Papers

Clinical decision-making often involves selecting tests that are costly, invasive, or time-consuming, motivating individualized, sequential strategies for what to measure and when to stop ascertaining. We study the problem of learning cost-optimal sequential decision policies from retrospective data, where test availability depends on prior results, inducing informative missingness. Under a sequential missing-at-random mechanism, we develop a doubly robust Q-learning framework for estimating optimal policies. The method introduces path-specific inverse probability weights that account for heterogeneous test trajectories and satisfy a normalization property conditional on the observed history. By combining these weights with auxiliary contrast models, we construct orthogonal pseudo-outcomes that enable unbiased policy learning when either the acquisition model or the contrast model is correctly specified. We establish oracle inequalities for the stage-wise contrast estimators, along with convergence rates, regret bounds, and misclassification rates for the learned policy. Simulations demonstrate improved cost-adjusted performance over weighted and complete-case baselines, and an application to a prostate cancer cohort study illustrates how the method reduces testing cost without compromising predictive accuracy.

We develop parameter-free algorithms for unconstrained online learning with regret guarantees that scale with the gradient variation $V_T(u) = \sum_{t=2}^T \|\nabla f_t(u)-\nabla f_{t-1}(u)\|^2$. For $L$-smooth convex losses, we provide fully-adaptive algorithms achieving regret of $\widetilde{O}(\|u\|\sqrt{V_T(u)} + L\|u\|^2+G^4)$ without requiring prior knowledge of comparator norm $\|u\|$, Lipschitz constant $G$, or smoothness $L$. The update in each round can be computed efficiently via a closed-form expression. Our results extend to dynamic regret and find immediate implications for the stochastically-extended adversarial (SEA) model, which significantly improves upon the previous best-known result (Wang et al., 2025).

We study an empirical interpretation of the Pitman efficiency in testing for uniformity in the two-parametric family of the beta distributions. We show that for contamination models the Pitman efficiency approximates relative efficiency very well.

Recent theory suggests that reward-model-first methods can be more sample-efficient than direct policy fitting when the reward function is statistically simpler than the induced policy. We propose DDO-RM, a finite-candidate decision-optimization method that converts reward scores into an explicit target distribution. Unlike PPO-based RLHF or DPO, DDO-RM performs a KL-regularized mirror-descent update to project the policy toward a reward-improved distribution over a candidate set. Preliminary experiments on Pythia-410M show that DDO-RM outperforms DPO in pair accuracy (0.52 to 0.56) and mean margin (0.13 to 0.53). Our framework provides a principled connection between reward learning and mirror-descent policy improvement.

K-means clustering is a workhorse of unsupervised learning, but it is notoriously brittle to outliers, distribution shifts, and limited sample sizes. Viewing k-means as Lloyd–Max quantization of the empirical distribution, we develop a distributionally robust variant that protects against such pathologies. We posit that the unknown population distribution lies within a Wasserstein-2 ball around the empirical distribution. In this setting, one seeks cluster centers that minimize the worst-case expected squared distance over this ambiguity set, leading to a minimax formulation. A tractable dual yields a soft-clustering scheme that replaces hard assignments with smoothly weighted ones. We propose an efficient block coordinate descent algorithm with provable monotonic decrease and local linear convergence. Experiments on standard benchmarks and large-scale synthetic data demonstrate substantial gains in outlier detection and robustness to noise.

Family-wise error rate control is essential when even one false rejection is costly, but distribution-free procedures do not exploit information in a specified alternative model. Existing dual theory characterises the maximum-average-power procedure under strong family-wise error control, but its optimal multipliers had been computed only up to $K = 3$. Under an exchangeable product model of independent $p$-values with a common non-increasing density under the alternative, we develop a general-$K$ statistical methodology that removes this computational barrier. An elementary-symmetric-polynomial representation yields global monotonicity of every constraint function and converts the coupled multiplier problem into monotone coordinate-wise searches. The resulting algorithm, symmetric-polynomial optimal testing (SPOT), uses bisection coordinate descent and has polynomial per-sweep cost in $K$ and the Monte Carlo size. Under stated conditions, its exact population objective values converge to the optimum and every limit point is optimal, without contraction or strong-convexity assumptions. Additional local regularity gives linear convergence of the exact population iterates, and an achieved $O_p(N^{-1/2})$ residual gives $\sqrt{N}$-consistent Monte Carlo output. In a truncated-normal scaling experiment, the relative average-power gain over Hommel's method increases from 15% at $K = 3$ to 83% at $K = 12$. Applications with up to 21 hypotheses demonstrate SPOT's practical reach.

Generalized linear mixed-effects models (GLMMs) are widely used to analyze grouped and hierarchical data. In a GLMM, each response is assumed to follow an exponential-family distribution where the natural parameter is given by a linear function of observed covariates and a latent group-specific random effect. Since exact marginalization over the random effects is typically intractable, model parameters are estimated by maximizing an approximate marginal likelihood. In this paper, we replace the linear function with neural networks. The result is a more flexible model, the neural generalized mixed-effects model (NGMM), which captures complex relationships between covariates and responses. To fit NGMM to data, we introduce an efficient optimization procedure that maximizes the approximate marginal likelihood and is differentiable with respect to network parameters. We show that the approximation error of our objective decays at a Gaussian-tail rate in a user-chosen parameter. On synthetic data, NGMM improves over GLMMs when covariate-response relationships are nonlinear, and on real-world datasets it outperforms prior methods. Finally, we analyze a large dataset of student proficiency to demonstrate how NGMM can be extended to more complex latent-variable models.

Data leakage remains a recurrent source of optimistic bias in biomedical machine learning studies. Standard row-wise cross-validation and globally estimated preprocessing steps are often inappropriate for data with repeated measurements, study-level heterogeneity, batch effects, or temporal dependencies. This paper describes bioLeak, an R package for constructing leakage-aware resampling workflows and for auditing fitted models for common leakage mechanisms. The package provides leakage-aware split construction, train-fold-only preprocessing, cross-validated model fitting, nested hyperparameter tuning, post hoc leakage audits, and HTML reporting. The implementation supports binary classification, multiclass classification, regression, and survival analysis, with task-specific metrics and S4 containers for splits, fits, audits, and inflation summaries. The simulation artifacts show how apparent performance changes under controlled leakage mechanisms, and the case study illustrates how guarded and leaky pipelines can yield materially different conclusions on multi-study transcriptomic data. The emphasis throughout is on software design, reproducible workflows, and interpretation of diagnostic output.

In this paper, we study Sibson's $\alpha$-mutual information in the context of the additive Gaussian noise channel. While the classical case $\alpha = 1$ is well understood and admits deep connections to estimation-theoretic quantities, such as the minimum mean-square error (MMSE) and Fisher information, many of the corresponding structural properties for general $\alpha$ remain less explored. Our goal is to develop a systematic understanding of $\alpha$-mutual information in the Gaussian noise setting and to identify which properties extend beyond the Shannon case. To this end, we establish several regularity properties, including finiteness conditions, continuity with respect to the signal-to-noise ratio (SNR) and the input distribution, and strict concavity/convexity properties that ensure uniqueness in associated optimization problems. A central contribution is the development of an $\alpha$-I-MMSE relationship, generalizing the classical identity by relating the derivative of $\alpha$-mutual information with respect to SNR to the MMSE evaluated under appropriately tilted distributions. This connection further leads to a generalized de Bruijn identity and new estimation-theoretic representations of Rényi entropy and differential Rényi entropy. We also characterize the low- and high-SNR behavior. In the low-SNR regime, the first-order behavior depends only on the input variance. In the high-SNR regime, for discrete inputs, $\alpha$-mutual information converges to the Rényi entropy of order $1/\alpha$, while for general inputs we connect it to $\alpha$-information dimension. Overall, our results show that many fundamental relationships between information and estimation extend beyond the Shannon setting, in a form involving $\alpha$-tilted distributions.

We study the Kullback–Leibler (KL) divergence approximation theory of Gaussian mixture models (GMMs) by isolating an abstract mechanism behind several necessary-and-sufficient statements. The necessity direction is universal: if a density is approximable in KL divergence by finite GMMs, then it must have finite second moment. The sufficient direction is reduced to the construction of approximating GMMs whose likelihood ratios converge pointwise and whose finite log-ratios form a uniformly integrable family. We verify this mechanism on a finite log-moment class of continuous strictly positive target densities, from which bounded, $\mathcal L^p$ $(p>1)$, and Orlicz-dominated subfamilies follow immediately. We also show that a countable-scale support-aware target density class, which allows zero density regions, satisfies the same equivalence. Finally, we give counterexamples showing that the countable-scale class strictly extends the fixed-scale class, that the finite log-moment and countable-scale support-aware classes do not contain one another, and that their union is not exhaustive.

Inferring directed acyclic graphs (DAGs) from data via Markov chain Monte Carlo (MCMC) is computationally challenging in moderate-to-high dimensional settings because their discrete sampling space grows super-exponentially with the number of nodes. To address scalability, several recent MCMC-based graph inference methods restrict the search space to a subset of edges, at the cost of introducing error into the inference procedure. In this work, we derive sharp lower and upper bounds on the total variation distance between the unrestricted posterior distribution and the posterior distribution induced by a state-of-the-art restricted search space MCMC method. These bounds characterize regimes in which the approximation error is negligible and regimes in which it is not. In order to reduce the error, we propose a flexible transdimensional MCMC sampler which allows the search space to expand or contract dynamically as the chain progresses. The sampler is defined by birth-and-death rates that induce a prior distribution on the set of search spaces, rather than assume a fixed restricted search space throughout. We outline an efficient implementation of the proposed algorithm and demonstrate its finite-sample performance through simulation studies.

Diffusion models generate samples by iteratively querying learned score estimates. A rapidly growing literature focuses on accelerating sampling by minimizing the number of score evaluations, yet the information-theoretic limits of such acceleration remain unclear. In this work, we establish the first score query lower bounds for diffusion sampling. We prove that for $d$-dimensional distributions, given access to score estimates with polynomial accuracy $\varepsilon=d^{-O(1)}$ (in any $L^p$ sense), any sampling algorithm requires $\widetilde{\Omega}(\sqrt{d})$ adaptive score queries. In particular, our proof shows that, within any polynomial total-query budget, successful sampling requires searching over $\widetilde{\Omega}(\sqrt{d})$ distinct noise levels, providing a formal explanation for why multiscale noise schedules are necessary in practice.

It is well known that estimating the expectation of any given bounded random variable with values in $[-B, B]$ has two basic properties: (1) the number of samples needed scales at the order of $O(B^2/\epsilon^2)$, where $\epsilon$ is the prescribed target precision, and (2) such a sample complexity is independent of the underlying probability measure. We show that neither of these properties can no longer hold when evaluating the worst-case expectation of the random variable, where the probability measures defining the expectation belong to a $\phi$-divergence ball centered at some nominal measure $P$. Specifically, the sample complexity and its dependence on the nominal measure can be completely characterized by the growth of the divergence function. When the divergence function $\phi$ exhibits superlinear growth, a $P$-independent sample complexity can be obtained for sample average approximation, which depends only on the growth of $\phi$, the radius of the divergence ball, and the target precision. We also provide sample complexity lower bounds and demonstrate the optimality of the obtained bounds for commonly used $\phi$-divergences. On the other hand, when superlinear growth does not hold for $\phi$, we show that for any estimation method, evaluating the worst-case expectation has a $P$-dependent sample complexity lower bound that can be made arbitrarily large by changing $P$. In this case, we discuss the approach of adopting the generalized definition of $\phi$-divergence, through which $P$-independent sample complexity can be recovered, and hence demonstrate the essential role of absolute continuity in affecting statistical tractability.

In many problems of data-driven modeling for dynamical systems, the governing equations are not known a priori and must be selected phenomenologically from a large set of candidate interactions and basis functions. In such situations, point estimates alone can be misleading, because multiple model components may explain the observed data comparably well, especially when the data are limited or the dynamics exhibit poor identifiability. Quantifying the uncertainty associated with model selection is therefore essential for constructing reliable dynamical models from data. In this work, we develop a Bayesian sparse identification framework for dynamical systems with coupled components, aimed at inferring both interaction structure and functional form together with principled uncertainty quantification. The proposed method combines sparse modeling with Bayesian model averaging, yielding posterior inclusion probabilities that quantify the credibility of each candidate interaction and basis component. Through numerical experiments on oscillator networks, we show that the framework accurately recovers sparse interaction structures with quantified uncertainty, including higher-order harmonic components, phase-lag effects, and multi-body interactions. We also demonstrate that, even in a phenomenological setting where the true governing equations are not contained in the assumed model class, the method can identify effective functional components with quantified uncertainty. These results highlight the importance of Bayesian uncertainty quantification in data-driven discovery of dynamical models.

Conjoint experiments randomize multidimensional profiles, yet political science applications typically report only nonparametric averages that do not recover counterfactual choices or individual tradeoffs. We develop a hybrid structural approach for recovering individual preferences from conjoint data. The estimator combines a flexible machine-learning mean preference function, via a deep netural network in our applications, with respondent-level empirical-Bayes updating in a logistic random utility model, allowing preferences to vary with observed characteristics while learning residual heterogeneity from repeated choices. Double/debiased machine learning delivers valid inference for population-average preference parameters with any sufficiently accurate first-stage learner. Across three applications, the method reveals heterogeneity reduced-form averages obscure: opposition to undemocratic behavior is broad but uneven in intensity, progressive tax preferences are widespread across partisan subgroups, and partisan polarization offsets the average gender effect in candidate choice. The framework opens the door to core theoretical questions in political science by recovering substantively interpretable structural parameters.

This study applies the Causal Fairness Analysis (CFA) framework of Plecko and Bareinboim (2024) to decompose the total variation in STEM outcomes attributable to ADHD status into direct, indirect, and spurious components using Pearl's Structural Causal Model. Drawing on nationally representative data from the High School Longitudinal Study of 2009, this study examines two outcomes: cumulative STEM GPA and science identity. Total variation decomposition reveals a statistically significant ADHD penalty on STEM GPA (TV = -0.670), of which 63.3% is attributable to the direct effect (x-DE), indicating that the majority of the disparity operates through pathways not mediated by observed sociodemographic or academic confounders. In contrast, the effect on science identity is small and non-significant (TV = -0.068). Counterfactual direct effect analysis using the one-step debiased estimator further reveals that the direct effect is structured by race, with notable variation across racial and ethnic subgroups. Sensitivity analyses confirm robustness to moderate unmeasured confounding. These findings advance the understanding of ADHD-related inequities in STEM education and highlight the need for fairness-aware policies that address both direct institutional barriers and their differential impact across intersecting social identities.

High-dimensional and complex discrete distributions often exhibit multimodal behavior due to inherent discontinuities, posing significant challenges for sampling. Gradient-based discrete samplers, while effective, frequently become trapped in local modes when confronted with rugged or disconnected energy landscapes. This limits their ability to achieve adequate mixing and convergence in high-dimensional multimodal discrete spaces. To address these challenges, we propose Hyperbolic Secant-squared Gibbs-Sampling (HiSS), a novel family of sampling algorithms that integrates a Metropolis-within-Gibbs framework to enhance mixing efficiency. HiSS leverages a logistic convolution kernel to couple the discrete sampling variable with the continuous auxiliary variable in a joint distribution. This design allows the auxiliary variable to encapsulate the true target distribution while facilitating easy transitions between distant and disconnected modes. We provide theoretical guarantees of convergence and demonstrate empirically that HiSS outperforms many popular alternatives on a wide variety of tasks, including Ising models, binary neural networks, and combinatorial optimization.

This paper studies a finite reversible lumpable Markov chain for which relaxed spectral compression yields a larger determinant than partition-constrained compression. For a symmetric six-state lumpable chain and the positive operator $T=P^2$, I compare the relaxed benchmark \begin{equation*} \mathfrak D^{\mathrm{rel}}_3(T):=\sup_{U^*U=I_3}\det(U^*TU) \end{equation*} and the partition-constrained benchmark \begin{equation*} \sup_{\mathcal A\,\mathrm{3\text{-}partition}}\det Q_{\mathcal A}(T), \qquad Q_{\mathcal A}(T)=H_{\mathcal A}^*TH_{\mathcal A}. \end{equation*} Here the partition-constrained benchmark is the compression induced by normalized indicator vectors of genuine partitions of the state space. I derive closed formulas for the two analytically central partition families, prove strict upper bounds for both in a local-mode-dominated regime, and combine these bounds with an exhaustive enumeration of all $90$ partitions into three nonempty cells in an explicit six-state model. For this model, one obtains a strict global gap: \begin{equation*} \sup_{\mathcal A}\det Q_{\mathcal A}(T)<\mathfrak D^{\mathrm{rel}}_3(T). \end{equation*} Thus, in this model, indicator-based partition frames are strictly weaker than relaxed orthonormal frames even after global partition-constrained optimization.

We study online covariance matrix estimation for Polyak–Ruppert averaged stochastic gradient descent (SGD). The online batch-means estimator of Zhu, Chen and Wu (2023) achieves an operator-norm convergence rate of $O(n^{-(1-\alpha)/4})$, which yields $O(n^{-1/8})$ at the optimal learning-rate exponent $\alpha \rightarrow 1/2^+$. A rigorous per-block bias analysis reveals that re-tuning the block-growth parameter improves the batch-means rate to $O(n^{-(1-\alpha)/3})$, achieving $O(n^{-1/6})$. The modified estimator requires no Hessian access and preserves $O(d^2)$ memory. We provide a complete error decomposition into variance, stationarity bias, and nonlinearity bias components. A weighted-averaging variant that avoids hard truncation is also discussed. We establish the minimax rate $\Theta(n^{-(1-\alpha)/2})$ for Hessian-free covariance estimation from the SGD trajectory: a Le Cam lower bound gives $\Omega(n^{-(1-\alpha)/2})$, and a trajectory-regression estimator–which estimates the Hessian by regressing SGD increments on iterates–achieves $O(n^{-(1-\alpha)/2})$, matching the lower bound. The construction reveals that the bottleneck is the sublinear accumulation of information about the Hessian from the SGD drift.

Sociological research has framed collective action in science, innovation, and culture as tripartite networks connecting teams of actors, lists of prior works, and sets of labels (e.g., keywords, topics). While methods for multipartite social networks were proposed decades ago, and have received a recent surge in interest, none of the suggested solutions scale to the size and granularity of contemporary data sets (scientific publications, patents, filmmaking) and at the same time allow for testing multiple competing hypotheses about the drivers of collective production. In this paper, we address this gap by applying Relational Hyperevent Models (RHEM) to dynamic tripartite hypergraphs. Using scientific networks as a case study, we model events linking any number of actors, references, and keywords, testing and controlling for inter-dependencies within and between each set.

Variable-length Markov chains on finite quivers provide a natural framework for context-dependent stochastic growth under incidence constraints. I study quiver-valued variable-length Markov chains observed through finite boundary windows and develop a first-order theory of visible-depth identifiability via stationary visible one-step transition laws and their restricted differentials on prescribed tangent blocks. For visible depth $m$, the main object is the stationary one-step informative map $q_{\mathcal{Q}}^{(m)}$. In the edge-homogeneous regime, once the local visible support is fixed and the representation hypothesis holds, all admissible visible depths encode the same edge-level extension law and hence have the same first-order rank. In the exact-depth regime of context length $r$, the depth-$r$ boundary process is the canonical finite-state Markov chain, smaller visible windows are deterministic truncations, and every coarser informative map factors $C^1$-smoothly through the depth-$r$ informative map on the relevant affine transition-array neighborhood. Hence rank cannot increase beyond depth $r$. After quotienting a tangent block by directions already invisible at depth $r$, I characterize strict coarse-depth loss exactly by coarse rank deficiency, equivalently by strict rank drop from depth $r$ to depth $m$ on the original block. I also give subspace-based and global selected-coordinate criteria, a global one-coordinate branching criterion, and an explicit depth-two example. Under full fine-depth rank and strict coordinate-rank loss at every smaller depth, a global coordinate-rank theorem yields $m_*(T,\theta_0)=r$. Reduced local coordinates remove stochastic redundancies, first-order criteria are invariant under $C^1$ reparameterization, and the statistical and LAN consequences remain conditional on additional estimation and likelihood-level hypotheses.

I formulate an entropy-rate maximization problem at the observable level for stochastic processes observed through an information-reducing observation map. For a visible stationary law, the map determines an observational fiber of hidden stationary laws generating that law. In the finite-state finite-memory setting, retained visible constraints determine a feasible class of stationary $(r+1)$-block laws, and the entropy maximizer is defined as the entropy-rate maximizer on this class. The paper formulates entropy-rate maximization on feasible classes induced by partial observability and develops a structural theory for the resulting maximizer. I prove existence and uniqueness of the maximizer, with uniqueness under a fixed-context-marginal hypothesis and, more generally, via a strict-concavity characterization by row proportionality. Two global characterization regimes are central: a fixed one-point marginal yields the i.i.d. maximizer, and a fixed $r$-block law yields the $(r-1)$-step Markov extension. The gap functional equals a conditional mutual information and vanishes exactly at the maximizing completion. I also derive optimality conditions, local geometry of the maximizer, a latent random-mapping realization that leaves the visible law unchanged, and a local empirical consistency theorem, and illustrate the framework by an aliased hidden-state example.

Classical information-theoretic learning bounds typically rely on KL mutual information and moment-generating-function (MGF) arguments, which are well matched to bounded or sub-Gaussian losses but can be ineffective when losses or rewards are heavy-tailed. We develop a tail-aware information-theoretic framework for sub-Weibull data, where the tail parameter $\theta$ controls the tail heaviness: $\theta=2$ corresponds to sub-Gaussian, $\theta=1$ to sub-exponential, and $0<\theta<1$ to genuinely heavy tails. Our key technical ingredient is a decorrelation lemma that bounds change-of-measure expectations using a shifted-log $f_\theta$-divergence, which admits explicit comparisons to Rényi divergence without MGF arguments. On the empirical-process side, we establish sharp maximal inequalities and a Dudley-type chaining bound for sub-Weibull processes, with logarithmic and entropy terms raised to the power $1/\theta$. These tools yield tail-adaptive selection bounds and a multiscale information-theoretic Dudley inequality based on shifted-log and Rényi mutual information. We apply our theory to large language models (LLMs) in the context of reward hacking within reinforcement learning from human feedback (RLHF). We show that Rényi-regularized alignment provides finite reward guarantees and ensures that best-of-N policies remain well-controlled, thereby mitigating the catastrophic Goodhart effects where standard KL-regularization fails. We illustrate Rényi-regularized RLHF by experiments, including controlled heavy-tailed rewards and token-space reward attacks.

An individualized treatment rule (ITR) tailors treatments to a patient's specific characteristics. However, randomized controlled trials (RCTs) are often underpowered to detect the treatment effect heterogeneity needed for reliable ITR estimation. To address this limitation, there is growing interest in leveraging information from multiple studies to improve statistical power and support individualized decision-making. A key challenge in this context is that available RCTs may not evaluate the same set of treatments. In this paper, we propose an integrative learning framework that synthesizes evidence across multiple RCTs that share a common comparator but differ in their alternative treatment arms. Our method integrates information through a regularized weighted misclassification risk function and adaptively determines the contribution of each study to the ITRs of the others. We rigorously study the excess risk of the resulting estimator. Simulation studies demonstrate that the proposed approaches improve the estimation of both value and benefit functions. We illustrate the utility of our methodology using data from two landmark studies of major depressive disorder: the Establishing Moderators and Biosignatures of Antidepressant Response in Clinical Care study and the International Study to Predict Optimized Treatment in Depression study, both of which include a selective serotonin reuptake inhibitor as a common treatment arm. We find that the separate learning method outperforms one-size-fits-all methods, and our integrative methods further improve performance.

Causal mediation analysis in cluster-randomized trials (CRTs) is complicated by the presence of multiple mediators, intracluster correlation, and within-cluster interference. Existing mediation methods often fall short in accommodating these features simultaneously, and semiparametric efficient estimators that fully address them remain unavailable. We develop a unified framework that defines a class of mediation effect estimands, including exit indirect effects, exit spillover mediation effects, and their interaction effects, to investigate causal mechanisms in CRTs with an arbitrary number of mediators under an unknown causal structure. We introduce a set of interpretable causal assumptions for point identification of each estimand. For optimal inference, we first derive the efficient influence functions for the proposed estimands and construct corresponding one-step and debiased machine learning estimators. In particular, to flexibly model the joint mediator density, we employ an elliptical copula marginal regression model that combines a nonparametric marginal regression with an interpretable association structure. We assess the finite-sample performance of the proposed estimators through simulation studies and illustrate the methodology by reanalyzing the PPACT CRT data with three causally unordered mediators.

Missing values in electronic health record (EHR) data pose a significant challenge for epidemiologic research. Traditional methods for handling missing data, like mean imputation, may introduce bias. Multiple imputation (MI) offers a principled solution by generating multiple plausible values based on statistical models. However, MI requires careful model specification and validation of imputations, ideally using multivariate graphical tools. We demonstrate the application of such tools to validate MI in a study of chronic kidney disease, assessing cardiovascular outcomes linked to neighborhood socioeconomic status (nSES). This study used data from Duke University Health System (DUHS) and Lincoln Community Health Center (LCHC). Eligible patients had at least one encounter within DUHS or LCHC and had two estimated glomerular filtration rate (eGFR) values <60 mL/min per 1.73 m2 more than 90 days apart between January 1, 2007 and July 1, 2008. Socioeconomic status was assessed using the Agency for Healthcare Research and Quality (AHRQ) index based on census data. The main outcome was a cardiovascular disease-related hospitalization. Participants were mostly older (mean age 73 years), female (64%), and Black (43%). Participants living in lower nSES neighborhoods had higher mean systolic blood pressure (SBP: 140 mmHg) and hemoglobin A1c (HbA1c) levels (7.1%) as compared to participants living in higher nSES neighborhoods. A machine learning based approach, Classification and Regression Trees (CART), was the preferred approach to impute missing data. The distributions of imputed values of SBP and HbA1c were impacted by whether marginal or conditional values of SBP and HbA1c were imputed. The choice of MI had minimal impact on inference and prediction. Future research may want to extend our results and consider how results may differ when using EHR data from multiple health systems.

Density ratio estimation (DRE) is a useful tool for quantifying discrepancies between probability distributions, but existing approaches often involve a trade-off between estimation quality and computational efficiency. Classical direct DRE methods are usually efficient at inference time, yet their performance can seriously deteriorate when the discrepancy between distributions is large. In contrast, score-based DRE methods often yield more accurate estimates in such settings, but they typically require considerable repeated function evaluations and numerical integration. We propose One-step Score-based Density Ratio Estimation (OS-DRE), a partly analytic and solver-free framework designed to combine these complementary advantages. OS-DRE decomposes the time score into spatial and temporal components, representing the latter with an analytic radial basis function (RBF) frame. This formulation converts the otherwise intractable temporal integral into a closed-form weighted sum, thereby removing the need for numerical solvers and enabling DRE with only one function evaluation. We further analyze approximation conditions for the analytic frame, and establish approximation error bounds for both finitely and infinitely smooth temporal kernels, grounding the framework in existing approximation theory. Experiments across density estimation, continual Kullback-Leibler and mutual information estimation, and near out-of-distribution detection demonstrate that OS-DRE offers a favorable balance between estimation quality and inference efficiency.

While the manifold hypothesis is widely adopted in modern machine learning, complex data is often better modeled as stratified spaces – unions of manifolds (strata) of varying dimensions. Stratified learning is challenging due to varying dimensionality, intersection singularities, and lack of efficient models in learning the underlying distributions. We provide a deep generative approach to stratified learning by developing two generative frameworks for learning distributions on stratified spaces. The first is a sieve maximum likelihood approach realized via a dimension-aware mixture of variational autoencoders. The second is a diffusion-based framework that explores the score field structure of a mixture. We establish the convergence rates for learning both the ambient and intrinsic distributions, which are shown to be dependent on the intrinsic dimensions and smoothness of the underlying strata. Utilizing the geometry of the score field, we also establish consistency for estimating the intrinsic dimension of each stratum and propose an algorithm that consistently estimates both the number of strata and their dimensions. Theoretical results for both frameworks provide fundamental insights into the interplay of the underlying geometry, the ambient noise level, and deep generative models. Extensive simulations and real dataset applications, such as molecular dynamics, demonstrate the effectiveness of our methods.

Synthetic face generators can produce many nominal identities, but nominal count does not determine how many are jointly distinguishable under a specified verification rule. We define finite-dimensional capacity as the supremum of codebook sizes over distinct latent identity codes whose induced identity-conditional embedding distributions satisfy per-identity genuine acceptance and pairwise impostor non-match constraints. For deterministic view-invariant pipelines, fixed-code capacity equals the spherical-code cardinality over the realizable embedding set and reduces to the classical spherical-code cardinality when every sphere direction is realizable. For stochastic identity-conditional embedding distributions concentrated with probability at least $1-\eta$ in spherical caps of angular radius $\rho$, we derive a sufficient center-separation condition, spherical-code capacity lower bounds under full angular expressivity, and positive asymptotic lower-bound exponents for dimension-indexed pipeline families. We also derive prior-constrained random-code lower bounds from pairwise center-separation failure probabilities. When each identity-conditional embedding distribution has support equal to a spherical cap of angular radius $\rho$, we derive necessary zero-error geometric conditions and, for $2\rho<\arccos(\tau)$ under full $\rho$-cap angular expressivity, show that the restricted zero-error capacity equals the classical spherical-code cardinality at minimum angle $\arccos(\tau)+2\rho$. For finite repeated-view samples, a maximum clique in the resulting compatibility graph identifies the largest sampled subset satisfying all empirical genuine and pairwise impostor constraints. We evaluate this sample-restricted quantity on a deterministically selected DigiFace-1M subset under three fixed recognizers with identity-disjoint in-domain threshold calibration.

Existing studies indicate that complex system degradation is characterized by degradation of multiple dependent parameters. Capturing the dependencies is crucial for accurate degradation modeling and effective degradation control. This work aims to uncover these dependencies through causal analysis, focusing on pairwise causal discovery. Firstly, considering the steady-state characteristic of physical dependencies between parameters, a causal discovery strategy using degradation increments is proposed combined with non-temporal causal discovery techniques. Then, five types of non-temporal causal discovery techniques, including constraint-based, score-based, functional causal model-based, gradient-based and the emerging ordering-based technique, are selected as benchmark methods to identify the most suitable approach. Numerical studies based on Wiener process are first conducted to investigate the method effectiveness on both independent and causally dependent degradation paths. Additionally, sensitivity analysis is performed to evaluate how degradation process characteristics affect the accuracy of causal discovery. Then, two engineering applications are given to show the practical applicability of the approach, including a second-order multiple-feedback band pass filter and a turbofan engine. Our findings indicate that the proposed strategy, which uses degradation increments, outperforms methods that rely on raw degradation data. Among all evaluated techniques, stable Peter-Clark and greedy equivalence search exhibit robust and accurate performance across both numerical and engineering cases, which are recommended for causal discovery between degradation paths. The code is available on GitHub: https://github.com/dirge1/causal_deg_data.