-distribution sunflower plot of Dupont and Plummer addresses this by drawing raw points in sparse regions and sunflowers, whose petals count observations, in dense ones. It is not available in R, and its single petal weight for dense cells fails when cell counts span more than an order of magnitude: small dense cells collapse to one petal, and the count a reader decodes can be wrong by 150% or more. We propose geometric petal tiers, in which each cell uses the smallest weight among 1, b, b2, . . . observations per petal that keeps its flower within P petals. Every flower then has between about P/b and P petals, and the relative error of every decoded count is below b/(2P), 17.9% at the defaults b = 5 and P = 14. We also define tie diagnostics and decision rules that use them to choose between an ordinary scatter plot, jitter or transparency, a classic sunflower plot and geometric tiers. Both are implemented in the ggplot2 extension ggsunflower. We illustrate the methods on survey, automotive, historical and large commercial data, and in a simulation over 35 conditions geometric tiers keep the worst-cell error at or below 14% where a single weight reaches a median of 251%.

Computing local-level variable importance (VI) is critical for understanding machine learning models. Local permutation-based VI metrics offer a simple yet powerful approach, measuring how much a model's output changes when perturbing its inputs. However, existing perturbation strategies often ignore feature dependence or account for it only through restricted approximations. Perturbing a feature without fully respecting this dependence can generate unrealistic counterfactuals, thereby evaluating the model in regions unsupported by the observed data distribution. We address this gap by introducing a new model-agnostic local VI metric and estimator, Local Understanding of Covariate Importance with conditional Density estimation (LUCID). LUCID perturbs features of interest using plausible values drawn from the full conditional distribution given the remaining observed features. We establish theoretical connections between LUCID and existing VI measures, and provide an estimator with convergence guarantees and uncertainty quantification. Finally, we demonstrate LUCID's stable performance and practical utility across simulated and real-world examples.

Health surveys often sample clusters within strata and then individuals within clusters, and flexible outcome and treatment models complicate inference for adjusted contrasts from such surveys. We establish finite-population limit theory for survey-weighted augmented inverse-probability-weighted estimation within the double machine learning framework, using cross-fitting that keeps whole clusters together; a cross-fitted targeted maximum likelihood estimator inherits the results under additional targeting conditions. With first-stage clusters drawn by simple random sampling without replacement within strata, the theory allows arbitrary within-cluster dependence, appreciable sampling fractions, and many small strata, a few large strata, or both. Under bounded cluster sizes, a cluster-generation model and nuisance convergence-rate conditions, the estimator is asymptotically normal about the finite-population contrast, and a stability property derived from the cluster model controls the cross-fitting error without entropy conditions. The stratified ultimate-cluster variance estimator, without a first-stage finite-population correction, consistently estimates a with-replacement variance that is at least the without-replacement design variance of the normal limit, so the resulting intervals are asymptotically conservative, and exact when the excess is negligible. In simulations, near-nominal or higher coverage coexisted, in different settings, with remainder diagnostics above the study's negligibility benchmark and with inaccurate standard errors. In three Service Provision Assessment surveys from Malawi and Haiti, each frame treated as a fixed population and resampled 500 times, mean reported standard errors were 4%-20% larger than the empirical standard deviations, while coverage of whole-frame reference values was near or above nominal.

We study the minimax regret rate for episodic kernel-based reinforcement learning, where the expected reward and transition density belong to a Sobolev reproducing kernel Hilbert space, with continuous state and action spaces, and adversarial initial states. For Sobolev smoothness parameter $\nu>0$, state dimension $d_S$, and action dimension $d_A$, we prove that the minimax expected regret over $T$ episodes is $\widetilde\Theta(T^{\max\{e_A,e_B\}})$, where $e_A=\frac{\nu+d_A}{2\nu+d_A}$, $e_B=\frac{\nu+d_A+3d_S/2}{2\nu+d_A+2d_S}$ with the other model parameters fixed. The two exponents reflect the distinct difficulties of action selection and learning across states. We introduce Sobolev Optimal Value Iteration (SOVI), which attains this rate from observed rewards and transitions. For fixed smoothness and reference-kernel lengthscale, its regret bound has a prefactor polynomial in the state-action dimension $d=d_S+d_A$, horizon $H$, transition and reward RKHS norm bounds $u,B_r$, and kernel norm-comparison factor $\Lambda$, up to logarithmic factors. The algorithm separates initial action selection from continuation-value estimation and combines local polynomial approximation with the shared RKHS structure of the approximation errors. Our matching upper and lower bounds determine the optimal dependence on the number of episodes for Sobolev kernel-based reinforcement learning. Thus, we partially address the open problem in Vakili(2024) under the condition of the Sobolev kernel setup.

We generalize cubic smoothing splines to signals with a finite number of discontinuities at unknown positions. Rather than changing the roughness penalty, we deform the domain over which it is computed: every interval between consecutive observations can be stretched by a non-negative warp, subject to a total budget, and the optimizer spends the budget only where a jump makes stretching cheaper than bending. The estimator inherits the state-space form of smoothing splines, so it can be fitted by maximum likelihood with the Kalman filter, and it also admits a fast alternating algorithm that solves a banded linear system and takes a projected-gradient step on the warps. The budget is selected by an extended Bayesian information criterion whose constant is calibrated by simulation. In a Monte Carlo comparison the new estimators reconstruct a piecewise-smooth signal more accurately than the cubic smoothing spline with discontinuities of Storath and Weinmann, at the cost of more false alarms. Two energy applications illustrate the method: the temperature response of electricity demand in Northern Italy, and retail fuel prices, whose discontinuities, changes in excise duties, are dated by public record. The three supplements follow the references.

Contextual dueling bandits learn from pairwise comparisons, the same feedback used to fine-tune language models from human preferences. Standard algorithms explore aggressively and can show users two poor options. We study conservative contextual dueling bandits, which require that, at every round, the cumulative score of the better displayed item stay above a $(1-\alpha)$ fraction of a trusted baseline's. An item's score is its probability of beating the best available item, so the constraint depends only on utility differences and needs no absolute rewards. This also changes how safety can be certified: we show that transferring the safety test of the conservative bandit algorithm CLUCB unchanged can incur linear regret, even when the baseline's utilities are known exactly. For linear utilities, we develop MaxInP-BC, which certifies scores through utility gaps and, when it must fall back, pairs the baseline with an informative partner so that fallback rounds still drive learning. It is safe with high probability at all times, with regret $\tilde{O}(d\sqrt{T}/\kappa)$ plus an additive cost of conservatism that does not grow with the horizon. For general utility classes, we develop BTIGW-BC, which turns the guarantee of an online logistic-regression oracle into cumulative safety certificates. It is safe at all times, makes finitely many fallbacks, and has sublinear regret whenever the oracle's logistic regret is sublinear. In experiments, both satisfy the constraint, which their unconstrained counterparts often violate, at a moderate cost in regret.

Clinical prediction models are typically developed in a single finite dataset, which is one of many possible datasets that could have been collected. Applying the same model development procedure in another sample will provide different predictions for an individual. The degree to which an individual's prediction varies across development sets is referred to as prediction instability and is not captured by average performance measures. Resampling of the training data offers a practical way to assess instability, but its ability to do so accurately and reliably for different model choices is unclear. In a simulation study, we considered non-parametric, parametric, and out-of-bag bootstrap methods and repeated 10-, 5-, and 2-fold cross-validation (CV) to estimate metrics of prediction instability for Elastic Net logistic regression and Random Forest. The same metrics estimated in independent development samples provided the reference. We assessed individual-level prediction variance, empirical 95% range width, and mean absolute differences from the original prediction and true outcome probability. Instability increased in smaller development samples, was greater for Random Forest, and varied across the probability range. Repeated 5- and 10-fold CV underestimated prediction instability across scenarios. Default non-parametric bootstrap more closely approximated independent sampling for Elastic Net but often overestimated instability for Random Forest, as did parametric bootstrap. Out-of-bag bootstrap and repeated 2-fold CV generally agreed most closely with independent sampling, although no procedure was uniformly best. Resampling procedures should not be used interchangeably, nor should less variable predictions be taken as evidence of greater stability. To estimate prediction instability, prefer procedures that predict on held-out data and limit overlap between training resamples.

Randomized unbiased estimation can be highly efficient for approximating expectations of functionals of stochastic differential equations. However, in practice, the optimal computational efficiency is rarely attained due to errors arising from parameter estimation and truncation. The effects of these errors on computational efficiency have not been investigated. In this article, we study the error analysis and optimal sample allocation for the single term estimator. Specifically, under mild assumptions on the estimated parameters, we develop a unified error analysis that incorporates both sources of error, and establish the relevant convergence rates for the computational efficiency loss. Based on this analysis, we further investigate the optimal allocation of the computational budget for parameter estimation, and propose several efficient sample allocation strategies. Numerical experiments support the theoretical convergence rates and demonstrate the effectiveness of our proposed methods.

This paper studies randomised quasi-Monte Carlo (RQMC) methods for Metropolis--Hastings (MH) algorithms, whose accept--reject discontinuity complicates RQMC analysis. To circumvent some of these complications, we study Array-RQMC for MH, with $N$ replicated chains evolved over a fixed time horizon $T$, where we make use of Hilbert sorting for joint multidimensional updates at each step. For one-dimensional MH and bounded-variation test functions, we prove a sampling mean squared error bound of $\mathcal O_T(N^{-4/3})$ relative to the finite-horizon expectation. For product targets, coordinatewise Array-RQMC Metropolis-within-Gibbs retains this exponent for additive marginal bounded-variation tests, with constants that may depend on dimension. With Gaussian random-walk proposals, the bounds extend to scalar and additive marginal polynomials with an additional factor $(\log N)^\ell$, where $\ell$ is the maximum polynomial degree. For joint updates in state dimension $p$ using $p$ proposal uniforms, we provide a conditional $\mathcal O_T(N^{-1-1/(p+2)})$ framework under regularity assumptions that are not verified for the joint multidimensional examples. Numerical experiments support the verified one-dimensional theory and demonstrate variance reduction for joint Hilbert-sorted updates, including Bayesian logistic regression, while illustrating the limitations of these gains as dimension increases.

Web platforms use randomized experiment logs to choose budget-feasible treatment allocations while limiting losses to a designated group of units. Under network interference, each allocation requires expected outcomes at treatment-exposure pairs that may be sparsely observed. Existing safe policy selection methods require valid inference for each candidate, but pooling nearby exposures to improve estimability can introduce allocation-specific bias that variance-only bounds omit. We propose exposure-stratified guardrail selection (ESGuard), which aggregates unit-level bias bounds into a bias bound on each candidate's guardrail effect. It incorporates these bounds into simultaneous lower confidence bounds to select an allocation or retain the status quo. Under randomized rounds, a correct exposure mapping, and valid curvature and slope bounds, the conditional bias bound holds in finite samples. With independent rounds and additional regularity conditions, the unconditional probability of selecting an unsafe allocation is asymptotically at most the target level. We evaluate ESGuard on a simulator built from New York City ride-hailing trips. With six months of simulated rounds, we compare estimators that pool nearby exposures on the same logs and candidate sets. Adding the bias bound raises simultaneous coverage from below the nominal 95 percent level to at least 95 percent for every estimator in this comparison. In the underserved-only scenario, ESGuard violates the guardrail in 0.06 percent of its selections, compared with 16 percent for the point-estimate rule under the same gain ranking, and abstains in about a quarter of replications. Omitted spillovers can invalidate the protection. Code is available at https://anonymous.4open.science/r/esguard-4D2E/.

In stochastic multi-objective bandits, Pareto optimal arms may represent different trade-offs among objectives. Balanced sampling across them is important in practical settings such as online advertising. Yet, there seems to be little consensus on how to measure fairness, to our best knowledge. Achieving fairness is also tough as the set of Pareto optimal arms is unknown, and early sampling decisions may create imbalances that are difficult to erase afterwards. We measure unfairness by the KL divergence from the uniform distribution over the true Pareto set to the empirical measure induced by the sampling trajectory. Nicely, this measure admits an exact decomposition that separates the cost of sampling suboptimal arms from the imbalance among Pareto optimal arms. With a recursive form for unfairness, we develop \emph{adaptive top-two sampling}, which combines sampling counts and empirical evidence of Pareto membership to define a gain for each arm and guide sampling without committing to an estimated Pareto set. When the Pareto set is stable under small perturbations, we establish an $O(\log T/T)$ and $O(T^{-1})$ bound on expected unfairness and variance, respectively. Information-theoretic lower bounds show that the $\log T/T$ rate can be unavoidable for uniformly fair policies on stable instances, matching our upper bound. More surprisingly, stability is indeed crucial, as the expected unfairness can be $\omega(\log T/T)$ without it. Finally, we create a variant that encourages exploration of undersampled arms to reduce the risk of persistent underallocation. At sufficiently large horizons, it preserves the rate on the expected unfairness while admitting a sharper bound of $O(\log^2 T/T^2)$ on variance. Numerical studies support these findings and suggest practical insights.

We develop a nonparametric spectral estimation method for variance component covariance matrices in high-dimensional multivariate mixed effects models. Although classical MANOVA estimators are unbiased entrywise, their eigenvalues can be severely dispersed and biased when the number of traits is comparable to the number of realizations of the random effects. Building on a system of self-consistent equations that characterizes the limiting spectral distribution of a MANOVA estimator, we construct a kernel-based nonlinear transformation of the sample eigenvalues that targets the optimal rotation-invariant estimator under Frobenius loss. The resulting procedure corrects for high-dimensional eigenvalue spreading without imposing a parametric model on the unknown covariance spectrum. Our work is motivated by applications in quantitative genetics, where estimating genetic covariance components across many phenotypic traits is of central interest. We implement the method in a new Python package mlmm-shrinkage and compare the proposed estimator with classical estimators across several common experimental designs in this domain through simulations and a real-data application.

Chain-of-thought (CoT) monitoring aims to detect potentially harmful behaviors in large language models (LLMs) from their CoTs. Intervention-based evaluations measure this property by asking whether a monitor can detect the role of an intervention from the CoT when the targeted behavior occurs, typically using minimal-criterion sensitivity ($\mathsf{TPR_{mc}}$) proposed by~\citep{guan2025monitoring}. However, a target behavior observed under intervention is not necessarily caused by it, and we show that this metric can overestimate how often a monitor detects target behaviors actually caused by the intervention. Specifically, we establish $\mathsf{TPR_{mc}}$ as an upper bound on this causal detection rate and derive a complementary lower bound, which provides a tighter evaluation framework for CoT monitorability. With this framework, we study whether LLMs can be trained to make their CoTs easier to monitor. We first construct a training dataset of paired control--intervention tasks spanning multiple intervention types. Using the dataset, we design and apply three intervention-specific training methods based on off-policy self-distillation, reinforcement learning, and on-policy self-distillation to train separate variants of each base model. Across various model-monitor configurations, all three methods generally improve CoT monitorability over the corresponding base models under our extended evaluation framework. Our code is available at https://github.com/xinyue-lxy/learning-to-be-monitored.

We study online inference for averaged Riemannian stochastic gradient descent (RSGD) when the unknown parameter lies on a manifold. Although the averaged RSGD iterate is asymptotically normal, practical inference requires estimating the unknown covariance operator in its limiting distribution. Existing online batch-means methods cannot be applied directly because a general Riemannian manifold lacks an addition operation, and the tangent space containing the Riemannian gradient varies across iterations. Through vector transport, we construct overlapping and non-overlapping covariance estimators that adaptively combine information from successive batches while accounting for the changing tangent spaces. The proposed estimators are online, require neither Riemannian Hessian computations nor a prespecified terminal sample size, and attain the same convergence rates as their Euclidean counterparts (Zhu et al., 2023). Together with asymptotic normality, these estimators enable online statistical inference for RSGD on manifolds. We illustrate the proposed framework through a detailed study of streaming principal component analysis on Grassmann manifolds.

This paper presents a framework for pricing catastrophe (CAT) bonds in incomplete markets through Riemannian geometric optimization of financial entropy. We formulate pricing-measure selection as a maximum financial entropy problem on a finite-dimensional manifold of equivalent measures, parameterized by the non-hedgeable market prices of catastrophe frequency and severity risk, with discounting fixed by the default-free term structure. Since unconstrained entropy maximization recovers the physical measure, we incorporate observed secondary-market spreads, both pooled and date by date. For a compound Poisson-generalized Pareto loss model with Wang distortion, we derive closed-form expressions for financial entropy and the Riemannian metric. We show that CAT bond prices depend on frequency and severity premia primarily through the risk-neutral trigger intensity, making the two premia unidentifiable from market data alone; the entropy criterion resolves this ambiguity. Using a hypothetical earthquake CAT bond calibrated to quarterly spreads of Golden State Re II Ltd. (Series 2014-1), we obtain risk-neutral trigger intensities of 6.58 to 10.45 times the physical intensity (9.20 pooled and 18.26 at the end of 2015). The model-implied risk multiple aligns with the market spread-to-expected-loss ratio (8.87 versus 8.81 at issuance). The minimal-entropy measure attributes 53.5% to 63.7% of the log risk multiple to frequency risk, reflecting the selection principle rather than market identification. Finally, equating the coupon with the quoted spread understates the risk multiple by approximately half.

The Minimum Covariance Determinant (MCD) estimator is one of the most widely used high-breakdown estimators of multivariate location and scatter. Its practical success relies on suitable corrections, which remove the asymptotic and small-sample biases introduced by trimming. While explicit consistency factors are available under the multivariate normal distribution and, more recently, under multivariate Student-t models, these corrections only address the asymptotic behavior of the estimator. In the Gaussian case, it is known showed that the MCD also exhibits a non-negligible finite-sample bias, particularly for small sample sizes and relatively large dimensions. Such bias propagates to robust Mahalanobis distances and may affect subsequent procedures, including multivariate outlier detection and robust inference. This paper investigates the finite-sample behavior of the MCD estimator and of the related robust distances under multivariate Student-t distributions, a potentially relevant scenario in many application fields. Extensive Monte Carlo experiments are carried out over a broad range of sample sizes, dimensions, trimming levels and degrees of freedom. Based on these experiments, empirical correction factors are obtained and approximated by smooth functions, providing a computationally efficient calibration suitable for practical implementations to heavy-tailed data. The proposed corrections are developed both for the scatter bias and the robust distance quantiles, thus leading to improved approximations for robust inference.

Prediction models degrade when the deployment population differs from the one used to train them, and training data cannot say which feature-response relationships will transfer. Expert elicitation is slow and unscalable. We ask whether coefficient priors elicited from large language models (LLMs) improve out-of-distribution (OOD) prediction as the shrinkage target of a regularised regression instead of zero, across five tabular datasets split on real covariates, selecting providers and prior structures by source-domain cross-validation alone. The priors help on four of five datasets, by up to 7.9% OOD mean squared error, and ten of twenty-four prior-informed results resolve the mean improvement at conventional confidence; single-deployment intervals are much wider. On classification the gain is in probabilistic accuracy, not ranking, and control tests with corrupted priors attribute the benefit to feature-specific content rather than scale. Two findings temper this. Variation across candidate priors exceeds the average benefit, yet cross-validation margins fall below their own noise, so selection avoids the worst prior rather than finding the best. On three datasets the prior improves in-domain accuracy as much as out-of-domain, making it better regularisation than shift correction. Only where the training data are large enough that the in-domain gain disappears does elicited knowledge demonstrably improve transfer. Code, priors and results: https://github.com/hugohammer/llm-priors-ood.

In this paper, we firs tderiv eth eunifor mstron gconsistenc yan dth ecorrespondin gconvergence rates for kernel estimators of both the distribution function and the probability density function for sequences of m-linearly extended negative quadrant dependent (m-LENQD) random variables under mild regularity conditions. The obtained results significantl yimprov ean dexten dexisting ones in the literature. As their theoretical applications, we further establish the uniform strong consistency and convergence rates for estimators of the hazard rate function and the mode of the density function. Numerical simulations and a real data analysis are conducted to assess the finite- sample performance of the proposed estimators, confirmin gth easymptoti cpropertie sderive din the theoretical analysis.

Gradient equilibrium (GEQ) is a recent framework in online learning that requires the time-averaged gradients along the learner's trajectory to vanish. We study GEQ when the learner is constrained and ask when such an equilibrium remains achievable in an adversarial environment. We first show that constrained GEQ (CGEQ) can be impossible even under simple and slowly varying constraints, motivating imposing local temporal structure on the adversary. We then develop an approachability framework for CGEQ under locally constrained adversaries, and obtain necessary and sufficient characterizations in the spirit of classical Blackwell approachability. Beyond these theoretical characterizations, we provide an algorithmic principle that converts window-wise no-regret learners into CGEQ algorithms under a local feasibility condition with margin. Experiments on synthetic and real data demonstrate that the resulting algorithms perform strongly in both achieving GEQ and respecting the constraints.

Semiparametric Bayesian networks (SPBNs) are flexible probabilistic graphical models for representing joint distributions of continuous variables through lower-dimensional conditional probability distributions (CPDs). Existing SPBN models use kernel density estimators (KDEs) for non-parametric CPD modeling, but their one-component-per-observation mixture representation can lead to increased inference costs and density estimation errors, which can be limiting. In this work, we define a new CPD modeling approach for non-parametric components based on Bayesian non-parametric Dirichlet Process Mixtures (DPMs) within SPBNs. DPMs provide a data-adaptive clustering of observations, yielding a more compact representation while retaining the flexibility of non-parametric density estimation, unlike KDE. This results in lower inference costs while improving density estimation performance. We implement the proposed DPM-based SPBN (DPM-SPBN) by extending the open-source PyBNesian package and evaluate it on synthetic and real-world data. For a given network structure, DPM-based CPDs significantly outperform state-of-the-art Bayesian network models for continuous variables in density estimation, achieving lower absolute log-likelihood errors on synthetic data. We further investigate the impact of the proposed CPD modeling on structure learning methods relying on CPD-based scores. DPM-based SPBNs achieve lower structure learning and density estimation errors than KDE-based SPBNs. Experiments on real-world data corroborate these findings, identifying DPM-based CPD modeling as the most effective alternative among the evaluated non-parametric approaches for continuous variables.

Concentration inequalities for time series usually involve unknown quantities measuring dependence. This prevents their direct use as confidence bounds, or as criteria for learning algorithms. Recent works proved empirical concentration inequalities for stationary Markov chains, where the unknown quantities are estimated. We extend this approach beyond the Markov case, by considering stationary chains with infinite memory. We first establish a bounded-differences inequality on bounded Polish metric spaces under a summable Wasserstein contraction condition. We then estimate its dependence coefficients from a single trajectory, using either transition counts when the state space is finite, or derivatives of kernel conditional-density estimates for a continuous state space. We give general error bounds and illustrate their dependence on the memory tail. Finally, a fixed multiplicative safety factor turns these estimators into empirical concentration bounds. While some regularity assumptions are required to make the bounds empirical, our procedure remains fully adaptive in the sense that it does not require the knowledge of the constants involved in these assumptions. The empirical confidence bounds derived from our results are asymptotically as tight as the theoretical ones.

Time series data often exhibit variation, clustering, and persistence in uncertain environments. Based on an uncertain measure, existing studies have primarily focused on conditional mean structures while ignoring variance and persistent dynamics. To address this gap, this paper develops an uncertain generalized autoregressive conditional heteroskedasticity (UGARCH) model that incorporates lagged conditional variance terms. We first prove sufficient conditions to ensure mean stationarity of the variance process. Then, we propose recursive constrained least squares to estimate the model parameters and use cross validation to determine the orders of the mean and variance equations. In addition, we also employ uncertain hypothesis tests to determine whether a time series exhibits a UGARCH effect and to assess the goodness of fit. Point and interval forecasts are subsequently constructed from the recursively obtained next-period conditional variance and the expected value of the mean equation. Numerical simulations demonstrate the parameter estimation performance of the proposed method and show that UGARCH achieves smaller conditional-variance fitting errors than UARCH under the considered settings. In the empirical analysis, we apply the proposed method to West Texas Intermediate crude oil futures closing prices, which exhibit time-varying volatility and support short-term point and interval forecasts. Keywords: Uncertain measure, Uncertain time-series analysis, UGARCH model, hypothesis test

Whether a federated predictor benefits an individual client depends on both population heterogeneity and estimation uncertainty. We study this question for Nadaraya--Watson regression in a triangular array with potentially different fitting and evaluation distributions. Each client transmits local kernel summaries, allowing the server to construct a covariate-density-weighted federated estimator without centralizing individual observations. A client combines its fitted local predictor with the federated predictor through a prescribed weight, and gain is the paired reduction in squared prediction loss on an independent audit observation. Exact identities characterize both the gain of the realized fitted pair and its repeated-fitting mean. We also establish pointwise inference for the federated estimator. A two-stage analysis of client gains separates fitting variability from audit variability, retains the covariance created when a client's observations enter both its local estimator and the common server estimator, and yields fixed-block simultaneous inference across clients. Robustness is examined under two perturbation models. A Kullback--Leibler neighborhood provides a variance-sensitive lower certificate in a regular positive-gain-variance regime, while an $L_2$ neighborhood yields an exact least-favorable regression direction and robust value. Exact-model, vanishing-variance, and exact-zero-direction boundaries are kept separate because their scalar inputs are nonregular. The framework thus distinguishes decisions about a realized fitted pair from claims about the learning procedure under repeated fitting.

Manifold-valued data arise in a wide range of scientific applications. Existing statistical methods typically follow either an intrinsic approach, which characterizes the manifold itself via quantities such as geodesic distance or the Fr\'echet mean, or an extrinsic approach, which embeds the manifold into an ambient Euclidean space, performs unconstrained statistical estimation, and projects the result back to the manifold. While the latter is conceptually and computationally simple, likelihood-based extrinsic solutions remain scarce, due to the lack of a general recipe for choosing an embedding map and for characterizing its impact on the transformed distribution. In this article, we tackle the problem using an augmented likelihood approach: we view the manifold-valued data as the Fenchel mirror map of a latent variable, which is nearly unconstrained, and model the latent variable with a canonical distribution, such as Gaussian. The key transform, the Fenchel mirror map, is a generalization of the Euclidean projection and can effectively map into the interior of the manifold. Taking a Bayesian approach, the proposed framework enjoys a generative model interpretation and an augmented likelihood free from intractable normalization constants, and it supports principled uncertainty quantification and flexible model-based extensions. We study the theoretical and computational properties of the framework and demonstrate its utility in analyzing neuroimaging data of high-dimensional correlation time series, where each correlation matrix is rank deficient.

Offline maximum entropy reinforcement learning seeks to estimate the optimal soft action-value function \(Q^*\) from a fixed dataset without further interaction with the environment. Although \(Q^*\) is characterized by the soft optimal Bellman equation, the corresponding Bellman operator cannot be evaluated directly because both the reward distribution and the transition kernel are unknown. Consequently, estimating \(Q^*\) requires the joint treatment of environment modeling and value function learning. We propose a framework that addresses these tasks separately. Conditional diffusion models based on probability flow ODEs estimate the conditional reward distribution and transition kernel. The resulting distributional estimators define a data driven approximation of the soft optimal Bellman operator, whose empirical Bellman residual is minimized over a class of deep neural networks to estimate \(Q^*\). We establish nonasymptotic guarantees for the proposed method. For operator learning, we derive total variation error bounds for the conditional flow estimators that account explicitly for score estimation, numerical discretization, and terminal smoothing. For soft value learning, we first use the soft Bellman equation to show that $Q^*$ inherits H\"older regularity from the conditional reward and transition densities. For any H\"older index $\beta$, the value learning stage achieves an excess Bellman residual risk bound of order $\widetilde{\mathcal O}\big(n^{-\frac{2\beta}{d+2\beta}}\big),$ where $d$ is the total dimension of the state and action spaces. In contrast to many existing theoretical analyses of deep reinforcement learning, our results do not require Bellman completeness. Numerical experiments support the theoretical findings and demonstrate the effectiveness of the proposed method.

We study the early training dynamics of nonlinear residual networks with blocks of zero hidden layers as a model of more general residual networks. We analyze networks in a sequential limit where first width and depth diverge, then the number of training data points diverges, and finally the input dimension diverges. This is a more refined setup compared to the typical mean-field theory analysis in which depth and width diverge but the number of training datapoints and input dimension are fixed. Our main technical result is a diagrammatic computation of both the network output and the hidden representations after the few steps of gradient descent in terms of the statistics of inner products of inputs from the training set. One of our principle conceptual findings is a negative result: in the limit of large input dimension the network output can learn only a linear function of its inputs early in training since higher learned monomials are more and more suppressed in the input dimension and vanish in the large dimension limit. This suggests that mean-field limits of ResNets are surprisingly rigid in the regime of high input dimension and may not be sufficient toy models for understanding feature learning. Finally, we investigate how to ensure hyperparameter transfer for the learning rate as a function of dataset size and input dimension. We test our predictions numerically and find excellent agreement with our theory.

Statistical software makes methods widely available. Good statistical work, however, often requires conversations that clarify the scientific question, how the data were collected, and what conclusions the evidence can support. Coding agents and reusable AI infrastructure create opportunities for statisticians to incorporate these processes into agentic tools for research, practice, and education. A review of agentic AI applications for data analysis and study design distinguishes general human participation from human interaction guided by statistical principles. We examine how statistical expertise can guide the design and evaluation of domain-specific interactive harnesses. Two practical entry points illustrate how statisticians can begin building: an implemented causal-consulting skill within an existing agent environment and Model Forge, a custom harness for interactive model and algorithm development. These design examples connect statistical principles to choices about context, memory, tools, and interfaces, and motivate evaluation of both computational correctness and the scientific consequences of interaction. We conclude with a broader interdisciplinary research agenda on adaptive human-agent interaction, including selective information acquisition, evolving user preferences, and evaluation under uncertainty.

Low-rank adaptation (LoRA) is now a standard approach to parameter-efficient fine-tuning of large neural networks. Its low-rank factorization is inherently non-identifiable, making LoRA a singular statistical model whose learning complexity cannot be described by parameter dimension alone. We study this complexity using the real log canonical threshold (RLCT) from singular learning theory and how it depends on both the low-rank parameterization and the pretrained network. Under a local linearization, LoRA induces a low-rank trace regression model whose Kullback--Leibler geometry is controlled by a forward-backward covariance, constructed from activations and backpropagated sensitivities at the adapted layer. This covariance determines which low-rank perturbations are statistically observable. When it is nondegenerate along the relevant low-rank directions, we recover the classical learning complexity of reduced-rank regression; when its kernel meets these directions, additional statistically indistinguishable parameters change the singular geometry. We characterize this geometry locally through observable tangent directions and a residual singularity, and globally through affine linear sections of determinantal varieties. The global learning complexity need not be determined at a ground-truth parameter, as singularities on another rank stratum can dominate. We also introduce a curvature-corrected multi-temperature estimator of the RLCT. Experiments on multilayer perceptron and transformer architectures reveal strongly layer-dependent forward-backward geometry and show that ambient covariance degeneracy does not imply degeneracy along the low-rank directions relevant to adaptation.

Background: Proportions are common in interrupted time series analysis (ITSA), but bounded outcomes and serial dependence require distinct modeling considerations. This article provides guidance on selecting and evaluating models for proportion outcomes. Methods: Two published healthcare ITSA studies were reanalyzed. The first examined weekly proportions near the boundaries of the unit interval using segmented linear regression. The second examined monthly proportions without reported assessment or adjustment for autocorrelation. Both were reanalyzed using models appropriate for their outcome distributions and temporal dependence. Results: In the first study, segmented linear regression produced six fitted values below zero, and 35 of 66 confidence intervals extended beyond the permissible outcome range. Fractional logistic regression constrained fitted means to the unit interval while accommodating zeros and ones. In the second study, beta autoregressive regression identified significant first-order serial dependence (rho1 = 0.263, 95% CI 0.093 to 0.432; P = .002). After adjustment, residuals showed no evidence of remaining autocorrelation (Ljung-Box Q12 = 15.59; P = .211). Conclusions: Boundedness and serial dependence are distinct considerations in ITSA of proportions. Accounting for autocorrelation does not ensure predictions remain within the permissible range, and modeling bounded outcomes does not ensure serial dependence is addressed. Investigators should assess both features and select models appropriate for the outcome distribution and temporal dependence.

The hazard ratio is the most commonly reported measure in survival analysis, yet it suffers from well-documented limitations, including non-collapsibility and susceptibility to selection bias as treatment-specific risk sets change over time. In this article, we revisit the survival ratio and the cumulative incidence ratio as alternative measures that directly compare survival and failure probabilities. We introduce a novel modeling framework that targets these ratio functions and constructs coherent nuisance models so that the target and nuisance parameters are variation independent. The framework yields valid monotone survival and cumulative distribution functions and supports likelihood-based and doubly robust estimation of the target parameters, together with estimation of covariate-specific survival and cumulative incidence curves. We illustrate the proposed frameworks using two clinical-trial datasets. In a prostate cancer trial, the estimated survival ratio for 1.0 mg estrogen versus placebo increased over follow-up, while a descriptive selected-cohort analysis of a breast cancer screening trial yielded a cumulative incidence ratio that crossed one over time.

Negative electricity prices create market-relevant risks for trading, bidding, renewable revenues, and flexibility valuation. Quantile regression-based postprocessing is a widely used approach for converting point forecasts into predictive distributions. We show that the standard specification, which imposes a single linear relationship between the point forecast and predictive quantiles, can become structurally incoherent when the conditioning point forecast is negative. In such cases, the resulting predictive distribution may exhibit not only ordinary quantile crossing, where adjacent quantiles are in the wrong order, but also a severe version of crossing, which we refer to as tail-quantile inversion. This can reverse downside and upside risk signals. To address this problem, we propose sign-dependent QR-based postprocessing specifications that allow the quantile function to differ between positive and negative point forecasts. Using German day-ahead electricity prices as an empirical illustration, we show that these specifications improve predictive accuracy, particularly for negative point forecasts and for extreme probability levels. Moreover, when combined with joint estimation across quantiles, the sign-dependent specifications largely eliminate tail-quantile inversion and all quantile crossing. The results demonstrate that the sign of the point forecast should be explicitly accounted for when constructing probabilistic forecasts in markets where negative prices can occur.

This paper develops a foundational measure-geometric theory of robustness in machine learning. Robustness is formulated as escape from a retained task state relative to a reference measure on a data-bearing manifold, a perturbation gauge, and an admissible realization mechanism. The theory distinguishes the structural possibility of escape on the reference manifold from its ambient realization under mechanism-dependent observable readouts, while placing both within a unified geometric account. It introduces a mesoscopic organization of escape geometry, through which vulnerable mass, finite-scale morphology, delivered progress, and branch coherence are characterized by explicit laws as linked but non-identical robustness phenomena. By recasting robustness as the geometric formation, organization, realization, and persistence of escape, the theory provides a structural basis for analysis beyond scalar criteria.

Competing risks analysis is an important framework for studying time-to-event data in the presence of multiple, mutually exclusive events. Traditional methods, such as non-parametric estimators and proportional hazards models, may not adequately capture complex hazard patterns or provide stable estimates in practical settings. In this study, we propose a parametric approach based on the generalized exponential regression model for competing risks data. The model allows flexible hazard shapes and incorporates covariate effects through a regression structure. To improve estimation stability, we integrate elastic net regularization into the likelihood framework. The proposed method is evaluated through simulation studies and applied to a real dataset. The results indicate that the elastic net approach provides more stable and precise estimates compared to standard maximum likelihood estimation. Overall, the proposed framework offers a flexible and practical approach for analyzing competing risks data.

Two-slope mixed-effects models with subject-specific changepoints are widely used to describe biphasic longitudinal trajectories. Treatment-arm summaries are often reported using a single "typical subject" curve obtained by plugging mean parameter values into the subject-specific model. This can be misleading when changepoint timing varies across individuals, because averaging subject-specific curves is not the same as evaluating one curve at average parameter values. A related issue is that individual trajectories may bend gradually rather than change slope through a literal kink, with transition sharpness varying between subjects. We propose population-average estimands for two-slope mixed models with random changepoints and smooth subject-level transitions. These estimands target the treatment-arm mean trajectory and instantaneous slope by averaging over fitted heterogeneity in changepoint timing and transition sharpness. We use a baseline-anchored normal-convolution soft hinge, which preserves the broken stick interpretation, includes the abrupt changepoint model as a limiting case, and allows analytic evaluation of the changepoint-marginalized kernels under a truncated normal changepoint distribution for fixed transition sharpness. Averaging over the fitted transition sharpness distribution is then performed by deterministic one-dimensional quadrature. In an application to HIV-1 RNA data from ACTG Protocol 315, with censoring and heavy-tailed residuals, the marginal summaries reveal a more gradual population-level transition than conventional plug-in summaries. The marginal slope remains more negative for longer, and the marginal mean continues to decline after the plug-in curve has begun to flatten. These results illustrate the main reporting implication: in random changepoint models, representative subject curves should not be treated as population-average trajectories or slopes.

Spectral clustering is a widely used clustering method with strong empirical performance and favorable computational scalability. Existing theory has established its consistency and risk guarantees under Gaussian and related mixture models, primarily in settings with relatively simple latent structure. In this paper, we study spectral clustering under a semiparametric Gaussian mixture model, where the observed data become Gaussian-mixture distributed only after unknown monotone coordinate-wise transformations. We derive non-asymptotic bounds for its cluster recovery error. As a byproduct of the theory, we show that spectral clustering provides a theoretically justified initialization for a broad class of high-dimensional mixture-based learning procedures.

Well-documented failure modes of machine learning (ML) systems, including bias, toxicity, hallucination, and leakage of sensitive information, has driven the rise of machine unlearning (MU). Most existing work, however, targets parameter-space indistinguishability, seeking unlearned models that are as close as possible to models retrained from scratch on retained data. In this paper, we argue that this objective is misaligned with practical and user-relevant concerns: MU should prioritize statistically guaranteed output-space control rather than parameter-space similarity. We contend that output control with statistical guarantees is the operationally meaningful notion of unlearning, as it supports rigorous and robust objectives, scales to modern deployment settings, and enables interpretable guarantees. We substantiate this position with empirical evidence and with recent theoretical results that, although not originally formulated in this framing, are consistent with it.

Before every neutral objective in professional League of Legends, each team decides which of its five players walk to the pit, which hold a lane, and when to commit. The leading analysis of such setups, TAGNet for Counter-Strike 2, clusters graph-network embeddings of player positions and reads cluster win rates as tactical quality. We treat a setup as an allocation on a walled map: formations are set partitions on the lattice $\Pi_5$, control is a geodesic Voronoi split of a navigation graph, and commitment is an allocation game solved as a quantal response equilibrium priced in win probability. On all 164 games of MSI and Worlds 2025 (2,994 team-setups, minimap tracks validated against Riot positions), trailing teams leave the pit empty far more often than leading teams, by +34.8 percentage points at the first baron and +30.4 percentage points at Atakhan. Our model, HASSE-QRE, finds six setup patterns that whole-model series-bootstrap refits reproduce (ARI 0.819, comparable to 0.813 for TAGNet's $K$-Means step alone) and indicates that teams price fights, travel and committed players, not the objective itself (weight 0.018; fixing it to one costs 59.9 log-likelihood points). The equilibrium reproduces a quarter of the commitment gap, almost all of it through state variables rather than its strategic channels. Weighting a team-level contrast by exposure moves it across $p=0.05$ (0.046 against 0.052). We release the tracks, setups and code.

Interpretability in Exploratory Factor Analysis is characterized by the fundamental trade-off between structural sparseness and explained common variance. Lacking formal evaluation metrics, researchers frequently sacrifice parsimony in the pattern matrix to maximize the model's explanatory coverage. For this reason, we extend Hoyer's sparseness ratio into a flexible \texorpdfstring{$\mathcal{L}_{p_1}/\mathcal{L}_{p_2}$}{Lp1/Lp2} formulation to introduce the Interpretability Index $I$, defined as the geometric mean of explained common variance and the generalized sparseness index. We analyze the asymptotic behavior of $I$ as norm orders increase, establishing that raising $p_2$ alone yields a stable limit, whereas increasing $p_1$ and $p_2$ jointly causes the measure to saturate. The canonical choice $(p_1,p_2)=(2,4)$ balances numerical stability with a smooth, tractable objective. We finally demonstrate, on five benchmark datasets from the statistical literature, how the Interpretability Index provides a systematic criterion for selecting the optimal parametrization of the latent space according to the user's analytical goals.

Two important indicators of climate change are the decline in Arctic sea ice area and the increase in average temperatures globally. Both sea ice and temperature exhibit seasonal patterns, making circular regression a natural approach for their analysis. In this paper, we construct computationally tractable Nadaraya-Watson type kernel regression estimators on the unit circle, using rate-optimal kernel density estimators. We introduce a leave-one-out cross-validation procedure for the data-driven selection of the smoothness parameter and evaluate the performance of the proposed estimators through a simulation study covering 48 scenarios, including comparisons with existing local polynomial estimators. We then apply our methodology to historical Arctic sea ice area data and daily temperature observations in Stockholm. The fitted regression functions and their derivatives or finite differences allow us to investigate changes in both the levels and seasonal dynamics of these environmental indicators. In particular, we observe a decline in fitted sea ice area and substantial changes in its seasonal rates of growth and retreat, together with higher fitted temperatures across most of the year in Stockholm during the more recent period in comparison to historical data.

Kernel techniques are among the most powerful methods in data science and statistics. One of the key properties of kernels contributing to their numerous successful applications is that the associated reproducing kernel Hilbert space is expressive enough to encode probability distributions without loss of information or to capture the joint independence of $M\ge 2$ random variables, under mild conditions. Computationally, these measures (i) rely on the so-called Gram matrix which contains the pairwise similarity of sample points, and (ii) converge at a rate $O \left(n^{-\frac{1}{2}}\right)$, where $n$ denotes the sample size. We focus on measuring the independence of latent position random graphs (LPRGs), a model widely-applied for instance in neuroscience, sociology, and epidemiology. In LPRGs a latent representation is associated to each node. The similarity of these representations (meant via a kernel) determines whether two nodes are connected, making the Gram matrices only implicitly observable and the estimation challenging. Despite the fundamental importance of quantifying the independence of $M$ LPRGs, very little is known about the solution of this problem; existing techniques are restricted to $M=2$ components and only guarantee consistency without rates. In this work, (i) we define an independence measure called graph Hilbert-Schmidt independence criterion (GHSIC) for $M\ge 2$ LPRGs, (ii) we prove that it can be estimated at a rate $\tilde{O}\left(n^{-\frac{1}{3}}\right)$ relying on spectral decomposition of adjacency matrices, (iii) without the need to choose kernels and their hyperparameters. Numerical experiments demonstrate the effectiveness of the proposed GHSIC estimator both on synthetic examples and in the analysis of brain functional connectivity subnetworks in natural language perception.

We study how contemporaneous equity-return transmission shapes systemic exposure among 51 listed European banks. Using a recursive non-Gaussian network model on volatility-standardized returns, we construct two market-capitalization-weighted measures based on direct and indirect transmission effects: the Systemic Impact Score (SIS), capturing a bank's potential influence on other institutions, and the Systemic Vulnerability Score (SVS), capturing its exposure to disturbances originating elsewhere. The fitted network reveals distinct transmitting and receiving roles, highlighting differences between institutions with substantial outward influence and those with high incoming sensitivity. Receiving roles are generally more reproducible than transmitting roles across repeated network discovery, and selected connections remain informative about lower-tail dependence in subsequent observations. Volatility standardization substantially reduces detected dependence among recovered shocks, although the full-sample model does not satisfy the designated criterion for supporting shock independence. Alternative recursive orderings also yield comparable directional tail asymmetries, limiting the extent to which these patterns validate a particular ordering. The results provide complementary measures of model-implied systemic transmission and vulnerability, with explicit attention to uncertainty and identification limits

Variables can predict an outcome jointly while each is individually independent of it, causing structure learners to discard informative links. We investigate how to operationalise weaker faithfulness through two approaches: selective protection in an adjacency learner and explicit retention of alternative directed graph explanations. Matched experiments show that changes at both an early candidate-removal stage and later pruning can be necessary. In noisy XOR with a descendant (N = 10,000, 20 matched samples), combining reactivation and protection raises complete parent-pair recovery from 0/20 for either change alone to 20/20, while adding false edges. A confirmation study shows that the family estimator can preserve explanations lost by orienting the protected learner's selected skeleton: in one weaker signal XOR condition (N = 10,000) it retains the entire reference family in 48/50 samples, compared with 0/50 for the skeleton adapter. These gains are conditional. Protection can add only error, weaker effects produce false exclusions, and an ordinary faithful-chain comparison favours the PC learner. Complete nonempty family outputs decline at six/eight variables, and all 400 twelve/twenty-variable searches remain unfinished at the reporting cutoff. Neural controls show engineered retention, while the family estimator returns no accepted graph within its application budget. This operationalisation identifies two deletion stages that must be coordinated in the motivating example and shows when direct family search preserves explanations lost by selecting one skeleton. Finite evidence and computation remain substantial barriers to reliable general inference.

A scalar measurement can determine a robust decision without identifying the latent covariance or every decision risk. We study finite Gaussian projection experiments with a fixed mean-plus-standard-deviation target. The observation laws define a set of indistinguishable moment models. We exhibit decisions that agree throughout that set despite unidentified risk values, and give a confidence-set certificate that returns a decision or an unresolved outcome. We prove soundness, necessary abstention for indistinguishable models with incompatible optima, and a fixed-checkpoint certification guarantee. A scalar profile reduces nonlinear risk comparisons to convex inner problems. Across seven synthetic instances, jointly calibrated moment regions give sufficient budgets 21 to 52 times explicit expected-sample information lower bounds. Separately measured stopping times are lower, with zero decision errors in 1,900 simulated streams. These comparisons do not constitute matching bounds, and the guarantees require Gaussian replication.

Grey incidence analysis ranks the factors of a system from short sequences without assuming a probability model, but the grade it returns has no sampling theory: nothing indicates whether an ordering reflects the system or the few data at hand. This paper reduces trend agreement between two sequences to the signs of their centred increments and analyses the resulting counts with Lovegrove's likelinesses, averages of probabilities over an underlying set of distributions that replace the elicitation of a prior density by a statement of what is possible. The result is a concordance incidence grade with closed-form inference: estimation, the likeliness that one factor outranks another, prediction of future concordance and a baseline for searches over many candidates, with bounded underlying sets encoding partial knowledge. Comparisons whose sign falls below the resolution of the data are treated as indeterminate; discarding them is exact when they carry no information, and otherwise the grade becomes a grey number whose width measures what the data cannot decide. The grade is symmetric in any system and resolves the conflict between the totality and pair-symmetry axioms by moving system dependence into inference. Simulations show that plug-in prediction underestimates variance by a factor of up to 1 + m/n regardless of sample size, and that a fair-coin baseline yields up to 63% spurious patterns in searches, where a correctly bounded set keeps them below 5%. Worked textbook examples and a regularisation benchmark with known truth illustrate the method.

We characterize the high-dimensional convergence of the arithmetic mean from the independence Metropolis-Hastings algorithm initialized at its proposal. Convergence is measured in the worst-case expected absolute error over bounded functions. We prove necessary and sufficient conditions for the mean to converge in relation to the importance weight. We precisely describe the logarithmic cutoff profile of the sharp decay to convergence where the relative entropy between the target and proposal determines the required iteration length. These results make connections between the required iterations of independence Metropolis-Hastings and importance sampling. For concentrating Bayesian posteriors, we illustrate how the required iteration time can be estimated providing concrete guidance for simulations. We justify the logarithmic cutoff analysis by showing a standard cutoff phenomenon is absent and prove that popular convergence diagnostics based on simulation output can validate inaccurate simulations.

Dimension-reduction methods such as PCA identify a subspace capturing most of the variation in the data, but not a distinguished basis within it. Selecting a basis in which the samples are sparsely represented is a recurring task in unsupervised learning, addressed by methods in factor analysis, dictionary learning, and subspace clustering. We pose this task as Wasserstein approximation of the data distribution by measures supported on the axes of an orthogonal frame, which, in turn, reduces to a nonsmooth optimization problem over the orthogonal group $\mathrm{O}(d)$. We show that, under mild assumptions, both empirical risk minimizers and a locally initialized stochastic gradient method recover the population-optimal frame at the near-parametric rate $\widetilde{O}_d(n^{-1/2})$, where $n$ is the sample size. The main difficulty is that, although the population objective is smooth, the empirical objective is not-this mismatch precludes standard covering arguments. We overcome this issue by replacing Lipschitz continuity of the empirical gradient with a coarsely Lipschitz bound, a tool that may be of independent interest. Numerical experiments support our theoretical findings.

The Poisson kernel-based (PKB) distribution is a flexible and numerically convenient family on the unit sphere used for model-based clustering, uniformity testing, and directional data analysis. Available random variate generators rely on rejection from a von Mises-Fisher, angular central Gaussian, or projected Saw envelope and therefore require a rejection loop, tuning or root-finding, or a univariate setup phase. We show that none is needed. A uniformly random line through the interior point $\rho\mu$ meets the sphere in two points; selecting one with probability proportional to the opposite segment length returns exactly the target density. The sampler is exact, rejection- and tuning-free, uses one uniform direction and one Bernoulli draw, and costs $O(d)$ operations. We place the distribution in the power-Saw family of densities proportional to $\|x-\rho\mu\|^{-\xi}$ and show that, for $d\geq3$, the same construction is exact for one further exponent. This introduces the Newton kernel-based (NKB) distribution, whose normalizing constant is identically one. Across a wide grid of dimensions, concentrations, and sample sizes, the direct sampler is faster than the better of two previously recommended methods in every case, with the largest gains at moderate concentration.

To our knowledge, this paper provides the first systematic quantitative evaluation of the ICT Accumulation-Manipulation-Distribution (AMD) model on CME E-Mini NASDAQ-100 futures (NQ). The AMD model proposes that institutional participants structure intraday price delivery in three phases: overnight accumulation in the Asian session, a deliberate false breakout during London, and directional delivery to the opposing liquidity pool during the New York morning session. Using four years of real 15-minute OHLCV data from Databento (January 2022 to July 2026), we operationalize the AMD rules into a strict mechanical strategy and analyze the statistical properties of the resulting signal across N = 484 manipulation signal days using chi-square testing, Benjamini-Hochberg multiple comparisons correction, and regime-conditional analysis. Three principal findings emerge. First, the AMD manipulation signal fires on 44% of trading days. Second, 57% of valid signals see full target delivery before the 9:40 AM New York entry window opens, revealing a structural mismatch between the model's timing assumptions and actual price delivery. Third, within this operationalization of the AMD model, no directional information beyond the daily trend regime was observed: when the signal aligns with the trend it is correct 62-71% of the time (Cohen's h = 0.23-0.43); when it opposes the trend it is correct only 25-30% of the time (Cohen's h = 0.42-0.53). All four trending-regime cells survive Benjamini-Hochberg correction. Key findings are robust to variation in the manipulation breach threshold (5-20 NQ points), Asia range filter percentile, and stop buffer. The regime-dependent signal pattern constitutes a specific, testable hypothesis for follow-on research.

The integration of external information can substantially improve the efficiency of statistical inference in an internal study, but is often complicated by between-study heterogeneity. Additional challenges arise when external studies examine auxiliary outcomes rather than on the target outcomes and provide only summary statistics. To address these challenges, we propose a semiparametric framework that incorporates external summary statistics into the internal analysis via the generalized method of moments. To accommodate between-study heterogeneity, the external summaries are formulated within a random-effects model centered on the internal parameter estimates and subsequently embedded into the internal estimating equations. We establish the asymptotic properties of the proposed estimator. While uncertainty from the first-stage estimation vanishes as the number of external studies increases, it remains non-negligible where the number of external studies is moderate. In this setting, we show that a hybrid bootstrap procedure correctly propagates this uncertainty and consistently estimates the asymptotic covariance of the proposed estimator, thereby ensuring valid inference. We further derive theoretical efficiency bounds and demonstrate robust finite-sample performance through simulation studies. In an application to colorectal cancer tumor sequencing data, we show that integrating external information of auxiliary outcomes reveals molecular subtype-specific etiologic associations that would not be detected using the internal data alone.

In global sensitivity analysis, Shapley effects provide a game-theoretic allocation of output variance among potentially dependent inputs. Their computation typically requires repeated conditional sampling, which can be challenging even when independent samples from the joint input distribution can be generated exactly. This difficulty arises naturally in hierarchical models, where conditioning on arbitrary subsets need not follow the generative order. To efficiently estimate Shapley effects under such dependence between input variables, we propose a permutation-based framework using coupled unbiased Markov chain Monte Carlo (MCMC). Our proposed method combines two complementary coupling mechanisms: (i) a joint coupling of conditional distributions along a random permutation, designed to reduce the variance of estimated marginal contributions, and (ii) a meeting-based coupling of Markov chains that removes finite-time MCMC bias through a telescoping correction. Under suitable conditions, we prove that the resulting estimators for the unnormalized Shapley effects are unbiased, and have finite variance and finite expected computational cost. Numerical experiments with various test cases illustrate the computational efficiency of the proposed method.

The Gaussian Mixture Model (abbrv. GMM) is a classic generative model for clustered data, for which spectral clustering based on dimension reduction via PCA has been an effective framework. In this work, we investigate kernelised covariance matrices as an alternative mechanism for dimension reduction and spectral clustering, focusing on truncation kernels that select pairs of data points with suitable separation constraints. The structure of these matrices is motivated by considerations from statistical mechanics of repulsive particle systems. Our approach is particularly relevant for the setting of imbalanced clusters, where the sizes of the largest and smallest clusters have a diverging ratio with the problem size, and existing PCA-based methods turn out to be of limited effectiveness. We demonstrate efficient dimension reduction and clustering performance of these modified covariance matrices, substantiated by quantitative bounds. In particular, they achieve superior dimension reduction guarantees compared to PCA for a wide range of parameters in the regime of imbalanced clusters. For spectral clustering, our approach exhibits improvements over PCA in the imbalanced setting, in particular guaranteeing accurate clustering in a much larger regime of intra-cluster separations. Our theoretical guarantees reflected in a range of simulation experiments that exhibit these improvements for unbalanced clustering problems. Our analytical approach, based on kernel Laplacians, provide a toolbox as well as a relevant conceptual ingredient for clustering problems at large. We believe that our work will motivate the application of such kernelised covariance matrices for related classes of statistical problems, as well as spur new investigations in the theory of random matrices.

The Bayesian approach to inverse problems provides a powerful framework for inferring unknown parameters from noisy observations. Among the computationally most challenging statistical inference problems are problems with intractable likelihoods, especially if no unbiased likelihood estimator is available. We focus on a class of such problems, where a biased likelihood estimator is obtained through a stochastic approximation of a forward map (stochastic solver). Obtaining an empirical posterior approximation in applications is very expensive due to the repeated stochastic solver evaluations required by standard sampling methods. To address this computational bottleneck, we develop a novel algorithm based on a density-tempered marginalized Sequential Monte Carlo (SMC) sampler. Our approach (hierarchical SMC) accelerates inference by leveraging a hierarchy of stochastic solvers with increasing accuracy both along the tempering sequence and within mutation stages of the SMC sampler. We establish, under additional assumptions, that the asymptotic speedup is proportional to the number of tempering stages $T$. We demonstrate computational gains on several examples, including a nonlinear, high-contrast inverse problem arising in photoacoustic medical imaging. On this problem, we demonstrate a speedup by a factor of 14.3, reducing the inference wall-clock time from about 4.6 days to 7.7 hours on an NVIDIA A40 GPU.

Blind random search is a simple global optimization method that only requires the value of the loss function, without needing gradient information or an explicit loss model. However, its classical formulation assumes that the loss evaluation is noise-free, which limits its direct application to practical problems with noisy measurements. We extend blind random search by combining a vanishing acceptance threshold with the running average of noisy loss measurements. Under the stated assumptions, we prove that the proposed algorithm converges almost surely to the global minimizer. We further analyze the rate at which the loss at the current estimate approaches the minimum loss. For low-dimensional cases, we obtain an almost sure asymptotic upper bound on the difference between the two, and for high-dimensional cases, we prove that this difference, suitably scaled, has an exact Weibull limiting distribution.

Density estimation with generative models is a common task in the physical sciences, for example in background estimation in particle physics. Existing machine learning techniques, however, lack methods to quantify uncertainties that can be propagated into a statistical test. That limits the validation of generative models to standard goodness-of-fit tests, in which the generative model is treated as exact (point-null hypothesis). But a model fitted to a finite training set is never exact, so if its uncertainties are neglected, any sufficiently powerful goodness-of-fit test will eventually reject it once the test set is large enough to resolve this estimation error. We propose SIGMA, an end-to-end framework combining density estimation with principled uncertainty quantification. We use Gaussian mixture models, for which uncertainties are tractable, and further construct a new goodness-of-fit test in which these uncertainties are profiled as nuisance parameters to account for the statistical error of the fit, promoting the conventional point-null test to a composite one. A rejection then signals genuine mismodeling rather than a test set large enough to resolve the error of the fit, provided the propagated uncertainties faithfully cover that error. We demonstrate the framework on a controlled two-dimensional toy model and on a collider physics benchmark modeling the QCD background process. With uncertainties included, the same model can be trusted on test sets up to an order of magnitude larger than the standard test allows. Our work thus provides a statistically grounded basis for a robust use of generative models in the physical sciences.

Standard reinforcement learning (RL) algorithms based on Markov Decision Processes (MDPs) are vulnerable to sim-to-real gaps. To address this issue, distributionally robust RL algorithms based on robust MDPs (RMDPs) have been proposed, for ensuring that agents can perform reliably to unpredictable conditions in real-world environments. Despite of the rich literature, \revision{an efficient model-free method is still lacking} for robust RL, particularly for Markovian data, which is more relevant in practical applications. In this work, we take a step towards settling this problem by proposing a sample efficient (asynchronous Q-learning) algorithm for robust RL. In particular, consider a $\gamma$-discounted infinite-horizon RMDP with state space $\mathcal{S}$ and action space $\mathcal{A}$, where the uncertainty set is specified by the total variation (TV) distance with uncertainty level $\sigma$. Assuming access to a single trajectory of Markovian samples induced by a behavior policy and the nominal transition kernel, the sample complexity of the proposed Q-learning algorithm --- to achieve an entrywise $\varepsilon$-approximation of the optimal Q-function --- is bounded by $$ \frac{1}{\mu_{\mathsf{min}}\varepsilon^2(1-\gamma)^3\max\{1-\gamma,\sigma\}} + \frac{t_{\mathsf{mix}}}{\mu_{\mathsf{min}}(1-\gamma)} $$ up to logarithmic factors. Here, $t_{\mathsf{mix}}$ and $\mu_{\mathsf{min}}$ denote the mixing time and the minimum state-action occupancy probability of the sample trajectory, respectively. This result is nearly minimax optimal, with an additional factor of $\frac{1}{1-\gamma}$. Moreover, it achieves a significant improvement over existing bounds.

As an experiment in agentic assisted analysis, we used an agentic AI model to analyse the effect of agentic AI on arXiv submissions; data acquisition, analysis and the first draft were carried out by the model under the author's direction. Monthly arXiv submissions have grown exponentially for three decades, at about $9\%$ per year. Fitting a piecewise log-linear trend with seasonal modulation to the full record, we find two post-pandemic breaks in growth rate. The first, in mid-2022, precedes ChatGPT and is carried by the expansion of AI research within computer science (CS); outside CS, growth from 2022 to 2025 remained at $8\%$ per year. The second, between December 2025 and February 2026, follows the release of models capable of extended autonomous work and is absent from the AI categories themselves. Since December 2025 submissions without a CS listing have run $17\%$ above trend, and in September 2026 total submissions reached a record $40\,363$. The excess is largest in mathematics, where submissions doubled within a year, and follows a gradient from formal and theoretical fields to experimental ones that is not ordered by surveyed rates of AI-tool adoption. In mathematics the increase is distributed across subfields and authors, two-thirds of it arising from a larger number of distinct authors rather than from a few prolific submitters. Across fields the clearest signature is a shift to single authorship: solo-authored papers rose by $70\%$ within a year in mathematics and condensed matter, and by $44\%$ in quantitative biology even as its total output fell, but not in experimental particle physics. The analysis is deliberately agnostic of LLM-detection methods: it uses only submission counts and metadata, with treatment timing fixed by calendar dates, and is therefore free of the outcome-triggered timing bias of author-level studies.

Estimation and inference under boundary constraints arises in many areas of statistics, but the available asymptotic theory remains limited and fragmented relative to the classical case, where the parameter lies in the interior of the parameter space. In this paper we derive a boundary-constrained analogue of the standard asymptotic expansion for a minimiser of a loss function, in which the deviation of the estimator from the population parameter is decomposed over lower-dimensional subspaces determined by the directional derivatives at the minimiser. Using this decomposition we show that the limiting distribution of the estimator is a mixture of point masses and truncated skew-normal distributions, with mixture probabilities determined by the directional derivative conditions. A consequence of this result is that the constrained and unconstrained coordinates do not separate: the presence of parameters on or near the boundary affects the sampling distribution of the unconstrained parameter estimators. We apply the expansion to obtain the asymptotic distribution of the quasi log-likelihood ratio statistic under a range of null hypotheses, and since the resulting mixture is not pivotal in the quasi-likelihood setting, we propose a simulation-based procedure for approximating critical values. The framework unifies and simplifies several existing results. We illustrate the methodology by testing for interactions in exponential family graphical models, where the parameters are subject to compatibility constraints that ensure a joint distribution exists.

For the estimation of a normal mean in $\mathbb{R}^m$ under quadratic loss, Brown (1971) proved that a generalized Bayes estimator $\delta_F$ is admissible if and only if a diffusion attached to the marginal $f^*$ of the prior $F$ is recurrent, and gave two radial tests: a sphere-average test for recurrence and a ray test for transience. He observed that both tests can be silent. We read his theory as the potential theory of the weighted Euclidean space $(\mathbb{R}^m,f^*\,dx)$ and ask what the modern theory of recurrence on weighted manifolds adds inside the zone where neither test applies. Three results. (A) If $f^*\,dx$ is doubling and supports a scale-invariant Poincar\'e inequality, the sphere test is already complete: recurrence, Brown's sphere test and the ball-volume test are equivalent, so geometric criteria can add decidable cases only on the inadmissible side. (B) They do add one. An explicit prior on $\mathbb{R}^3$ whose marginal is small along a lattice of circles at dyadic scales with spacing equal to the square root of the scale, is doubling and Poincar\'e, silences both of Brown's tests, and is transient by the ball-volume test; its generalized Bayes estimator is inadmissible. (C) Outside the doubling-Poincar\'e class the zone is two-sided: the same circles placed on one surface per scale give an admissible estimator, decided by a flux bound along a non-spherical exhaustion, and a logarithmic product prior gives an inadmissible one, decided by product heat kernels. A comparison lemma shows that any positive weight with Lipschitz logarithm is, up to constants, the marginal of a prior, so that explicit weights may be used throughout.

We prove accelerated mixing for a Metropolis-adjusted kinetic Langevin sampler: the OBABO splitting with friction $2\sqrt m$, adjusted with Neal's non-reversibly updated uniform, which is redrawn after every epoch of $E$ steps. For targets $e^{-U}$ on $\mathbb{R}^d$ with $mI\preceq\nabla^2U\preceq LI$ and an $L_H$-Lipschitz Hessian, started from a Gaussian at an approximate minimizer, the chain reaches total variation accuracy $\varepsilon$ after $\tilde O\big(\sqrt{\kappa d}+\sqrt{L_H}(d/m)^{3/4}\big)$ gradient evaluations, where $\kappa=L/m$ and $\tilde O$ hides logarithmic factors. The $\sqrt\kappa$ term is necessary, already on Gaussian targets. The proof discretizes Lu's entropic hypocoercivity by means of an entropy inequality for free transport, so that the relative entropy of the unadjusted chain decays at a rate of order $\sqrt m$ up to a discretization bias. With the non-reversibly updated uniform, the leading energy errors of velocity Verlet telescope along a path. A coupling argument combines the two. On Bayesian logistic regression posteriors for two standard data sets, which satisfy these assumptions, the gradient cost of the chain with the non-reversibly updated uniform grows roughly like the square root of the condition number at the posterior mode, and that of standard adjusted OBABO roughly like its $3/4$ power.

Random jump forests are introduced as a nonparametric tool for estimating conditional cumulative transition rates and occupation probabilities in multi-state models. The main idea is that a single tree partitions the covariate space for all transitions simultaneously, using partially observed trajectories, and with Nelson--Aalen estimators at the terminal nodes. Occupation probabilities are recovered through the product integral without requiring the Markov assumption. Uniform consistency and a functional central limit theorem for the estimators of cumulative rates can be established under the assumption that the covariate vector takes finitely many values, and under suitable splitting conditions. In particular, pointwise confidence intervals and simultaneous bands are available in the Markov case. Simulation studies show a predictive accuracy close to the correctly specified benchmark models and marked improvements when benchmarks are misspecified, with further illustration on four biomedical data sets. In the coverage study, confidence bands attain near-nominal coverage in several settings, with pooling of heterogeneous rates being the main source of possible undercoverage. The method, including prediction assessment and variable importance, is implemented in the R package JumpForests.

We consider a bivariate contamination model in which the distribution of a gold standard is known, while the mixing weight and the distribution of the contaminant component are unknown, and we test whether the two coordinates of the unknown component are independent. Classical independence tests are meaningless here, since a mixture of product distributions is not a product distribution. Following the Inversion-Best Matching principle, we invert the model for every candidate weight and compare the inverted distribution of the unknown component with the product of its marginals. The weight is estimated where they are closest in a Cramer-von Mises sense, and the scaled minimal distance is the test statistic. An exact quadratic structure makes the estimator computable without numerical optimisation and yields a transparent asymptotic theory under both hypotheses. The null distribution is calibrated by a multiplier bootstrap. Simulations and applications to gene expression data and spam filtering illustrate the method.

This paper considers model-based clustering of diffusion processes using high-frequency data. We formulate a finite mixture model for diffusion processes and develop a clustering method based on their constant diffusion coefficients. Using a quasi-likelihood, we propose estimators of the number of clusters, the diffusion coefficients and the cluster labels, and establish their consistency and the asymptotic normality of the diffusion coefficient estimators under high-frequency asymptotics with the number of observed processes fixed. Simulation studies are conducted to investigate the finite-sample performance of the proposed method.

Complete stepped-wedge designs guarantee eventual intervention delivery, but their final periods contain no concurrent controls. Under exposure-time heterogeneity this creates a design-analysis tension: a constant-effect analysis can be precise while targeting a design-dependent contrast, whereas an exposure-time indicator (ETI) analysis targets a prespecified time-averaged effect but may be much less precise. We develop an adequacy radius that quantifies the size of exposure-time heterogeneity compatible with a prespecified tolerance for estimand mismatch, and combine it with precision calculations to compare complete wedges with rollout-constrained hybrid designs that retain a wait-list arm. For a balanced wedge with a wait-list under the Hussey-Hughes model, the wait-list share and the correlation act on the constant-effect GLS estimand through a single scalar; we prove that the radius increases strictly with the wait-list share and decreases strictly with the within-cluster correlation, but remains bounded. Numerical optimisation and a model-based redesign of a weekend allied-health trial show that intermediate hybrids make estimand-aligned ETI analysis materially more affordable, giving a robustness-precision frontier rather than a binary choice between a complete wedge and a parallel trial. A second trial, of digital adherence support for tuberculosis treatment, has the same radius but no detectable heterogeneity; its test of a constant effect would detect heterogeneity large enough to halve the coverage of the nominal interval only 37% of the time. The contribution is a diagnostic and design principle: when exposure-time heterogeneity is plausible, retaining concurrent controls reduces the price of analysing the estimand one intends to report.

We study score matching over finite-dimensional approximation spaces generated by gradients of eigenfunctions of the infinitesimal generator of a forward diffusion. We compare polynomial approximation under one-dimensional Ornstein-Uhlenbeck flow with trigonometric-polynomial approximation under heat flow on the torus. In the Ornstein-Uhlenbeck setting, we show that for every non-Gaussian initial distribution uniformly comparable to the standard Gaussian, the asymptotic score-estimation variance grows exponentially along a subsequence of polynomial degrees. Averaging the score-matching loss under the diffusion semigroup, which we call operator-informed score matching (OISM), reduces the variance at each fixed degree but does not eliminate this exponential instability at sufficiently small times. A local minimax lower bound shows that the same high-degree dependence arises when estimating the polynomial projection of the score, rather than being an artifact of OISM. In contrast, under heat flow on the torus, OISM over trigonometric polynomials admits finite-sample control of the score error along the forward diffusion and yields the classical minimax rate for estimating sufficiently smooth densities under expected Kullback-Leibler loss.

Uncertainty in the drift of a stochastic differential equation (SDE) induces uncertainty in statistical observables defined as expected functionals on path space. We propose a derivative-informed method for identifying low-dimensional structure in the drift-to-observable map, with the aim of improving the tractability of uncertainty quantification. This approach can be viewed as a generalization of the active subspace method to infinite-dimensional input spaces and to path-space observables of SDEs. Representing uncertainty in the drift by random perturbations in a Hilbert space, we construct a path-space sensitivity operator using first-order Fr\'echet derivatives of the observable with respect to the perturbation. We establish conditions under which the leading eigenfunctions of this operator correspond to directions of maximum mean squared sensitivity of the observable, thus enabling low-dimensional approximation with certified error bounds. We further introduce a martingale control variate of the Fr\'echet derivative that reduces error in sample-based estimators of the operator. Using a Galerkin scheme for numerical discretization, we show that reduced representations of the drift from the derivative-informed basis yield more accurate approximations of mean first hitting times of SDEs than the Karhunen-Lo\`eve expansion of the same truncation order.

The estimation of the extreme value index (EVI) plays a central role in extreme value theory (EVT), particularly in the statistical modelling of heavy-tailed phenomena. Classical semi-parametric estimators of the EVI are known to suffer from substantial bias, especially for moderate sample sizes and for common choices of the number of upper order statistics. In recent years, several reduced-bias (RB) methodologies have been proposed in order to improve finite-sample performance while preserving asymptotic efficiency. In this paper, we present new classes of nonlinear RB EVI-estimators based on power mean-type and Lehmer mean-type functionals of upper log-excesses. Under a unified higher-order framework, asymptotic representations are derived and bias and variance properties are compared relative to other RB EVI classes introduced in the literature. An asymptotic comparison at optimal levels is conducted to identify regions of the space parameter where the proposed estimators are more efficient. A comprehensive Monte Carlo simulation study assesses the finite sample behaviour of the new estimators for several heavy tailed models. The results show that nonlinear reduced-bias estimators can provide improvements in bias and root mean squared error, although their relative performance depends on the underlying distribution and the choice of the tuning parameter. An application to hydrological data illustrates their practical use and the importance of threshold selection in EVI estimation.

In observational survival studies, delayed treatment initiation can induce immortal time bias in effect estimation when treatment groups are defined by future uptake. Despite originating from distinct predictive or causal inference paradigms, time-dependent approaches (e.g. the time-dependent Cox model, landmark supermodel, multi-state model, and g-formula) are often treated as interchangeable in applied research when addressing immortal time bias. These models effectively reclassify person-time according to time-varying exposure, yet the interpretation of their target estimands depends on assumptions about the underlying disease process and clinically relevant time scales. We study effect estimation for allogeneic hematopoietic stem cell transplantation under baseline intention-to-transplant strategies in acute myeloid leukemia (i.e. transplantation is planned for a prespecified time, subject to the patient's eligibility), within a multi-state framework allowing time-inhomogeneous Markov, semi-Markov, and state arrival extended semi-Markov dynamics. Using target trial emulation, we benchmark, on the probability scale, estimates derived from Cox-type time-dependent models and the g-formula against the corresponding theoretical values implied by the stochastic data-generating process. Decomposing the partial likelihood score, we show that single-time-scale models can be systematically biased when death intensities from different originating states are not proportional on a common time scale, as can occur when the clock is reset after intermediate events. We quantify these findings in a phase III simulation study and illustrate their implications in an application to data from three acute myeloid leukemia trials.

We study the PAC-learnability of function classes that map inputs to probability distributions over a finite space. The learner observes i.i.d. input-label pairs, with each label sampled from the conditional distribution of an unknown target hypothesis in the class. We measure learning through excess risk under a bounded proper loss that satisfies two mild assumptions. We introduce the \textbf{fat$_\gamma$Natarajan dimension}, which generalizes both fat$_\gamma$ and Natarajan dimension, and prove that its finiteness at every positive scale is necessary and sufficient for learnability. We also introduce a \textbf{fat$_\gamma$Graph} dimension extending the classical \gra dimension and establish inequalities comparing the two dimensions at different scales, leading to an equivalent characterization of learnability. Finally, we show that a distribution-valued function class is learnable if and only if each of its coordinate classes is learnable in the probabilistic concept model.

In less than five years, generative artificial intelligence (AI) has dramatically reshaped the public-facing web. Using a sample of more than 5 million crawlable webpages, we demonstrate that the share containing AI-generated text has increased precipitously, but unevenly. Our analyses reveal substantial differences in AI composition across language and domain categories (e.g., .edu, .com, or .gov). A higher fraction of English-language webpages contain AI-generated text compared to pages in other languages, and prevalence is substantially lower on .gov and .edu domains compared to .com. These distinctions persist even after accounting for compositional effects. We also find heterogeneity based on a page's position in the connectivity structure of the web. AI-generated text is most common and has increased most rapidly in pages that are the least central in the graph, which drives much of the overall growth in AI-generated text. A slower increase has taken place for more central pages, with the most central tier emerging as the least likely to contain AI-generated text. Methodologically, we use multiple AI detection tools, and generate estimates using a Bayesian hierarchical model that incorporates corrections for the detectors' statistical properties (both false positives and false negatives) and allows for different detector performance across groups, while propagating uncertainty in detector error rates into prevalence estimates. Our work suggests that AI is a prominent, but varied, contributor to public information and internet-based communication.

We study maximum likelihood estimation in the $\beta$-model for random hypergraphs, in which a group of vertices succeeds with a probability that depends on the sum of its members' parameters. The data are independent binary outcomes of experiments on groups of vertices, tested according to an arbitrary, possibly incomplete design. Suppose that every tested group has $k$ vertices and success probability in $[\epsilon,1-\epsilon]$. For complete designs in which every group is tested at least $r$ times, a unique maximum likelihood estimator (MLE) exists with high probability if $r\geq C_k\epsilon^{-1}\log n/n^{k-1}$, and for $m$ experiments on uniformly random groups, it suffices that $m\geq C_k\epsilon^{-1}n\log n$, where $C_k$ depends only on $k$. Both conditions are linear in $\epsilon^{-1}$ and, for fixed $k$, cannot be improved in general beyond the constant; they follow from finite-sample bounds for arbitrary designs. We bound the mean squared error of the MLE over a parameter box by the smallest eigenvalue of the correlation matrix of the design, and show that the rate is minimax optimal for balanced complete designs. Among $k$-uniform designs with a given number of experiments, designs in which every pair of vertices takes part together in the same number of experiments maximize this eigenvalue, and uniform random designs nearly do so with high probability. Finally, the design matrix of a uniform random design has full column rank with high probability if $m\geq\max\{n,c(n/k)\log n\}$ for fixed $c>2$ and $k=o(n/\log n)$, or $m\geq n$ when $k\geq(1+\xi)\log n$. For fixed $k$, the threshold $(n/k)\log n$ is sharp, and above it rank deficiency is asymptotically as likely as some vertex taking part in no experiment.

Bayesian mixtures, and Solomonoff induction in particular, provide an idealized model of prediction learned by minimizing population log loss, and they underlie universal agents such as AIXI. An agent that plans with an approximation of such a predictor decides where that predictor is evaluated, and a planner can be drawn toward branches on which its model is optimistic. We study which approximation guarantees survive this feedback. For normalized finite-horizon causal mixtures, the transfer coefficient factors into a policy likelihood ratio and task evidence available before each prediction, so task rarity and action-branch rarity can compound. Within one fixed universal mixture, infinitely many computable environments admit excess prediction error of order $\kappa\eta\,2^{K(\mu)}$, where $\eta$ is the training KL budget and $\kappa$ measures policy coverage. This order is sharp, and for each target a computable predictor attains it at a suitable finite horizon, so computability alone does not remove the penalty. For bounded returns, value error is controlled by the probability of entering a poorly covered branch plus a square-root term in $\eta$. Both terms are necessary, and finite maximum coverage ratios are not required. Motivated by the mentor override of Golden Handcuffs, we derive margins for preserving deferral decisions and conditional KL budgets with an unavoidable inverse-history-probability penalty, and a counterexample separates agreement of deferral decisions from task-wise event avoidance. The guarantees apply to any predictor architecture that meets explicit population approximation and coverage conditions, which gives concrete targets for implementing idealized universal agents.

Objective: Variable selection for clinical prediction models is commonly done on statistical grounds related to model fit. Statistical approaches ignore the cost of misclassifications and the harm of measuring a variable. We aim to develop a decision-analytic variable selection algorithm based on net benefit. Materials and Methods: The proposed algorithm selects the set of predictors by exhaustively comparing the net benefit of all combinations using cross validation, with an extension based on backward elimination for settings with many predictors. Variables are selected if the increased net benefit from better predictiveness exceeds the net benefit reduction from the harm of variable measurement. Harm includes financial expense, time, resource allocation, and patient burden. We illustrate the algorithm through a simulated example and three real case studies. We compare its performance to traditional backward elimination and LASSO. Results: Across the case studies, parsimonious models were generated that optimized clinical utility with minimal impact on discrimination and calibration. Penalizing predictors based on their cost ensured that expensive or burdensome variables were only selected if they offer sufficient clinical utility. The algorithm was implemented in the R package NBvarsel. Discussion: Decision-analytic variable selection allows prediction models to maximize patient benefit while minimizing unnecessary clinical burden and costs. Conclusion: Clinical utility should not be an afterthought reserved only for implementation. By integrating a harm-adjusted net benefit metric directly into the variable selection process, NBvarsel allows researchers to select only the variables that sufficiently improve clinical utility.

We present general upper and lower bounds on the optimal error exponent for the Active Simple Hypothesis Testing (ASHT) problem. ASHT has been extensively studied across diverse fields including Information Theory, Bandits, and Operations Research, a precise characterization of this optimal exponent has remained elusive. This work combines a recent Partial Differential Equation (PDE) characterization of the optimal exponent by Vijayan (2025) with the comparison principle of viscosity solutions of Hamilton-Jacobi PDEs. This leads to bounds that are parametrized by convex functions, sets and lower Hamiltonian relaxations, a flexibility that generalizes many known results in the literature.

In survival studies, right censoring often prevents the observation of subjects' complete outcomes, complicating inference about the underlying distribution. Evaluating whether a given distribution adequately fits right-censored data is therefore essential. However, guidance on the relative performance of available goodness-of-fit tests under different data conditions remains limited. In this work, we present a comparative assessment of three classical goodness-of-fit tests adapted for right-censored data: Kolmogorov-Smirnov, Anderson-Darling, and Cram\'er-von Mises. Through an extensive simulation study, we investigate how factors such as the proportion of right-censored observations, sample size, and distributional symmetry influence the statistical power of each test. All procedures were implemented using the R package GofCens, and power estimates were obtained from extensive replicated simulations. The results identify the settings in which each test exhibits superior performance, providing practical, evidence-based recommendations for test selection in survival analysis. These recommendations are summarized as user-oriented guidelines and complemented by a concise graphical cheat sheet, enabling practitioners to quickly identify the most suitable test for their specific application.

Typed decision models return a probability distribution over options supplied by the caller, and software tends to act on that number as if it were a guarantee. We measure what it licenses, comparing the open-weight Laya, in five checkpoints, with Jev 1.13, available only through an API. A probability licenses a statement about groups of decisions, not about one decision, and a gate that promises an accuracy among the cases it decides alone keeps that promise only with a lower bound checked on data like its traffic. Across 190 exchangeable evaluations, a threshold chosen by point estimate breaks the promise in 147, Learn then Test keeps it in all of them, and a Beta lower-bound rule decides more items on average than Learn then Test in 128 and fewer in none, without its guarantee. Calibrated in English, Jev keeps a promise of 95% in 8 of 51 locales. Ordinary typing noise breaks a promise calibrated on clean text, and, in exploratory runs, training with noise raises the noisy accuracy with no loss on clean text. In four further domains, a fine-tuned Laya needs from 250 to 8,000 labels to be non-inferior to Jev, and does not get there on LIAR2. Under drift, the preregistered claim for a Bayesian retraining loop is not supported: the loop avoids needless retraining where the change does not degrade the model, at a cost equivalent to recalibration alone, and re-qualifying the threshold in every window keeps the promise more often where it does. On the monthly series of M3, neither decider predicts the direction of the next value better than a constant answer. Confidence separates right from wrong answers with an AUROC from 0.80 to 0.96, but it does not select faithful explanations, and exact coalition games of single decisions show what an explanation can still say through an API that rounds to two decimals. We release the frozen answers with a datasheet, a promise check and the code.

We propose a novel approach for clustering that is later extended to classification. Our methods are based on an underlying determinantal point process (DPP) that generates sparsity and separation of location parameters in a mixture of finite mixture models specification. A key aspect is the assumption of a DPP conditional on the number of groups and a prior on this number, which results in a dynamic mixture of repulsive mixtures. The proposed model also involves a sparse tensor decomposition of the covariance function used to define the DPP, which is coupled with a separate mixture model employed to handle the intra-cluster correlation structure. Further computational enhancement in the high-dimensional case comes from exploiting a mapping from Euclidean space to the space of (valid) correlation matrices. A suitable posterior simulation scheme involving a modified telescopic sampler is developed. The resulting approach is seen to perform satisfactorily in simulated and real data examples.

Early stopping is commonly used to prevent fitting label noise from degrading generalization. Does this mean that, when a positive fraction of the training labels is intrinsically unpredictable, the best test error must occur before interpolation? Experiments on a CIFAR10 classification task with randomly labeled examples motivate the opposite possibility: although fixed-feature and shallow models exhibit late-time deterioration of the test error, the need for early stopping essentially disappears in larger, deeper convolutional networks even after the random labels have all been classified correctly. We determine when late-time deterioration disappears entirely in a solvable setting. For kernel gradient flow on a synthetic noisy classification task, we derive sharp high-dimensional learning curves and obtain a phase diagram controlled by the balance between the linear kernel component that learns the shared relation between inputs and labels and the higher-order components that enable fitting the label noise. In one regime, generalization is optimized at finite time and early stopping is necessary. In another, generalization improves until interpolation, but the endpoint is suboptimal across kernels. At the boundary, the interpolating endpoint attains the lowest generalization error across the considered kernel family and all training times. Thus, even with extensive irreducible label noise, interpolation can outperform early stopping. The neural-network experiments leave open how learned representations acquire an analogous favorable geometry.

When a product is sold out, customers who prefer it may buy a substitute or leave without purchasing. Efficient demand estimation in this setting is challenging when inventory is observed only at replenishments, because the stockout times, and hence the products available to each customer, are unobserved. Estimating demand efficiently under stockout substitution has been a challenging problem since Anupindi et al. (1998), and it has become more pressing as the scale of such problem instances grows thanks to innovations in retailing such as unmanned stores. We provide a complete solution to this problem for the multinomial logit (MNL) model, the most widely used model of customer choice. Using a continuous-time representation of the purchase process developed in Zhou et al. (2026), we show that the exact likelihood and its gradient are one-dimensional integrals, no matter how many products, customers, or stockouts there are. Both can be evaluated at a cost linear in the numbers of products and periods. Our method estimates demand for 2,000 products from 2,000 selling periods in under ten minutes.

On January 5, 2025, New York City became the first U.S. city to operate a permanent cordon-based congestion toll, charging vehicles entering Manhattan's Central Business District (the Congestion Relief Zone, or CRZ). This study estimates the short-run causal effect of the toll on hourly traffic volumes using a two-way fixed-effects difference in-differences (TWFE-DiD) strategy that compares treated CRZ street segments to untreated Manhattan segments north of 60th Street. The empirical analysis exploits hourly entry counts from the MTA's toll-meter system (4.86 million ten-minute-block records) as the primary descriptive panel and high-frequency NYC DOT Automated Traffic Recorder (ATR) data as the within-segment identification panel. The headline TWFE-DiD point estimate indicates a 40 to 45 percent reduction in log hourly volume at treated CRZ locations, with the effect direction consistent across nine pre-registered alternative specifications and concentrated in peak hours. East River bridges in the balanced panel exhibit a statistically significant 25 percent post-period increase in volume, consistent with route substitution.

We introduce a Stein variational framework for sampling from a target density $p\propto e^{-U}$ when the target potential $U$ is possibly non-smooth. The key idea is to introduce a smoother reference potential $U_0$ and the anchor weight $a=e^{U-U_0}$. The resulting anchored Stein variational gradient descent (SVGD) algorithm is able to target $p\propto e^{-U}$ without computing the gradient of $U$ and include standard SVGD as a special case. In particular, we allow the target potential $U$ to be locally Lipschitz while the reference potential $U_0$ is continuously differentiable, along with other suitable assumptions. Under this low regularity regime, we establish a Kullback-Leibler first variation formula for anchored perturbations, identify the corresponding steepest descent direction in the reproducing kernel Hilbert space and prove a nonasymptotic convergence bound for the anchored SVGD algorithm in terms of an anchored kernelized Stein discrepancy. Finally, we also provide guidance on constructing a suitable reference potential $U_0$ based on $U$.

Learning high-level causal variables from low-level observations is a central problem for artificial intelligence. With known node semantics but an unknown graph, visual readout errors can misalign model settings with environment intervention values. We address this mismatch by pairing image readouts with verified values attained through fixed round-robin interventions. With encoder and decoder frozen, a dimension-wise invertible calibration map puts visual responses and target settings in shared numerical units. After freezing this map, the agent selects interventions to resolve graph directions and fits local mechanisms to predict consequences; inverse calibration returns counterfactual states to the decoder's original coordinates. Across Domino, Pick-and-Place, Balance, Gears, CAUSAL3D, and CausalWorld, we evaluate graph discovery, intervention consequences, and counterfactual images. Controlled domino experiments show that interventions resolve orientation ambiguity left by equal additional observations. In a matched-data domino control, calibration reduces the mean error of predicted changes in other variables within the same factual context, relative to an uncalibrated model.

Spatial prediction of non-Euclidean data, such as probability distributions, symmetric positive definite matrices, and networks, arises in many modern applications. Such data are naturally modeled as elements of general metric spaces, where classical kriging methods that rely on vector space structure are no longer directly applicable. We introduce Fr\'echet kriging, a general framework for spatial prediction of metric space-valued data. Fr\'echet kriging replaces the weighted average in classical ordinary kriging with a weighted Fr\'echet mean, while the weights are determined by characterizing the spatial dependence structure through a generalized variogram. This construction allows the proposed method to preserve the intrinsic geometry of the response space without requiring an embedding into a linear space. We establish theoretical properties of the proposed predictor, including a surrogate optimality property and consistency under variogram misspecification for a fixed working variogram under suitable regularity conditions. Simulation studies for univariate distributions and symmetric positive definite matrices demonstrate favorable predictive performance compared with existing approaches. We further illustrate the practical utility of the proposed framework through applications to exceedance probabilities of daily average temperatures, joint probability densities of daily minimum and maximum temperatures, and inter-industry monetary flow networks.

Per-point data-valuation scores -- Data Shapley, influence, leverage -- are routinely used to select training data: keep the points with the top $k$ scores. We argue that top-$k$ by a fixed, mode-blind per-point score is the wrong selection rule precisely in the regime where data selection matters. The reason is that a point's value is not a fixed number but depends on what else has been selected -- a near-duplicate of an already-chosen point adds almost nothing, however high its score -- and a per-point scalar cannot express this interaction. The theory quantifies this failure exactly: in an equal latent-mode model, fixed top-$k$ selection incurs a closed-form excess risk over adaptive coverage because it repeatedly spends budget on already-covered modes. The gap is largest when the budget is tight, the learner representation is redundant, and missing a mode is costly. The practical rule is simple: use balance when the labels available at selection identify the learner's modes, and use explicit coverage when they do not. On standard classification tasks with strong pretrained features, class-balancing the same per-point scores recovers the missing coverage and matches or outperforms generic set coverage across our datasets. Without labels, label-free coverage improves accuracy by up to +0.20 while raw top-$k$ can fall below random; when labels are coarser than the modes, coverage within coarse strata performs best. We turn this distinction into a pre-specified diagnostic and validate it across both favorable and stress-test settings. The limits are part of the claim: the advantage is representation-relative. Coverage fails when the relevant modes are not expressed in the learner geometry, as with spurious subgroups, or when data quality varies independently of coverage -- exactly as the theory predicts.

We revisit the variational characterization of Nesterov's celebrated method and study when Nesterov trajectories are eventually locally minimizing in a variational sense, a notion that we define precisely. We show that this question is equivalent to the nonnegativity of an associated Schr\"odinger operator acting on variations of the trajectory. For power objectives of the form \[ f(x)=\frac{1}{p}\|x\|^p \] (where $\| \cdot\|$ denotes the Euclidean norm), we give a complete characterization: Among all exponents $p\ge 2$, the Nesterov trajectory with non-zero initial position is eventually locally minimizing if and only if $p=4$. We further interpret this exceptional exponent through the stability theory of semilinear elliptic equations.

Models transporting overall survival (OS) effects across clinical trials fail when they ignore the "post-progression treatment regime"---the subsequent therapy rates across biomarker strata and randomized arms. Because this trial-level regime is systematically omitted from published reports, standard correction families are structurally inadequate: crossover adjustments target an unrealistic counterfactual with zero subsequent therapy, erasing critical environmental differences; population adjustments (including MAIC, STC, AIPW, and density ratios) match baseline individual covariates, leaving trial-level regimes mismatched. We show that these correction approaches carry near-identical bias, proving the failure is informational rather than methodological. Simulations demonstrate that ignoring the target setting's landscape can leave the directional sign of the transported effect completely undetermined, even when mechanisms are fully known. Conversely, providing the target regime to a regime-aware estimator successfully recovers the true treatment effect. Code is available at https://github.com/felizzi/post-progression-transport-WMHS-NeurIPS-2026.

We propose a joint sparse sample weighting and parameter estimation (SWAPE) framework that jointly optimizes model parameters and sample weights to train a model while simultaneously assigning an individual weight to each training data point. Our framework accommodates several widely used loss functions for the model parameters, including convex, weakly convex, and difference-of-convex loss, and weakly convex sparsity functions to control the complexity of the sample weights. To solve the resulting bivariate problem, we propose an alternating minimization method in which the subproblems for updating the model parameters and weights are each approximated using majorize-minimize techniques. We provide several approximation methods for the loss functions considered in this paper, and show the overall algorithm's convergence. Numerical experiments on various learning tasks and data anomaly settings demonstrate the effectiveness of the proposed framework in minimizing the impact of undesirable data points compared to benchmark methods.

Designing building energy systems is a high-dimensional, mixed-integer search problem: when each design is evaluated by a year-long simulation, black-box optimizers typically need hundreds of evaluations. This paper presents AutoEnergyDesign, a framework in which a Large Language Model (LLM) agent reasons over previous simulation results, proposes configuration changes and learns in context from each outcome, while a git-based mechanism automatically reverts unsuccessful changes. The agent couples a reasoning model with an explicit memory organisation (procedural, semantic, episodic and reflective), a constrained action interface and an environment-grounded keep/revert signal. The simulator covers eight technologies (photovoltaics, micro-wind, combined heat and power, solar thermal, gas boilers, batteries, heat pumps and inverters) and computes the 25-year Net Present Cost (NPC), including capital and operating costs and salvage value. On a case study with measured building load profiles, the agent, run with Claude Haiku 5.5, Sonnet 5.5 and Opus 5.5 (ten runs of 50 simulations each), is compared with random search, Tree-structured Parzen Estimator (TPE) Bayesian optimization and a $(1+1)$ evolution strategy sharing the agent's acceptance rule (300 evaluations, ten seeds). With all three models the agent reaches an NPC of about 3.53 MEUR, 1.20% below the best TPE result and 0.23% below the best evolution-strategy result. It comes within 1% of this design after a median of 18 (Opus) to 27.5 (Haiku) simulations, whereas among the baselines only the evolution strategy does so, in half of its seeds and after a median of 192 evaluations. Opus reaches the best design in all ten runs, Sonnet in six (two runs never come within 1%), and Haiku comes within 1% in every run but reaches it exactly only once. Every decision is recorded in an auditable trace.

Neural scaling laws describe the predictable power-law dependence of model performance on training time, dataset size, and model scale, with the exponents quantifying how performance changes with resources. Recent theory has shown how scaling exponents are shaped by the data distribution and the structure of the learning task, but whether network depth can improve training-time scaling has remained unsettled. Through a refined analysis of deep linear networks, we clarify that the task spectrum determines when increasing network depth improves training-time scaling. For spectrally difficult tasks, increasing depth improves the asymptotic training-time scaling exponent, whereas for easy tasks, the exponent remains independent of depth, with depth affecting only a logarithmic correction at the boundary between these regimes. Remarkably, we find that a reparameterization of time governed by the predictor norm provides a unified description of the dynamics across task difficulties. Under this reparameterization, the loss follows a common scaling law whose exponent is independent of depth. Numerical experiments on a task derived from benchmark image data and with nonlinear networks exhibit qualitatively consistent task-dependent effects of depth on training-time scaling. Our findings advance the theoretical understanding of how task structure shapes the effects of network architecture on learning and offer a basis for studying scaling behavior across broader classes of neural networks.

There is growing evidence that the performance of a neural network depends largely on a small subset of its weights, which are typically identified by training a dense network and pruning it afterwards. Within a Bayesian framework, pruning can be directly incorporated through a spike-and-slab prior, providing a proper treatment of the uncertainties involved, both with respect to pruning and to parameter uncertainty. In Bayesian neural networks (BNNs), sampling by MCMC is notoriously hard. We demonstrate that modern Monte Carlo methods based on piecewise deterministic Markov processes (PDMPs) can be used to sample posteriors of BNNs asymptotically exact whenever numerically constructed rate bounds hold. Spike-and-slab priors can be incorporated through sticky versions of the PDMPs, setting weights exactly to zero during sampling. We examine the predictive performance and convergence of the samplers, and show that sparse BNNs sampled this way can be more robust to input noise than dense ones.

Compiling implied volatility (IV) surfaces and forecasting derivative risk from high-frequency equity and options feeds requires addressing contract heterogeneity, schema shifts, volatility clustering, and non-stationary trading activity. Using a split-adjusted NVIDIA (NVDA) equity and options panel (2018-2025), this study integrates finite-population survey sampling, tempered inverse-probability sequence learning, and causal activity clocks. Four findings emerge. First, cross-sectional surface accuracy depends on matching the sampling design to the target functional: plug-in stratified allocations using lagged cell dispersion improve equal-weighted surface reconstruction with full grid coverage, whereas liquidity-weighted probability proportional to size (PPS) sampling excels on liquidity-weighted targets by construction yet inflates error when misapplied. Second, in longitudinal LSTM sequence training, PPS and Neyman designs retain more tail-volatility windows than simple random sampling (SRS); all three designs are equivalent to full-census training within chosen margins while cutting training time, though incremental forecasting gains of informative selection over SRS are not established. Third, out-of-sample evaluation shows a predictability contrast: short-horizon prices are not forecastable beyond a random walk with drift among tested models, whereas forward IV shows descriptive mean-reversion skill concentrated in a few post-shock reversal sessions (insignificant under session-clustered tests), with sequence models not beating a linear HAR-IV benchmark and feature pruning driving stability. Fourth, across a 6.3x start-to-end (nearly twentyfold trough-to-peak) price expansion, rolling volume and dollar clocks stabilize bar rates and compress intraday return kurtosis beyond diurnal scaling.

We study an accelerated Langevin equation in which the classical overdamped Langevin drift is supplemented by a nonlocal gradient velocity determined through a target-weighted Poisson equation. The target $p$ is only assumed to be strictly positive and need not be normalized. Writing the dynamics in terms of $\omega=\log(p/q)$ yields a coupled parabolic-elliptic system whose parabolic component contains the usual Langevin diffusion, whereas the elliptic component generates a global transport. The system is invariant under multiplication of $p$ by a positive constant. On a bounded smooth domain with homogeneous Neumann boundary conditions, we establish global existence and uniqueness of classical solutions for strictly positive $C^{2+\alpha}$ target weights and compatible $C^{2+\alpha}$ initial data. For long-time behavior, the oscillation of $\omega$ decays at the explicit rate $e^{-t}$ and the solution density exponentially converges to target density. Moreover, an exact entropy-dissipation identity holds relative to the target density, in which the relative Fisher information and the Jeffreys divergence give the two dissipation mechanisms associated with the Langevin and weighted-Poisson components. In particular, KL divergence also converges to zero with rate $e^{-t}$ without a convexity assumption on the target.

Despite the proliferation of sovereign Environmental, Social, and Governance (ESG) metrics, traditional linear macro-financial models systematically misprice Foreign Direct Investment (FDI) tail risks during systemic shocks. This paper demonstrates that sovereign risk transmission is fundamentally non-linear. Using a forward-chained global panel of 147 sovereign entities (2010-2024), we deploy a gradient boosting architecture that compresses forecast errors by over 39% against L1-regularized baselines precisely during extreme macroeconomic tail events. Moving beyond opaque predictive accuracy, we apply dual interpretability frameworks to mathematically map the discrete threshold effects of institutional quality. We identify a strict, non-linear "Governance Premium": incremental environmental and social improvements yield zero capital advantage until a foundational regulatory threshold is breached. Finally, we establish regime-conditional 95% uncertainty bounds that maintain valid marginal coverage even under severe ESG reporting failures. This framework equips policymakers with a distribution-free tool to quantify sovereign tail-risk, proving that global capital markets price institutional resilience as a discrete boundary rather than a continuous premium.

The tail dependence function $\Lambda$ of a bivariate copula, and the tail dependence order based on it, cannot distinguish between copulas whose tail dependence coefficient vanishes; for all of them $\Lambda\equiv 0$. Such copulas, among them the Gaussian copulas with correlation $\rho<1$ and the lower tails of all extreme-value copulas other than the comonotonicity copula, are nevertheless ordered by the much finer information contained in their tail expansion $C(u w)\sim u^{\kappa}\ell(u)\,b(w)$, where $\kappa$ is the tail order, $\ell$ is slowly varying and $b$ is the tail order function of Hua and Joe. We introduce an asymptotic tail order $\preceq$ that refines the tail dependence order and show that it is lexicographic in the three quantities $\kappa$, $\ell$ and $b$. As a consequence, every real-valued measure that is monotone with respect to $\preceq$ is constant on all but countably many tail-order classes; meaningful measures of intermediate tail dependence must therefore be defined level-wise, and we construct such measures. Finally we determine where the fine structure resides: if the joint tail is ray-regular on the exponential scale, then $b$ is determined by the tail order and the one-sided derivatives at $\omega=1/2$ of the angular dependence function of Wadsworth and Tawn; it is a (possibly broken) product of powers. If the joint tail is log-concave on the exponential scale, $b$ is squeezed between two such products.

Prioritized time-to-event outcomes arise frequently in clinical studies, where treatment comparisons may be based on a prespecified hierarchy of clinically important outcomes. The win ratio provides a natural summary of treatment benefit for such prioritized outcomes. However, methods for characterizing heterogeneous treatment effects by estimating how the win ratio varies with baseline covariates remain limited in this setting. We develop a flexible local inverse probability of censoring weighted estimator of the win ratio as a function of baseline covariates, without imposing the parametric shape or proportional win-fractions structure assumed by existing regression approaches. We use best linear projections to summarize the estimated log win ratio and develop formal tests of log-linearity of the win ratio function. We also propose a complementary model-based estimator of the covariate-specific win ratio assuming proportional hazards models for the marginal distributions, copula dependence between the margins, and Bernstein-polynomial sieves for the baseline hazards. Large-sample properties and bootstrap validity are established for both approaches. Simulations show that the local estimator is robust, while the model-based procedure can provide substantial efficiency and power gains under correct specification, and proportional win-fractions regression may lead to bias when its structural assumptions fail. The methods are illustrated by analyzing data from CALGB 9633, a randomized lung cancer trial.

Statistical graphical models are representations of the dependences between the components of a random vector. They encode these dependences via graphical separation criteria. We consider the algorithmic problem of checking separation statements for the various graph classes and separation criteria that have been developed by the community. We establish that all criteria we study can be checked by modular variations of reachability. For all separation criteria but $\sigma$-separation, this yields a linear-time algorithm. This is effectively the optimal attainable time since loading the graph itself is linear-time in the number of edges in the graph. Using the CIfly framework for deploying modular variations of reachability, we also provide R and Python code implementing our algorithms. We evaluate empirical runtime and scaling behavior in a simulation study.

We study distributed sparse linear regression with missing covariates and Byzantine workers. Correcting for missing observations can make the regression loss nonconvex, while Byzantine reports can distort gradient updates and impair estimation. A communication-efficient method is proposed that combines robust gradient aggregation with folded-concave regularization. Under appropriate regularity conditions, after $t$ communication rounds the estimation error is bounded by a transient component determined by the initial error and decaying geometrically with $t$, plus a residual component reflecting finite-sample variability and Byzantine contamination. Simulations and two real-data examples evaluate estimation accuracy, variable selection, and communication costs.

Multi-objective reinforcement learning is a fundamental problem for modern AI, with applications spanning post-training, reasoning models, and robotic control. A central challenge is to learn policies that balance competing objectives, such as helpfulness versus honesty in safety and alignment, or speed versus energy in robotics. The optimal policy can vary sharply with objective-weighting combinations, and the extent of this variation depends crucially on the degree of conflict among objectives, making per-preference policy learning sample-inefficient. In this paper, we introduce objective-conflict coefficients and derive near-optimal regret bounds for online settings using this measure, thereby quantifying the impact of misalignment on learning efficiency. Then, we design routed ensemble policy learning, an algorithm that maintains a small set of basis policies and dynamically routes trajectories to the policy with the highest potential reward. Finally, we consider trajectory selection as another approach to mitigate objective conflict, and design a kernel surrogate trajectory attribution algorithm that estimates how feedback signals from a set of trajectories influence policy learning on other trajectories. The kernel can capture nonlinear interactions such as synergy and antagonism between trajectories. We evaluate our approach across diverse environments and benchmarks: routed ensembles improve average performance by 4.3% over baselines for online environments from Meta-World and MO-Gymnasium, and kernel surrogate attribution improves correlation with leave-one-out analysis over existing baselines by 50%. When applied to trajectory selection, kernel-based attribution yields a 27% improvement in reward over baseline attribution methods.

We propose a test of whether two interventions induce the same outcome distribution when outcomes are structured and high-dimensional, such as images, text and graphs. We represent the interventional distributions as mean embeddings on learned deep features, such that the treatment effect is a doubly robust representation of the difference in mean embeddings. Our test, DR-KTE-Deep, optimises the learned features so as to maximise an estimated power criterion on this treatment effect representation. To motivate our learning objective, we show that the power criterion characterises the asymptotic signal-to-noise ratio of the test under fixed alternatives, and we derive a uniform finite-sample bound for its regularised estimate. We show that, with the learned kernel fixed, a held-out test is asymptotically valid under the null given suitable nuisance-estimation rates. Empirically, DR-KTE-Deep achieves substantially higher power than fixed and bandwidth-optimised kernel baselines across image, text and graph outcomes, while maintaining well-calibrated Type-I error.

Prediction is one of the most important research topics in statistics and machine learning. Moreover, in the domain of streaming data analysis, real-time prediction has become a rapidly growing research field in recent years, owing to the abundance of streaming data. In many fields of scientific research, such as biomedical research, weather forecasting, and traffic management, improving real-time prediction of streaming data is crucial in general. However, the complicated features of streaming data, such as large-scale and high-velocity, impose both theoretical and computational challenges to the development and evaluation of real-time prediction in this field, especially when the investigator has multiple models at hand. To overcome these challenges, we propose a novel Renewable Model Averaging method through Quadratic approximation (ReMAQ) for generalized linear model (GLM) for streaming data. By approximating the model averaging procedure for GLMs (which is typically a convex optimization problem) with a quadratic optimization formulation, ReMAQ allows us to update the model averaging prediction sequentially when a new batch of data arrives, without accessing the entire historical data. Moreover, ReMAQ is also computationally efficient as one only needs to consider a quadratic programming problem in each steps of updating. Under certain regular conditions, we also show that the proposed ReMAQ weight is asymptotically equivalent to that by the oracle model averaging based on the inaccessible full data, the asymptotic optimality of ReMAQ is then established accordingly. The proposed method is also assessed by simulation studies. We apply our proposed ReMAQ in the study of the Airline On-Time Performance Dataset and the results demonstrate the superiority our method in practical situations.

Estimating the conditional distribution of multivariate responses with continuous and binary components requires capturing their dependence, respecting their different supports, and accommodating potentially heavy-tailed continuous outcomes. We propose mixed engression, which fits a continuous generator using energy losses induced by bounded discrepancies and then thresholds its soft binary outputs to produce mixed-response samples. Because the discrepancies are bounded, the population objective remains well defined without response-moment assumptions. However, we show that the full-response energy distance for the complete response vector can tend to zero before thresholding yet remain bounded away from zero afterward as response dimensions grow. A separate energy loss on the entire binary vector captures discrepancies obscured by the full-response comparison and controls the effect of thresholding. The generator is therefore fitted using an augmented energy loss that combines the full-response and binary components. Under compositional smoothness of the structural functions and a uniform threshold-margin condition, we derive nonasymptotic bounds on the final sampler's full-response error for sparse ReLU estimators of models driven by Gaussian or Student-t structural noise, yielding polynomial convergence rates up to logarithmic factors. We also establish consistency when the covariate, response, and noise dimensions grow under suitable conditions. Simulations show improved accuracy relative to the comparison methods under the full-response energy distance, including in heavy-tailed settings. Moderate binary augmentation substantially improves binary-distribution accuracy while largely preserving full-response accuracy.

We present LatentLieODE, a novel neural ODE method to discover linear Lie symmetries and first integrals of the learned dynamics from timeseries observations such as video inputs. It starts with an autoencoder, which learns a low dimensional representation of the observed system and then maps trajectories to decorrelated latent coordinates $z \in \mathbb{R}^d$. A bank of linear infinitesimal generators $\mathbf{A}_k \in \mathbb{R}^{d \times d}$ is fitted jointly against a differentiable Lie-bracket residual $\mathbf{J}_f(z)\mathbf{A}z - \mathbf{A} f(z)$, while an MLP optimises the latent ODE $f(z) = \mathrm{d}z/\mathrm{d}t$ to reconstruct the observation-space timeseries. After suitable generators and $f$ are found, we fit a third network $H_\phi$ to generate the first integral of the system. Across six benchmark systems (harmonic oscillator, simple and twin pendulums, Lotka-Volterra, pixel pendulum, sphere geodesics) we find each system's relevant symmetries, and the discovered invariants match ground-truth conserved quantities at $|\rho| \geq 0.978$. On the sphere, the pipeline recovers a non-abelian $\mathfrak{so}(3)$, certified by the negative-definite Killing form of its recovered structure constants. This work presents significant advances in data-driven discovery of symmetries and provides a new pathway for neural surrogates of dynamical systems.

Accelerated dynamics methods, such as parallel replica dynamics, require estimates of the decorrelation time, after which a process trapped in one of its metastable states is distributed according to the quasi-stationary distribution (QSD). In a method proposed by Binder, Leli\`evre and Simpson, decorrelation times are defined by coupling a particle approximation of the non-linear conditioned probability flow of the process with MCMC-inspired diagnostics. We explore an alternative to MCMC diagnostics, based on coarse-graining of non-linear probability flows and simulation-based inference. We first develop a coarse-graining framework for killed Markov jump processes conditioned on survival, proving uniform-in-time, non-asymptotic estimates approximating the non-linear probability flow of the true process by the non-linear flow of a surrogate process, defined with a collective variable. The framework does not assume reversibility of the true process, and also gives bounds which scale explicitly in an adiabatic regime. Motivated by these results, we then construct a quasi-stationarity diagnostic using simulation-based inference: a recurrent neural network (RNN), trained on particle approximations of non-linear effective flows, estimates posterior probabilities of decorrelation. The learned diagnostic is then transferred to collective-variable traces of high-dimensional particle approximations. We investigate the behavior and robustness of this diagnostic on a set of numerical experiments, comparing it to the MCMC diagnostic of Binder et al.\ on both synthetic examples and a molecular system, in and out of its training prior distribution. When its prior assumptions hold, the RNN diagnostic outperforms the MCMC diagnostic in ROC analysis, is less conservative, fails less often, and has an interpretable decision threshold.

Quickest detection problems (QDPs) arise in a wide range of applications. However, the complementarity structure of the associated variational inequalities poses substantial challenges for traditional numerical methods, particularly in high-dimensional settings. We propose a PINN-based framework for estimating the optimal stopping boundary in QDPs with high-dimensional diffusion processes and spatial dependence, while overcoming the curse of dimensionality. Theoretically, we establish a novel equivalence between the free-boundary PDE and an infinite-dimensional stochastic optimization problem, rigorously justifying the use of finite-dimensional neural networks to approximate the true solution. Furthermore, we assess the robustness of the proposed framework by quantifying uncertainty in the learned stopping boundaries using deep ensembles and Monte Carlo dropout. The results show low variability across independent training runs and stochastic forward passes, indicating stable and consistent boundary estimates. We also compare our PINN results with reference numerical solutions where available and find close agreement.

We study minimax rates for conditional generative learning under integrated Wasserstein loss, with H\"older continuous conditional laws and an unrestricted covariate distribution. We evaluate unrestricted generators through their conditional output distributions. For expected Wasserstein loss, the minimax rate agrees with the rate under uniform covariates and follows from balancing localization bias against unconditional distribution estimation error. Higher moments of the loss can instead be dominated by unobserved covariate regions, which force an additional risk contribution of order $n^{-1}$, where $n$ is the sample size. An adaptive localization argument gives matching upper bounds. The results cover unrestricted response laws and possibly vanishing Besov densities for $W_1$, and extend to higher Wasserstein orders under uniform response density bounds and integrability conditions. A multiscale lower bound closes the logarithmic gap in unconditional $W_1$ minimax rates in dimension two for both unrestricted distributions and Besov density classes. We also establish minimax rates for conditional Brenier maps under density, integrability, and uniform curvature conditions.

In a membership inference attack, an attacker is given the data of a particular individual, and tries to determine if that individual was included in a particular dataset, given some kind of statistics derived from the dataset or a model trained on the dataset. Roughly, a membership inference attack is robust if it succeeds even when the statistics are highly noisy, regardless of how that noise is introduced. We give a simple presentation of robust membership inference attacks for mean estimation, covering both multivariate Gaussian distributions and binary product distributions. We also give a summary of the literature on robust membership inference, covering history, extensions, and connections to other research areas.

The CODIACS Vanguard randomized trial compared a depression care intervention with usual care for patients with persistent depressive symptoms following acute coronary syndrome. We assess whether its findings can be extended to broader patient populations represented by two cohorts constructed from the All of Us electronic health records. This extension is challenging when treatment effects vary across covariates whose distributions differ between trial and target populations. We develop Bayesian Propensity Score Predictive Inference (PSPI), an interaction-based outcome modeling framework that combines Bayesian additive regression trees for potentially high-dimensional covariates with natural cubic splines of the sampling propensity score. Two variants include a separate spline in the treatment effect component alone or in both the prognostic and treatment effect components. The spline components allow separate regularization of selection-related variation and thus improve potential outcome estimation within the observed range and allow an explicit extrapolation rule beyond their boundary knots under substantial covariate shift. An inverse Gumbel transformation of the sampling propensity score balances compression near zero and the extent of extrapolation. Simulation studies show that spline-based PSPI models outperform competing methods in the settings studied, achieving lower root mean squared errors and near-nominal coverage, particularly under substantial covariate shift. In the application, PSPI estimates larger intervention benefits in both target populations than the unadjusted trial estimate.

We propose a causal machine learning framework to integrate dynamic tolling for mobility, equity, and climate objectives, resolving a long-standing policy trilemma. Using Double Machine Learning, we estimate zone-type-specific treatment effects, showing that commercial zones yield nearly twice the delay reduction of residential zones. Our adaptive quota mechanism expands residential coverage 34-fold while preserving 93% of total delay savings and sustaining high return on investment. The framework delivers climate co-benefits, reducing emissions by 743.5 metric tons of CO2 (valued at 141 USD) via congestion-sensitive pricing under moderate traffic conditions with sensitivity bounds from 557.6 to 929.3 tons. Crucially, we uncover a policy-invariant toll range where performance remains robust across demand scenarios, simplifying implementation. This study reveals critical, previously unidentified insights, such as a policy-invariant toll range, that can significantly simplify real-world implementation. Validated on synthetic data mirroring real-world networks, our methodological advances lay the foundation for future public deployment.

Digital health interventions can provide scalable, personalized support for physical activity (PA), yet sustaining PA over the long-term remains difficult. Because affective experiences with PA are important for activity sustainment, we developed an online reinforcement learning (RL) system for the forthcoming ADAPT factorial trial that personalizes twice-daily activity suggestions aimed at improving affective associations with PA. This setting poses several challenges: participant-specific data are scarce relative to the candidate state space, important states and rewards are partially observed, interventions are frequent while the primary outcome is measured weekly, and activity suggestions can both improve PA-related experiences and increase intervention burden. We address these challenges using an expert-specified causal directed acyclic graph as an integrated RL design tool, guiding state construction, estimation of partially observed states, delayed credit assignment, and representation of beneficial and adverse intervention pathways. We further construct a longitudinal simulator calibrated from a prior micro-randomized trial to compare candidate RL designs under realistic state evolution, missingness, and participant heterogeneity before deployment. Together, this work illustrates how causal domain knowledge and prior trial data can make long-horizon, participant-specific RL more practical for digital health interventions targeting sustained behavior change.

Gossiping is a complex social phenomenon that unfolds through multiple types of relations. Among other things, individuals may initiate gossiping about others as a result of believing themselves to be the target of gossip. These two processes, i.e., gossiping and gossip perception, form a dynamic multiplex network, where self-reported and perceived gossip ties coexist and can influence each other over time. We develop a multiplex relational event model to study the joint evolution of these two processes. The proposed model includes endogeneous covariates that capture the influence of events occurring within one layer, as well as events occurring on the other layer, on the rate of self-reported and perceived gossiping. We analyze survey data collected annually from 44 secondary schools in Hungary. We show how a multiplex relational event model, including linear, smooth and random effects, is able to describe the social mechanisms operating within and between self-reported and perceived gossiping and to capture how these two layers influence each other over time.

Variational inference with normalizing flows provides a flexible posterior approximation for Bayesian model calibration, but struggles when the posterior has several modes separated by regions of low probability mass. We study this behavior in a two-parameter twin experiment and regularize the variational objective. Under Gaussian observation errors and a Gaussian prior, the corresponding minimization objective is the expected Tikhonov calibration objective minus the differential entropy of the approximating distribution. However, the expected Tikhonov term favors concentration near one of the modes, while entropy maximization does not guarantee multimodality. We introduce a regularization term based on R\'enyi divergence that encourages the approximate posterior to cover the prior support. In the connected bimodal case, a flow approximation represents both modes; in the case with well-separated modes, the flow and Gaussian mixture frequently miss one mode. The introduced regularization reduces the near-collapse fraction from 50% to 25%. While the regularization improves mode coverage in this experiment, it changes the Bayesian variational objective, introduces a trade-off in approximation accuracy, and does not provide a general solution to mode collapse.

Mamba-2 compresses all context into a fixed-size recurrent state that dominates decoding memory and bandwidth. Existing compressors quantize state values or prune state coordinates, scoring each channel independently in the native basis. We examine trained Mamba-2 states and find that energy is almost rank-one, but the output does not read the energetic directions: energy concentrates in a bias-like direction accumulated by the recurrence, while the content the output reads lives in a low-dimensional subspace that is not aligned with the state's coordinates. Even output-aware scores are computed coordinate by coordinate, so they miss the directions that determine the input--output map. Compression should therefore target the write--read subspace rather than individual coordinates. We propose \method{}, a post-training transform that instantiates balanced truncation with full controllability and observability Gramians and applies the rotation exactly via Mamba-2's scalar-identity transition. Because the useful rank varies widely across layers, we further introduce HARP, which allocates ranks per layer from the Hankel spectrum and protects retrieval-sensitive layers, improving both perplexity and retrieval over uniform budgets. Calibration takes a single forward pass, an order of magnitude faster than GHOST's released protocol, with lower peak memory. With HARP, \method{} halves the state dimension at an average WikiText-2 perplexity increase of $0.3$ points ($1.8\%$) across 130M--2.7B, versus $1.6$ ($9.3\%$) for GHOST, and the gap widens at 25\% retention.

During joint training, backpropagation allows later layers to reshape earlier representations, a feedback mechanism absent from purely layer-wise learning. We extend Neural Low-Degree Filtering (Neural LoFi), a recently proposed layer-wise spectral surrogate for feature learning, to capture this interaction. Starting from the exact backpropagation gradients, we derive a local coupling between a neuron's incoming weights and its corresponding next-layer column. In a controlled small-initialization regime with smooth activations and fixed surroundings, these dynamics amplify singular modes of a cross-layer correlation operator. Each growing mode pairs an earlier feature with an upper-layer response through shared neuron coefficients. This motivates a backward-coupling algorithm in which upper-layer sensitivities generate sample-dependent effective labels for earlier spectral steps, while corrected features refine upward transmission. A function-space variational principle characterizes the selected feature-response pairs by maximizing the label correlation of their product under kernel-norm constraints. For three-layer networks, we derive self-consistent continuous-time dynamics for the filtered hidden-layer features, the activation kernels, and the network output under the proposed backward coupling. In a hierarchical multi-index model, we establish a feature-recovery separation: backward coupling recovers lower-layer directions invisible to the original forward label-weighted statistic. The linear specialization recovers residual-dependent singular-mode dynamics, while paired modes provide a dynamical mechanism for linked low-rank covariance components consistent with the rainbow-network picture. Experiments with fully connected and convolutional networks on binary CIFAR-10 show improved accuracy over forward Neural LoFi at matched widths.

Isotropic Gaussian processes on networks have a covariance that is a function of a distance between locations, but valid constructions have so far required graphs with Euclidean edges and the resistance metric. We introduce a one-parameter family of metrics $d_\kappa$, $\kappa\ge0$, on arbitrary compact metric graphs, including those with loops and parallel edges, defined as the variograms of first-order Whittle--Mat\'ern fields, where $\kappa=0$ corresponds to an intrinsic limiting field. We show that these Whittle--Mat\'ern metrics are invariant under insertion and removal of vertices of degree two, that they decrease monotonically in $\kappa$, and that they can be evaluated by sparse linear solves. The metric $d_0$ coincides with the resistance metric on graphs with Euclidean edges, and thus extends it to general compact metric graphs without any modification of the graph. For every completely monotone function $r$ and every $\kappa\ge0$, we show that $r(d_\kappa)$ is a positive semidefinite function. This yields isotropic Gaussian processes on general compact metric graphs. For $\kappa>0$ the metric levels off beyond the screening length $\kappa^{-1}$, which can be used to induce long-range dependence in short-range covariance models, and $\kappa$ can be estimated from data together with the parameters of $r$. The models are compared on daily traffic maps from the street network of Paris and the motorway network around London. For both, $\kappa$ is estimated to be positive, $d_\kappa$ increases the maximised log-likelihood relative to $d_0$ for every covariance function, and it gives the best predictions at randomly held-out locations.

Hybrid forecasting systems combining machine learning and numerical weather prediction offer a promising route toward improved forecast skill while preserving physical consistency. In this study, we investigate online learning of state-dependent model-error corrections within the Integrated Forecasting System using 4D-Var. Building on previous work, we represent model error using a neural network embedded within the forecast model and trained from observations through the data assimilation cycle. First, we introduce a targeted variant of neural-network 4D-Var in which only the neural-network parameters are estimated, while the atmospheric state is provided by a precomputed analysis. We then propose a batched extension that processes assimilation windows in random order and averages parameter increments over multiple windows, drawing inspiration from mini-batch stochastic optimization. Experiments are conducted with the Integrated Forecasting System of the European Centre for Medium-Range Weather Forecasts at an intermediate resolution. Sequential online learning substantially improves forecasts within the training period, with error reductions exceeding 10%in some cases. However, these improvements fail to generalize to unseen data. In contrast, the proposed batched formulation produces hybrid models that consistently improve forecast skill in out-of-sample experiments. Larger batch sizes accelerate convergence and improve robustness by reducing the noise of parameter updates. Diagnostics further show that the online-trained corrections preserve realistic forecast variability and do not improve skill through artificial smoothing. These results demonstrate that the formulation of the online learning problem is critical for generalisation and establish batched online training as a promising framework for hybrid machine-learning/data-assimilation systems.

Source selection often favors similar data, yet marginal proximity need not identify the most useful pretraining source. We motivate a conditional alternative through domain-adaptation theory and a regression example in which broader source variation improves estimation despite increasing Wasserstein distance. We examine five-day cryptocurrency volatility using ten Binance spot assets, six forecaster families, three seeds, and shared temporal cutoffs. Selecting the three farthest rather than nearest Gaussian-Wasserstein sources improves mean persistence-relative skill by 12.52 percentage points for MLP and 20.90 for Transformer, with positive post-hoc paired date-block intervals. Forward pilot selection improves mean skill over random transfer in all six families. Compact recurrent models achieve the lowest absolute mean errors, separating selection utility from forecasting quality. Retrospective five-candidate screening shows additional gains from farthest-first geometry before pilot ranking for MLP and Transformer. We distinguish across-target associations from within-target decisions and evaluate target heterogeneity, training seeds, and temporal-block sensitivity. Findings are exploratory for a fixed historical panel; the screening analysis measures no runtime saving.

Large language models are trained to predict tokens from a finite vocabulary, each represented by a vector in a $d$-dimensional embedding space. Therefore, token prediction is a classification problem whose classes have a geometry determined by the embedding. We study how this geometry affects learning in a minimal solvable model: a linear predictor that maps a sequence of $N$ embedded tokens to a $d$-dimensional output through one $d\times d$ matrix per position, trained in the teacher--student setting. We call this the \textit{token perceptron}. We compute its generalization error with the replica method, taking $N\to\infty$ at fixed number of examples per position $\alpha=P/N$ and keeping $d$ and the vocabulary size $D$ finite. As a first reference case, we consider continuous embeddings on the sphere, which isolate the role of the embedding dimension. The natural load is then $\alpha/d$, and for $d>1$ a normalized label fixes the direction of the teacher output, so the problem becomes a noiseless regression: the teacher is recovered exactly above $\alpha_{\rm rec}=d^2/(d-1)$, and spherical cross-entropy reaches this threshold at large inverse temperature. In the same setting the storage capacity of random associations scales as $d^2$ for margins $O(1/\sqrt{d})$. We then turn to vocabulary embeddings, where a label only identifies the Voronoi cell of the teacher output, so that each example imposes inequalities rather than equations, as in the classical perceptron. We compute the Bayes-optimal error for a given embedding matrix through an approximate spectral closure and show that the optimal learner first resolves the directions in which the vocabulary has the largest variance, even when the teacher is isotropic.

Two compass bearings near north can have an ordinary arithmetic mean pointing south: the difficulty is that angles live on a circle, not a straight line. Using real wind observations, diffusion-MRI orientations, and geometric examples from machine learning, this article introduces circular and axial averages, directional variability, and probability models on circles and spheres. The guiding principle is to identify the geometry and symmetries of the observations before choosing a statistical method.