Papers

We establish general upper bounds on the maximum effective dimension and information gain of kernel Gram matrices over arbitrary designs in terms of approximation properties of the kernel or, equivalently, of the corresponding reproducing kernel Hilbert space. We show that across three regularity regimes, covering as special cases the Matérn and squared exponential kernels, these bounds cannot be improved beyond constant factors.

Learning operators from sequentially collected data arises in adaptive experimental design, Bayesian optimization, and dynamical-system modelling, where observations may be dependent, and future inputs or sensing operators may depend on preceding data. We derive time-uniform self-normalized concentration bounds for stochastic processes in Hilbert spaces with vector-valued noise. We use these bounds to obtain regression-error guarantees for linear operators, including targets outside the Hilbert estimation space, and for nonlinear parametric operators trained with strongly convex losses and regularizers. Our results allow possibly infinite-dimensional inputs and outputs without independence or mixing assumptions, providing a major step towards convergence guarantees for adaptive operator learning and learning from stochastic dynamical data.

Envelope methods improve estimation efficiency in multivariate analysis by isolating low-dimensional structures that contain all the information material to the parameter of interest. In multivariate linear regression with random predictors, predictor envelope models achieve this goal by removing variation in the predictors that is immaterial to the regression. In many applications, observations are naturally divided into several groups, such as treatment groups, regions, or demographic strata, and the regression relationship may differ between groups. Motivated by this setting, we propose a groupwise predictor envelope model for multivariate linear regression. The proposed model assumes that the group-specific regression coefficient matrices are represented through a common predictor envelope subspace while allowing group-specific regression effects and group-specific error covariance matrices. We derive an objective function for estimating the common predictor envelope, obtain the corresponding regression estimators, and establish asymptotic normality together with an explicit asymptotic variance formula. Moreover, we show that the proposed estimator is asymptotically more efficient than the estimator obtained by fitting predictor envelope models separately to each group. This theoretical advantage over the existing work is also demonstrated through simulation studies.

Cox regression with Lasso penalization is widely used for variable selection in time-to-event data, but reliable coefficient inference after selection remains difficult. We investigate bootstrap inference for the debiased Cox estimator after Cox Lasso selection. Two score-based procedures are considered: a wild bootstrap using independent multipliers and an Efron bootstrap using centered multinomial resampling weights. Both procedures keep the original Cox Lasso fit and debiasing matrix fixed and apply bootstrap weights to the score contributions that determine the first-order distribution of the debiased estimator. We establish asymptotic validity for both bootstrap schemes and study their finite-sample behavior in extensive simulations. The bootstrap intervals improve coverage accuracy over first-order debiased Wald intervals in many small- and moderate-sample settings, with the size of the improvement depending on the simulation setting and tuning rule. A SUPPORT2 application illustrates the procedures in practice.

Mixture-of-Experts (MoE) models provide a flexible framework for partitioning complex prediction problems into simpler local learning tasks through an input-dependent gating mechanism. Existing interpretable MoE approaches, such as Mixture of Decision Trees (MoDT), achieve transparency by employing homogeneous decision-tree experts, but this restricts the model to a single inductive bias across all regions of the feature space. We extend the MoDT framework by introducing heterogeneous expert families comprising decision trees, linear support vector machines, and quadratic discriminant analysis under a common probabilistic gating mechanism. To ensure coherent likelihood-based inference, non-probabilistic experts are calibrated to produce conditional class probabilities, allowing parameter estimation within the generalized Expectation-Maximization framework of MoDT. We further establish theoretical monotone ascent guarantees for the proposed heterogeneous gating updates, providing a justification for the optimization procedure. Experiments on a diverse collection of synthetic and real-world benchmark datasets demonstrate that the proposed framework adaptively specializes experts according to local data geometry, yielding interpretable expert assignments while achieving predictive performance competitive with homogeneous MoDT and Random Forests. The proposed approach combines interpretability, adaptive inductive bias selection, and probabilistic coherence within a unified mixture-of-experts framework.

Many time series are not stable over their full observation horizon, but may contain scientifically meaningful periods during which a signal remains stable up to a prescribed tolerance. We formulate this as an existence test for stable intervals in a non-stationary regression model with dependent, locally stationary errors. For a signal $d$ derived from the mean function, including deviations from reference levels and derivatives, stability over duration $\delta$ is encoded by $d_\infty=\inf_{t\in[0,1-\delta]}\sup_{s\in[t,t+\delta]}|d(s)|$. The hypothesis $d_\infty\ge\Delta$ states that no interval of length $\delta$ remains within the tolerance $\Delta$, while rejection provides evidence for the existence of a relevant stable period. We estimate $d$ by local linear regression and construct plug-in tests for $d_\infty$. The asymptotic distribution is determined only by near-extremal windows at which the minimax functional is attained. We formalize this localization through extremal sets and derive Gaussian and extreme value approximations for kernel estimators over possibly shrinking index sets with time-varying long-run variance. The resulting tests are consistent and have asymptotic level control. The theory also extends supremum-based relevant-change tests to time-varying long-run variance and derivative-based hypotheses. Simulations and applications to physiological and engineering time series illustrate the method.

In reinforcement learning policy evaluation, classic on-policy methods often suffer from high variance when estimating policy performance. To mitigate this issue, behavior policy search has been proposed to learn data-collecting policies tailored to reduce online evaluation variance. However, these approaches do not account for uncertainties in the transition functions. In practice, simulator transitions often differ from the real world due to modeling errors or approximation limitations. As a result, behavior policies trained in simulation may still yield high variance when deployed in real environments, leading to costly reliance on real-world evaluation samples. In this work, we propose a double-loop gradient-based algorithm for learning behavior policies that are both efficient and robust to transition uncertainty. Theoretically, we derive novel transition-variance gradient expressions and establish global convergence guarantees for the algorithm. Numerically, we demonstrate that our method is less sensitive to transition perturbations than existing approaches, providing supportive evidence for its practical utility.

Numerous methods have been developed to quantify feature attributions in individual predictions for tree ensembles. However, many applications require global measures of feature contributions to overall model performance. Although local attribution scores can be aggregated to characterize feature importance, such summaries do not directly decompose measures of predictive performance, such as $R^2$. This article introduces qshap, available in both R and Python, which provides Shapley decomposition of $R^2$ values for gradient-boosted decision trees (GBDTs) to quantify feature-specific contributions to model performance. By decomposing the quadratic loss of individual observations, qshap provides flexible tools to explore the importance of individual features and observations. qshap currently supports widely used GBDT implementations, including xgboost, lightgbm, and catboost, through a unified tree representation and efficient C++ backends. Its modular design can accommodate other GBDT implementations built from binary decision trees. In addition, we introduce a specialized backend for oblivious trees that exploits their symmetric structure to substantially accelerate computation.

Current LLM agent systems decide delegation before reasoning begins (a router picks a model) or after a response is complete (a verifier scores it and may retry). We study a third regime: an agent that recognises, during its own reasoning, that it is unlikely to succeed and transfers control to a stronger model. We formulate intra-generation delegation as a Bayesian optimal-stopping problem over a learned competence posterior – an online estimate of the agent's eventual task success whose sufficient statistics are learned from labelled trajectories, not read off raw entropy. We derive the myopic escalation threshold in closed form, characterise the optimal policy via dynamic programming, and prove that the optimal policy is a time-varying threshold with no shape assumption on the raw signal. We further prove exponential separation of the oracle belief at the Chernoff-information rate of the signal, a regret bound governed by the calibration of the posterior, and a finite-sample guarantee: with n labelled calibration trajectories the deployed plug-in policy's regret decays as 1/sqrt(n). A controlled simulation study confirms each prediction of the theory, including the predicted 1/sqrt(n) rate. We additionally report a real-model validation on a Qwen2.5-Coder 1.5B->7B code cascade (MBPP, 257 tasks), confirming two of three pre-registered predictions: the escalation frontier dominates post-hoc routing at equal cost, and the cumulative competence belief's discrimination rises over generation.

Individualized treatment rules (ITRs) map baseline characteristics to treatment recommendations, with the optimal ITR maximizing expected reward or policy welfare. Indirect methods may require restrictive modeling assumptions, whereas direct methods can be sensitive to nuisance estimation error and limited overlap. We propose orthogonal double residual learning (ODRL), a two-stage, cross-fitted framework that directly targets the optimal ITR through cost-sensitive classification using the product of treatment and outcome residuals. To our knowledge, ODRL is the first direct method with a universally Neyman orthogonal objective requiring neither restrictive modeling assumptions nor inverse propensity score weighting. Thus, nuisance estimation errors affect regret through a second-order product, and ODRL remains robust under limited overlap. The Fisher consistent objective accommodates general decision rule sieves. We establish nonasymptotic high probability value function regret bounds relative to the Bayes classifier for VC classes, including linear rules and decision trees, and calibrated regret bounds for surrogate relaxations using support vector machines and deep ReLU neural networks. We further show that generic surrogate relaxations need not preserve orthogonality, whereas bounded score hinge learning does. Simulations demonstrate strong performance across complex and linear decision boundaries, limited overlap, and working model misspecification. Applications to the Right Heart Catheterization study and the Oxford Net Zero experiment illustrate interpretable treatment or policy recommendations. The odrlITR R package implements ODRL.

Controlled-experiment Using Pre-Experiment Data (CUPED) reduces the variance of the treatment effect estimator in online experiments by adjusting the in-experiment outcome metric using its lagged pre-experiment value. This method can be strengthened by enriching its covariate set while keeping it automatable and guarding against overfitting and covariate leakage. In switchback and related clustered experiment designs, this enrichment may be especially fruitful since randomization-unit-level variation in the outcome amplifies the variance of the treatment effect estimator, so covariates that explain it yield disproportionately large efficiency gains. We develop an extended CUPED framework for switchback and related clustered designs that introduces sizeable improvement in variance reduction efficiency relative to conventional CUPED while remaining lightweight and free of assumptions beyond those already implicit in CUPED. The framework combines multiple historical lags with cyclic hour-of-day encodings and cluster- and hour-level fixed-effect dummies in the pre-period prediction model—covariates that target randomization-unit-level outcome variation—and uses cross-fitting so that realized variance reduction is not inflated by in-sample overfitting. In simulation, we analyze how each component affects validity and efficiency of this method and show that the full framework yields large gains in statistical power. We further validate the approach on publicly available trip records from the New York City Taxi and Limousine Commission. Finally, we derive a closed-form ceiling on the variance reduction attainable by any covariate adjustment and connect our empirical findings to the composition-dependent limits this ceiling imposes.

Time series classification underpins applications in healthcare, sensing, and industrial monitoring. Although time series foundation models support forecasting and transferable representation learning, classification still typically requires fitting a task-specific classifier on each target dataset, while individual channels of multivariate inputs are often encoded independently. We introduce ChorusTIC, a classification-native foundation model for in-context classification across heterogeneous channel configurations without target-task parameter updates. ChorusTIC combines episode-consistent Random Subchannel Slot Concatenation with a shared dual-axis encoder to model temporal and cross-channel interactions and map variable channel configurations into a fixed-width representation independent of the original channel count. It then calibrates feature axes using context-derived distributions and predicts query labels through leakage-protected in-context learning. We pretrain ChorusTIC solely on synthetic labeled episodes comprising context and query sets that share a task background, with classes distinguished by sparse temporal or cross-channel rules. Evaluations on the complete UEA-30 and UCR-128 archives show strong full-context and low-label performance without target-specific classifier fitting.

Understanding how neural networks learn and organize features is central to understanding their behavior. Much existing theory of feature learning has focused on the emergence of a global low-dimensional representation. We show that this picture is incomplete. In regression problems with clustered data, we demonstrate that multilayer perceptrons (MLPs) naturally develop monosemantic specialized neurons: individual neurons become strongly aligned with a specific predictive feature relevant to a particular region of the input space. Rather than learning a single global low-dimensional representation, MLPs learn a collection of local low-dimensional representations. We show that this ability to specialize gives MLPs a provable data-efficiency advantage over feature-learning methods based on a global low-dimensional representation.

Let $A_1,\ldots,A_T$ be independent uniformly selected subsets of a finite population of size $n$, with prescribed cardinalities $m_1,\ldots,m_T$. For each population element, define its occupancy level as the number of selected subsets containing it. Let $x_t$ and $x_{\geq t}$ denote the numbers of elements with occupancy exactly $t$ and at least $t$, respectively. The 2025 work introduced a general multi-set allocation occupancy (MAO) representation for higher-order occupancy moments. The present paper establishes its joint-probabilistic interpretation and provides a rigorous unified derivation for arbitrary joint occupancy categories and moment orders. Specifically, for arbitrary $B_1,\ldots,B_\ell\subseteq\{0,1,\ldots,T\}$, we prove the exact representation $F_\ell(B_1,\ldots,B_\ell)=G_T(B_1,\ldots,B_\ell)/(n)_\ell^{T-1}$, where $G_T(B_1,\ldots,B_\ell)$ is the corresponding generalized MAO transversal sum. This identity unifies the joint factorial moments of arbitrary occupancy categories within a single exact finite-population framework. In particular, writing $B_{\geq t}=\{t,t+1,\ldots,T\}$, the factorial moments of exact- and threshold-occupancy counts are obtained as the specializations $\mathbb{E}[(x_t)_\ell]=F_\ell(\{t\},\ldots,\{t\})$ and $\mathbb{E}[(x_{\geq t})_\ell]=F_\ell(B_{\geq t},\ldots,B_{\geq t})$. Mixed factorial moments, raw moments, variances, and covariances follow from the same representation through standard transformations. The formulas are verified by exhaustive enumeration over feasible parameter ranges and by Monte Carlo simulation. The resulting theory provides a rigorous and unified finite-population foundation for exact and threshold multi-set occupancy statistics.

Modified treatment policies (MTPs) are interventions based on each individual's natural treatment value. We study mean outcomes under MTPs for continuous treatments, including exposure mixtures. Positivity is the standard sufficient condition for identifying these mean outcomes without extrapolation: policy-generated values remain supported given covariates. With multivariate treatments or continuous covariates, treatment–covariate combinations can be sparse or unsupported. Retaining the policy, our partial-identification framework decomposes its mean outcome into a point-identified contribution inside a positivity region and one outside. We bound the latter by imposing Lipschitz continuity on conditional mean potential outcomes rather than a parametric extrapolation model. The restriction compares each outside mean with the mean at an anchor inside the region. Metric projection minimizes width among one-anchor intervals but concentrates anchors on a lower-dimensional boundary, making the endpoints not pathwise differentiable. Our novel interior-displaced projection moves anchors inward, restoring pathwise differentiability. With a known region, we derive influence functions, characterize when they are efficient, and obtain asymptotically normal estimators and confidence intervals. In simulations, our intervals attain at least nominal coverage where those assuming positivity undercover. In a pesticide-mixture application, protective associations suggested by methods assuming positivity are not robust to modest outcome variation beyond the estimated region.

Missing labels are usually regarded as a source of information loss in classification. We study a semi-supervised setting in which the probability of label missingness depends on the observed features through posterior classification uncertainty. In this setting, the missingness indicator is not only a record of an unobserved label, but also an observable signal generated by a mechanism linked to the classifier. We develop a likelihood-based information theory for such uncertainty-dependent missing labels. Under correct specification, we derive a Fisher-information decomposition that separates a partial-labeling component from a nonnegative mechanism-curvature term. Under joint misspecification of the label model and the missingness mechanism, we obtain the corresponding Godambe–Eicker–Huber–White sensitivity and sandwich-covariance partitions. We also clarify the relevant complete-data benchmark: favorable missingness can increase information relative to ordinary fully labeled or budget-matched non-informative labeling baselines, but cannot exceed the information in the augmented experiment in which labels and mechanism indicators are both observed. For plug-in classifiers, we connect the information decomposition to margin-based excess-risk bounds. In regular two-component mixture settings this yields the parametric \(n^{-1}\) excess-risk rate, with constants determined by the nuisance-adjusted information in discriminant directions. Gaussian-mixture calculations and a medical diagnosis example illustrate how uncertainty-dependent labeling mechanisms can improve estimation and classification under a fixed labeling budget.

Modern score-based generative models have achieved remarkable empirical success in high-dimensional tasks such as image, audio, and video synthesis. These models reduce distribution learning to a sequence of regression problems that, if solved exactly on finite data, would ultimately reproduce the training samples. Their ability to generalize must therefore arise from the implicit or explicit regularization during training. In this work, we develop a generative counterpart to the theory of benign overfitting and algorithmic regularization for overparameterized neural networks in the supervised lazy-training regime. We study denoising score matching in a vector-valued reproducing kernel Hilbert space with an inner-product kernel. In the proportional high-dimensional regime $n\asymp d$, we derive exact risk trajectories under gradient flow training. These trajectories exhibit three phases governed by qualitatively distinct estimators: a spectral estimator that generalizes, a pure-noise score with localized peaks that interpolate the training objective, and an empirical Bayes estimator that memorizes the data. We then analyze how these estimators combine along the reverse-time SDE and characterize the distribution of the resulting samples. The analysis reveals familiar mechanisms from supervised learning, including kernel linearization and self-induced regularization from the nonlinear part of the kernel, but also reveals a distinct phenomenology specific to generative modeling.

Econometric models offer parsimonious but inexact approximations to data-generating processes. This paper studies the generalized method of moments (GMM) when exchangeable specification errors of order $n^{-1/2}$ contaminate the moment conditions. I develop estimators for the mean and variance of these specification errors, establishing their consistency in an asymptotic framework where the number of overidentifying restrictions grows with the sample size. These hyperparameter estimates are used to develop a feasible bias-corrected estimator of target parameters. I also propose an empirical Bayes estimator that weakly improves precision by subtracting a best linear predictor of the first-order estimation error from the bias-corrected estimator. Using a combinatorial central limit theorem, I establish asymptotic normality of both estimators and provide variance estimators that enable misspecification-aware frequentist inference. Simulation exercises indicate the procedures can meaningfully improve on standard two-stage least squares estimation when exclusion violations are present. Revisiting the influential study of Angrist and Krueger (1991), I consider an instrument set where exchangeable excludability violations are plausible. Repairing the two-stage least squares estimates of the returns to schooling moves them in the direction of ordinary least squares and reduces sensitivity to the specification of controls.

Conformal Bayes combines Bayesian posterior predictive scores with conformal calibration, but under continuous label shift both the score and calibration weight depend on the unknown response-marginal density ratio. Existing methods typically estimate one shift parameter from pseudo-labels or predictive samples and plug it into calibration. We instead propose Joint Tilt-Sensitivity Conformal Bayes (JTS-CB), which performs sensitivity analysis over a prespecified set of plausible tilts; its split-conformal realization is JTS-SCB. Each tilt jointly determines the Bayesian conformal score and conformal importance weight. JTS-SCB forms a bounded sensitivity envelope over candidate tilts, but its calibration-only construction does not inherit the exact finite-sample weighted-conformal guarantee. We therefore study a separate candidate-weighted exact counterpart and show that its usefulness depends sharply on tail behavior. For scalar linear exponential tilts, any nonzero candidate tilt makes the exact set unbounded. More generally, tail-growing density ratios produce the same pathology, whereas quadratic tilts with a negative coefficient on \(y^2\) have vanishing tail weights and admit bounded exact inference on the original target. Ratio clipping provides a complementary bounded exact construction for a surrogate target when tails grow. Experiments show that strong plug-in predictive sampling can match the oracle when the shift is well identified, while sensitivity analysis is most useful for richer, weakly identified, or systematically biased shift models, at the cost of wider prediction sets.

We prove a strong-universality theorem at the exact fourth-moment threshold for finite families of non-Hermitian random matrices assembled from independent unordered-pair vector atoms. Within one atom, the matrix colors and the two endpoint orientations may have arbitrary joint real covariance, subject to reversal consistency across ordered type pairs and endpoint exchangeability in same-type blocks; the law may also depend on finitely many endpoint types. The matrices may be adjoined to an arbitrary deterministic tuple that converges jointly strongly with the type projections. For every fixed matrix amplification and fixed noncommutative \(*\)-polynomial, the resulting tuple converges strongly to an explicit covariance-matched free Gaussian family. Cross-type blocks are described by masked circular variables, whereas same-type blocks split into independent endpoint-symmetric and endpoint-antisymmetric semicircular sectors. Only a finite radial fourth moment is assumed off the diagonal, and a finite second moment suffices on the diagonal. The mode of convergence depends on the coupling: corners of one infinite array converge almost surely on a common event, while fixed-law nonnested triangular arrays converge in probability for each fixed test. As an application, we obtain exact-fourth-moment strong limits for finite-separable continuous left/right profiles, including separately weighted literal-transpose terms. For the nested almost-sure formulation, the fourth-moment threshold is sharp already on the Wigner subfamily; at this threshold arbitrary fresh rows admit no coupling-invariant almost-sure upgrade of our in-probability conclusion.

Fast, accurate algorithms for the Faddeeva function w(z), and hence the Voigt profile K(x,a), have existed for four decades, and analytic first derivatives are available in some implementations. What the established libraries reviewed here have not provided is the full likelihood calculus of the normalized Voigt distribution: applications still commonly resort to pseudo-Voigt approximations, finite-difference derivatives, or numerical convolution for parameter inference. Because w'(z) = -2z w(z) + 2i/sqrt(pi), every derivative of the Voigt log-likelihood is an algebraic function of K and the dispersion part L(x,a) = Im w(z), from the single complex evaluation that delivers the profile. This yields the score and Hessian in closed form, and the expected Fisher information by one-dimensional quadrature of an analytic integrand; for fixed interior widths sigma, gamma > 0, the MLE of the center and both widths is consistent and asymptotically normal at rate sqrt(n), despite the distribution having no finite mean or variance, so conventional likelihood-based standard errors apply. The conditional mean of the Gaussian component given an observation is (y - mu) - gamma L/K: a redescending function that attributes moderate deviations to the Gaussian (Doppler/resolution) component and extreme ones to the Lorentzian tail. The package voigtinference (Python, NumPy/SciPy, with a cross-validated Julia companion) supplies the toolkit: score, full parameter Hessian, expected information, Newton-based unbinned maximum likelihood with boundary diagnostics, conditional component moments, and evaluation validated against high-precision references at extreme width ratios. It applies directly to unbinned non-relativistic, constant-width Breit-Wigner x Gaussian resonance fits and supplies analytic Jacobians for line-shape refinement. Companion paper: arXiv:2605.01665.

We examine how strongly demographic-distribution diagnostics of LLM-based synthetic persona data depend on the official statistics chosen as the reference. We compared the sex $\times$ age-group $\times$ province joint distribution of the 1,000,000 Nemotron-Personas-Korea (NPK) records with Korean official statistics using total variation distance (TVD). Against the resident-registration population for April 2026, the time of use, the upper bound on the share discrepancy of any subgroup defined by the three variables was 1.81 percentage points, the margin of error of a survey with about 2,900 respondents. This value, however, depended on the reference date and population definition of the official statistics. The closest candidate examined, the 2024 census distribution for Korean nationals, gave 0.56 percentage points; no monthly resident-registration reference was closer. The reference distributions themselves also moved: between January 2025, when NPK's distance was smallest, and April 2026, the resident-registration distribution moved by more than twice that minimum distance, as did the census Korean-national distribution between 2024 and 2025. Raking reduced the range of the distance across reference months to about one fifth. Cell poststratification was applicable because none of the 204 cells was empty, and neither scheme produced extreme weights. Demographic alignment is necessary but not sufficient for response validity, and even this basic diagnostic depends on the choice of reference distribution. Providers should therefore disclose the source, reference date, population definition, and joint-cell allocation procedure of the official statistics used for demographic attributes, and users should repeat the diagnosis and adjustment against statistics current at the time of use.

Data mixing is a central design problem in large language model pretraining: given a fixed token budget, practitioners must decide how much data to allocate to each domain. Recent proxy-based methods address this problem by training small models on candidate mixtures, fitting a response model, and using the response to select mixtures for larger-scale training. We show that this workflow has the structure of a classical mixture experiment. Under this view, data domains are mixture components, token shares are component proportions, proxy-training runs are experimental design points, and validation loss defines a response surface over the probability simplex. We develop this formulation using sparse second-order Scheffé response-surface models and construct model-robust $\mathcal{I}$-optimal designs for proxy data-mixing experiments. Using RegMix as an empirical case study, we demonstrate how the framework can both interpret observed mixture responses and design more efficient proxy experiments. The Scheffé analysis shows that domain value is strongly relational: several domains that are weak under additive effects become favourable through pairwise interactions, especially through combinations with web-derived text. The sparse Scheffé model preserves mixture rankings across model scales and remains competitive with a flexible machine-learning predictor while providing an explicit decomposition of additive and interaction effects. In a simulation study calibrated to observed proxy-training responses, model-robust $\mathcal{I}$-optimal designs recover the relevant mixture ordering after removing about 25% of the original proxy runs. These results suggest that LLM data mixing should be treated not only as a prediction problem, but also as an experimental-design problem in which the proxy mixtures themselves can be chosen to improve statistical efficiency.

Heterogeneous compositional data may be simultaneously affected by missing values and atypical points, posing challenges for both clustering and outlier detection. We develop a mixture model for incomplete compositional data under Huber's contamination model, with contamination defined directly on the simplex and on observations that may be missing at random. The model provides a principled representation of outliers and allows the distribution of missing parts to be derived while accounting for contamination. We establish that maximisation of the observed log-likelihood constructed from contaminated, mean-parametrised Dirichlet densities is a convex optimisation problem. We then develop a tailored expectation-maximisation. The E-step incorporates the moments from the distribution of the missing parts of the data. Although the resulting parameter estimates are not available in closed form, the maximisation step admits tractable element-wise iterative updates. Numerical experiments demonstrate the performance of the proposed approach under varying percentage of missingness and contamination, and different sample size. An application to the American Time Use Survey identifies two interpretable clusters corresponding to work-intensive and sociable recreational days, while revealing atypical time-use compositions. In contrast, a conventional Dirichlet mixture model identifies four clusters, reflecting the influence of outliers and an artificial splitting of one cluster.

Revelation Control is the problem of choosing priced interventions that reveal hidden state only insofar as the revealed distinctions can change a consequential decision, while accounting separately for any useful progress created by the intervention itself. We develop this theory for learning systems, where states equivalent under declared current information can respond differently to future training and favor different actions. The framework defines decision-sufficient revelation and revelation depth, separates pure information value from productive reuse, embeds static Bayes refinement into state-dependent continuation value, and gives an exact cost-adjusted factorization criterion: an additional shallow coordinate is decision-nonredundant only when states sharing a scalar summary lie on opposite sides of the priced Stop/Continue boundary. We also give a target-independent protocol for model-specific instantiation and prove that bounded stop-flip risk alone cannot certify positive expected utility under unrestricted severity. Across Qwen2.5-7B and Mistral-7B-v0.3, deeper future-learning probes have positive decision value and productive reuse yields strict equal-compute utility advantages. Qwen additionally provides evidence for a decision-nonredundant shallow revealability regime; in Mistral, a scalar continuation architecture fit only on an independent development panel retains positive familywise-adjusted lower bounds on a disjoint target panel, consistent with scalar decision sufficiency within the tested architecture family and resolution. The evidence supports structural rather than numerical transfer: the decision theory, cost accounting, continuation logic, and evaluation protocol transport, while empirical proxies, coefficients, thresholds, and even the required shallow state dimension may be system-specific.

Sport leagues assign a fixed score-point to each match outcome and rank the players (or teams) by their accumulated scores. Since the score-points are mandated without a probabilistic model in mind, we investigate models that make the score the right summary of the data. We find the score to be a sufficient statistic if and only if the points rule is constant-sum, so that every match distributes a fixed number of points between its participants, and the outcomes follow the adjacent categories model whose slope parameters are the score-points themselves. The constant-sum condition fails in the National Hockey League and in football, which awards three points for a win instead of the two required by sufficiency. We also establish that when two players meet every other opponent equally often, and each plays as often at home as away, they are ordered identically by the score and by the estimated skills; a venue-balanced round-robin satisfies the condition for all pairs. The Elo ranking then produces the same order. In nine leagues from football, ice hockey and volleyball, the two orders differ only where the schedule condition fails, and the uniform rule awarding one point per outcome level agrees with the estimated skills more closely than each league's own rule, unless that rule is already the uniform one.

The Benter model for partial and complete rankings generalizes the well-known Plackett-Luce by attaching rank-level dampening parameters that permit some ranking stages to be noisier than others. This extension is empirically important for domains ranging from horse racing to ranked-choice elections. However, model fitting is made challenging by the fractional powers these dampening parameters introduce to the likelihood, breaking the conjugacy underlying existing Plackett-Luce samplers. This paper develops an efficient Bayesian estimation procedure for the Benter model. We introduce a two-part augmentation scheme using positive $\alpha$-stable and exponential auxiliary variables that linearizes the intractable normalizers in the Benter likelihood and yields closed-form Gibbs updates. A simulation study confirms efficient estimation, accurate parameter recovery, and nominal credible-interval coverage across sample sizes and item counts. We illustrate the algorithm on complete and partial rankings from survey preference and ranked-choice election datasets.

Data assimilation in the form of the ensemble Kalman filter (EnKF) is commonly used to combine large-scale physics-based models and real-world observations of a quantity of interest. However, the EnKF is known to be adversely affected by spurious correlations when applied to Numerical Weather Prediction (NWP) models with high-dimensional state spaces. To mitigate the effect of spurious correlations, various localization methods have been proposed in the NWP literature. Unfortunately, the efficacy of and need for these methods in the setting of general PDE-based forward models with large spatial mesh sizes is not well-understood. We conducted a numerical data assimilation experiment on an example PDE model to demonstrate the presence of spurious correlations and need for localization methods outside of the context of NWP. In addition, we compared the computational efficiency of various EnKF implementations in high-dimensional settings for which theoretical complexity bounds are available but empirical costs are unclear.

Many common data dependencies can be characterized by graphs: time series data are sequential (chain graph), images appear as pixels (lattice graph), areal data are defined by neighboring units (spatial adjacency graph), etc. Graph trend filtering seeks to smooth and predict such data. However, classical trend filtering only incorporates the graph for estimation of the trend, which limits its adaptivity, and is brittle in the presence of missing data. Further, it lacks uncertainty quantification and faces certain computing challenges. We address these limitations with a comprehensive Bayesian framework for (graph-) dependent data. Our approach leverages the graph at three critical junctures: 1) the trend, to enable smoothing, imputation, and prediction; 2) the local shrinkage, to enhance adaptivity and precision; and 3) the MCMC sampling algorithm, to deliver scalable posterior (predictive) inference via sparse and banded operations. For the proposed graph-dependent shrinkage priors, we study the local concentration and adaptivity properties and establish conditions for posterior propriety. Simulation studies demonstrate that, relative to state-of-the-art frequentist and Bayesian alternatives, this framework provides more accurate point estimates, more precise interval estimates, and highly competitive computing. We apply our methods for spatio-temporal modeling and forecasting of local area unemployment data for every county in the continental U.S. during the 2020 COVID-19 unemployment shock.

Ranked-choice voting is increasingly prevalent in elections. An extensive literature in social choice theory – the study of collective decision-making with the goal of making compromises among disparate opinions – considers theoretical and empirical properties of various election procedures. Recently, a sub-literature on rationalizability, led by social choice theorists and computer scientists, makes explicit connections between social choice rules and maximum likelihood estimation. This paper further illuminates connections between social choice and statistical summaries of data. We begin by studying rationalizability from a statistical perspective, expanding existing results to realistic voting contexts and deriving statistical details necessary for model estimation and assessment. We then apply our work to ranked-choice votes from the 2025 New York City Democratic mayoral primary election. We demonstrate how rationalizing models impose unrealistic distributional assumptions and thus exhibit poor fit to voting data. Additionally, we show that non-rationalizing models meaningfully elucidate heterogeneous voter preferences. Our work demonstrates the inability of rationalizing models to capture key features of distributions of preferences in political elections, depriving analysts of fundamental uses, such as inference, typically associated with statistical modeling. Conversely, we show that statistical modeling can capture nuanced voter preferences by paying careful attention to plausible data-generating mechanisms.