Papers

This paper studies information-theoretic inaccuracy measures for record values and their application to testing symmetry. We study the Kerridge inaccuracy measure, cumulative residual inaccuracy, cumulative past inaccuracy, and extropy-based inaccuracy measures to the distributions of nth upper and lower k-record values. Explicit expressions are derived for several common lifetime distributions (exponential, Pareto, Weibull, and uniform), and the monotonic behaviour of these measures with respect to the record order n, the parameter k, and the distributional parameters is analysed. Building on the known characterisation that equality of upper and lower record-based inaccuracy implies symmetry, we develop a nonparametric goodness-of-fit test for symmetry. The test statistic is the difference between the estimated Kerridge inaccuracy measures for upper and lower records, and its null distribution is obtained by a bootstrap procedure that enforces symmetry. A simulation study demonstrates that the test maintains its nominal level for a variety of symmetric distributions and achieves high power against skewed alternatives, while a real-data illustration on annual maximum temperatures confirms its practical usefulness.

Distributionally robust Markov decision processes provide a principled framework for sequential decision making under model uncertainty. We study how many samples are necessary and sufficient to learn an $\varepsilon$-optimal robust policy under the average-reward criterion. A generative model provides samples from the nominal transition kernel, whereas policy performance is evaluated over $(s,a)$-rectangular total-variation uncertainty sets of radius at most $\sigma$. Let $H_0$ and $H_\sigma$ denote the nominal and robust optimal bias spans, respectively. We identify $\sigma H_0$ as the perturbation scale separating high- and low-tolerance regimes. Our matching upper and lower bounds show that, up to logarithmic factors, the minimax total sample complexity is $$ NSA \asymp \frac{SA}{\varepsilon^2}\begin{cases} \min\{H_0,H_\sigma\}, & \varepsilon\gtrsim\sigma H_0,\\ \min\{H_0,H_\sigma\}+\sigma H_\sigma^2, & \varepsilon\lesssim\sigma H_0. \end{cases} $$ Here $S$ and $A$ are the numbers of states and actions, and $N$ is the number of samples per state-action pair. The sample complexity consists of a linear-span term that resembles the nominal AMDP results and a robustness-specific term that appears only in the low-tolerance regime. We attain these rates using reduction-based plug-in procedures that select the reduction—nominal or robust—and its discount factor: a span-informed procedure that makes these choices using known span parameters, and a span-agnostic procedure that calibrates both choices from data.

Random-projection tests for functional data depend on the probability law used to generate projection directions. A measure has to be selected to generate the random directions in which the data is projected. In $L^2$, probability measures defined by their moments can be discriminated using Gaussian measures. The covariance operator of the Gaussian projection law determines which regions and structures of the functional space receive appreciable probability, and consequently affects finite-sample power. We show that mixtures of non-degenerate Gaussian measures preserve the almost-sure separation property and induce a metric between functional probability laws. A concentration argument further makes explicit that the power of projection tests is governed by the integrated projected distance associated with the chosen law. As a concrete construction, we combine Haar and Fourier Gaussian components, which emphasize localized and oscillatory departures, respectively. The resulting permutation test retains consistency, while component labels and upper-tail separation scores provide a descriptive indication of the geometry of the detected discrepancy. Simulations illustrate the specialization of the two components and the robustness of their mixture, and an ECG5000 application identifies a predominantly localized difference between two classes. We also compare with a pooled functional principal component analysis covariance operator. This finite-rank, data-adaptive projection geometry preserves permutation validity through its label-invariant construction and improves power in several simulated settings, particularly under a change in covariance structure.

Multinational HIV cohort studies face regulatory barriers to cross-border sharing of individual participant data, limiting centralized pooled analyses. Federated statistical methods, which exchange only aggregated information, offer a privacy-preserving alternative but have rarely been examined in real-world distributed environments for HIV research. Here, we evaluate the feasibility and analytical performance of a communication-efficient federated framework within the Caribbean, Central, and South America Network for HIV epidemiology. Virologic failure and major regimen change after antiretroviral therapy initiation were analyzed as separate outcomes; for each, we estimated cumulative incidence functions (CIFs) and fit stratified cause-specific Cox proportional hazards models via a surrogate likelihood-based federated implementation. Each site imputed missing data, conducted local analysis, and shared only summary statistics according to a coordinated computation protocol. The federated approach exactly reproduced centralized CIFs and closely approximated centralized Cox regression estimates, outperforming conventional meta-analysis for both outcomes. These findings demonstrate that authentic federated analysis is feasible for multinational HIV research and can yield results closely aligned with centralized analysis while preserving data privacy. Post-hoc feedback from local analysts, however, indicated that broader adoption will require managing the logistical and coordination overhead and ensuring harmonized data collection and quality control across sites.

Latent trajectory analysis is a statistical method for explaining heterogeneity by partitioning patients into homogeneous subgroups based on similarities in outcome variables. In the context of clinical work, patients often do not follow the same course of illness or treatment response, and traditional analyses often average across patients, masking important subgroups. This work proposes a novel latent trajectory model for multivariate longitudinal outcomes with mixed types and uses the expectation-maximization algorithm as an estimation strategy. The proposed model can identify individual-level, time specific latent class memberships and a latent trajectory membership that describes how the latent class memberships change over time. By capturing these dynamic changes, we can highlight patients at higher risk of poor outcomes, reveal early indicators of improvement or decline, and ultimately support more individualized treatment planning. We present an application of our methodology to the Alzheimer's Disease Neuroimaging Initiative (ADNI), a longitudinal, multi-center, observational study to validate biomarkers for Alzheimer disease (AD) clinical trials.

Randomized experiments are often run in one population to guide decisions in another. Allocating by experimental proportions wastes budget on groups that rarely appear in deployment, whereas allocating by deployment proportions under-samples groups that are hard to measure precisely. We propose TWNA (Target-Weighted Neyman Allocation), a two-stage stratified design that uses pilot estimates of group–arm outcome variances to allocate final-stage sample sizes and treatment probabilities for target-weighted group average treatment effect (GATE) precision. The oracle rule has a closed form and balances deployment importance with statistical difficulty; the plug-in rule recovers it as pilot variance estimates stabilize. We also extend TWNA to handle uncertainty about deployment composition, remaining robust whether the target mix is roughly known or entirely unknown. Finally, we distinguish this weight robustness from a pilot-robust variant for skewed, rare-event, or contaminated outcomes. Simulations and real-covariate benchmarks show the largest gains when groups are both deployment-important and difficult to measure.

Latent subgroup analysis is central to fields such as genomics, precision medicine, and social science, where the goal is to identify heterogeneous populations with distinct covariate structures or response behaviors. Mixture models provide a natural probabilistic framework for this task, representing the data-generating distribution as a weighted combination of subgroup-specific laws with unobserved labels. In high-dimensional regimes, these analyses face significant challenges. Often, only a small subset of covariates drives meaningful subgroup separation; the remaining variables may introduce noise or redundancy. Standard clustering methods typically treat all dimensions as equal, but in high-dimensional spaces, irrelevant coordinates can distort distances and obscure the low-dimensional structures defining latent classes. This paper introduces a Bayesian distilled clustering framework for high-dimensional mixture models. We propose that clustering should occur within a statistically justified subspace rather than the full ambient space. Our method utilizes a Bayesian variable selection model to estimate posterior inclusion probabilities, quantifying the evidence that each covariate contributes to subgroup separation or response behavior. A "distilled" covariate set is then identified by controlling the expected false-discovery proportion. Clustering is performed on this reduced subspace, followed by conditional independence diagnostics to examine subgroup-specific dependencies among selected variables. Critically, this framework is model-based: the distillation step is tied directly to the mixture structure and response model rather than a generic dimension-reduction criterion. This ensures the resulting subspace remains aligned with the scientific objective: identifying latent subgroups that differ in both distributional structure and behavior.

Let $H\subseteq\{-1,+1\}^X$ be a class of finite VC dimension $d\ge1$. Writing $L$ for the binary risk and $L^*=\min_{h\in H}L(h)$, we construct a learner achieving the statistically optimal risk bound: from an i.i.d.\ sample of size $n$, for every $0<\delta\le 1/2$, with probability at least $1-\delta$, \[ L(\widehat h) \le L^*+ 7\cdot10^8\left( \sqrt{\frac{L^*(d+\log(1/\delta))}{n}} +\frac{d+\log(1/\delta)}{n} \right). \] This settles the sample complexity of agnostic PAC learning up to universal constants at every fixed $L^*$, matching the lower bounds of Devroye, Györfi, and Lugosi [A Probabilistic Theory of Pattern Recognition, Springer, 1996].

Vector autoregressive moving-average (VARMA) models have long been considered impractical beyond moderate dimensions: the likelihood is non-convex, the parametrization is identified only up to equivalence, and every evaluation costs a pass over the entire series. Yet their moving-average term captures with a few parameters what a pure autoregression matches only with many lags. We introduce an estimation framework that removes this computational barrier: each optimization iteration is independent of the series length $T$. The framework combines a partial-autocorrelation reparametrization that guarantees stationarity and invertibility by construction, Gaussian priors on the reparametrized coefficients with separate scales for diagonal and off-diagonal entries, and losses that depend on the data only through fixed-size sufficient statistics, evaluated by a Parseval (Fourier) identity at near-linear cost in the truncation length. This yields two point estimators: a regularized least-squares fit and a covariance-marginalized maximum-a-posteriori estimator. We prove that both recover the infinite-autoregressive representation of the true process at a near-parametric rate in fixed dimension, so the truncation introduces no asymptotic bias. The same machinery extends, at the same leading cost, to seasonal dynamics, exogenous regressors (VARMAX), and rolling-window refits. Empirically, the estimators stay close to the oracle forecast error from $d=10$ to $d=40$ (where classical conditional MLE returns non-invertible fits whose forecasts diverge) and match or beat VAR, Bayesian-VAR, component-wise ARMA, and sparse-VARMA baselines on retail-demand, meteorological, and air-quality data. This brings likelihood-based VARMA estimation, at a per-iteration cost independent of the series length, to the problem sizes where practitioners have so far relied on VAR models.

A monotone adversary observes an i.i.d. labeled sample and appends a finite number of further examples of its choice, every one of them labeled correctly by the target hypothesis. The learner sees a uniform shuffle of the combined sample and is scored on the original distribution. Every example is correctly labeled, but the insertions depend on the clean sample, so the combined sample is not exchangeable. Larsen, Pabbaraju, and Shetty, who introduced this model, showed that empirical risk minimization attains expected error $O((d/n)\log(n/d))$ for classes of VC dimension $d$, and that every known optimal learner can be pushed away from the $\Theta(d/n)$ rate, optimal for PAC learning. They asked whether the extra logarithm is an artifact of those particular algorithms or an inherent consequence of the lack of exchangeability. We show that this additional cost is inherent beyond VC dimension one. In the worst case over classes of VC dimension $d$ and over known finite insertion budgets, the minimax expected error is $\Theta(1/n)$ at $d=1$ and $\Theta((d/n)\log(n/d))$ for $d\geq 2$. The same rates hold with Littlestone dimension $d_{\mathrm L}$ in place of $d$, so the clean online-to-batch rate $O(d_{\mathrm L}/n)$ is unattainable as well. Thus, somewhat counterintuitively, adding correctly labeled examples can make learning harder by a logarithmic factor, even for classes that admit finite mistake bounds in online learning. The dimension-one upper bound is achieved by a simple improper learner whose analysis adapts the leave-one-out argument underlying the one-inclusion graph. All of our lower bounds are elementary and come from a single construction: an explicit class and prior on which two target hypothesis, which differ a point of nonnegligible mass, produce the same sample.

We study multinomial and biased-progressive No-U-Turn Samplers for strongly log-concave targets satisfying $mI_d\preceq \nabla^2U(x)\preceq LI_d,$ and $\|\nabla^2U(x)-\nabla^2U(y)\|_{\mathrm F}\le \gamma L^{3/2}\|x-y\|$ with \(\kappa\coloneqq L/m\). We introduce profile separation, a sufficient sign condition on the stationary mean U-turn diagnostics, and combine it with diagnostic concentration, leapfrog fidelity, and whole-orbit energy control to show that on a high-probability certification event, every doubling realization reaches a common terminal depth through a genuine U-turn. If \(T_\star\) is the selected physical trajectory length and \(a_\star=\sqrt m\,T_\star\), a terminal-depth transfer argument yields restricted conductance and warm-start mixing without lazifying either kernel. Up to logarithmic warm-start and accuracy factors, the transition bounds are \[ \widetilde O\!\left( 1+a_\star^2\kappa^2(1+\gamma)^{4/3} \right) \quad\text{and}\quad \widetilde O\!\left( 1+a_\star^4\kappa^3(1+\gamma)^2 \right) \] for multinomial and biased-progressive selection, respectively. These transition bounds are unconditional. Gradient-work bounds are deterministic when the maximum-depth cap is comparable to the certified depth and otherwise take cap-aware expected and high-probability forms. The framework recovers the Gaussian dimension dependence under these work-accounting conditions, provides population-profile and exact-diagnostic verification for nonlinear product targets, a near-isotropic specialization of the practical-tree certificate, and quantifies when a fixed post-warmup metric removes linear anisotropy.

The autocorrelation function (ACF) and partial autocorrelation function (PACF) are foundational tools for identifying autoregressive moving-average (ARMA) models, yet they are often introduced in ways that appear disconnected from the regression concepts students already know. This paper develops a unified, regression-based instructional framework for the ACF, PACF, Durbin–Levinson recursion, and one-step-ahead prediction for weakly stationary time series. We show that the ACF is the coefficient from a simple linear regression of a mean-zero stationary process on one of its lagged values, while the PACF is both the coefficient of the newest predictor in an expanding multiple regression and the corresponding partial correlation. Using partial regression, we derive the Durbin-Levinson updates for the newly added coefficient, the existing regression coefficients, and the prediction-error variance from familiar ordinary least-squares principles. Worked MA(1) and AR(1) examples show how the characteristic cutoff and tailing-off patterns of the ACF and PACF emerge naturally from this regression perspective. The same recursive regression coefficients also determine the optimal linear one-step-ahead predictor. The resulting framework provides a coherent instructional pathway from regression to model identification, recursive estimation, and prediction and suggests practical ways to connect introductory regression and time series courses.

We propose a new unit of analysis for longitudinal data: the Latent Memory Table. The scientific contribution is not the encoder. It is that table, treated as a reusable statistical object on the same footing as a matrix of principal-component scores, a table of estimated random effects, or a table of predicted probabilities. We estimate a statistical table that summarizes recent longitudinal history and is intended to be stored, queried, analysed and reused throughout the statistical workflow. A memory operator maps each masked windowed history to a finite-dimensional state; collecting those states with uncertainty yields the Latent Memory Table. Validation is organized around six properties—recoverability, personalization, temporal coherence, interpretability, stability and reusability—summarized by a composite quality index \(Q\); the Transformer, the SoccerMon case study and the simulations exist to argue that this table deserves that status. Classical exponentially weighted moving averages and related short- and long-horizon scalar summaries arise as restricted, typically univariate special cases of the same operator class. A simulation study with known memory mechanisms shows that \(Q\) and rotation-invariant recovery scores discriminate genuine multivariate or personalized memory from negative controls and from misspecified windows, whereas regime classification accuracy alone does not. SoccerMon serves as an empirical case study: a constructed Latent Memory Table attains \(Q\approx 0.73\) versus about \(0.40\) for classical and lagged principal-component baselines, with incremental held-out value for some wellness targets and Procrustes ensembles for row-wise reliability.

This work presents a novel attention-based framework for estimating the Individual Probability of Treatment Benefit (IPTB) in survival analysis contexts. The proposed model, called Surv-IPTB, directly quantifies the probability that a specific patient will experience extended survival time under treatment versus control. We reformulate IPTB estimation as a binary classification problem, leveraging pairwise patient comparisons across treatment and control cohorts. The framework incorporates a principled handling of right-censored observations through imprecise probability representations, where uncertain treatment effects are characterized by interval-valued probabilities. An attention mechanism with learnable query-key transformations enables flexible, data-driven aggregation of pairwise comparisons, while simultaneously learning soft class probabilities for censored cases. Through extensive experiments on synthetic datasets with complex nonlinear structures, including spiral, bell-shaped, and circular feature spaces, we demonstrate that our approach maintains robust performance across varying censoring rates and treatment effect strengths. The model consistently outperforms meta-learner baselines (T-learner and S-learner) equipped with random survival forests, Cox proportional hazards, and Beran estimators, particularly in challenging nonlinear scenarios where conventional methods exhibit significant degradation. The results establish the proposed attention-based framework as a scalable and statistically principled solution for personalized treatment benefit assessment in survival settings. The code implementing the model is publicly available.

We study the problem of sampling from target distributions whose potentials are simultaneously non-smooth, subject to superlinear gradient growth, and non-convex. We introduce the Subgradient Tamed Unadjusted Langevin Algorithm (SG-TULA), a discretisation of the Langevin diffusion that operates directly on subgradients, without relying on computationally demanding smoothing procedures. To handle the superlinear regime, taming techniques are employed to produce a stable, explicit scheme. We derive non-asymptotic convergence bounds in Wasserstein-2 distance, with all constants tracked explicitly in terms of dimension and inverse temperature, improving upon the currently known rates for subgradient-based Langevin algorithms. We further provide excess risk estimates for the associated optimisation problem. We verify the assumptions, with explicit constants, for the regularized pretraining potential of a LLM in the GPT-2 lineage and the boosted coordinate-wise variant of SG-TULA pretrains the former competitively against finetuned AdamW and Muon, for which no comparable non-asymptotic guarantees are presently available.

Persistence diagrams (PDs) provide stable and interpretable summaries of multiscale topological structure. While substantial progress has been made in the statistical analysis of PDs, existing literature often treats diagrams as static objects and provide limited frameworks for probabilistic modeling and stochastic evolution on PD space. We introduce a reinforcement learning framework for stochastic dynamics on PD space, where diagrams evolve through topology aware local edit operations. The dynamics define controlled Markov processes on spaces of finite PDs with variable cardinality. We establish conditions under which the induced Markov chains are irreducible, aperiodic, and geometrically ergodic, implying the existence of unique stationary probability laws on PD space. To guide the dynamics toward scientifically relevant topological targets, we formulate objectives that encompass distribution matching, task specific topological statistics, and structure-preserving compression. The resulting rewards balance task specific distributional targets, diagram fidelity, and complexity reduction, and yield a framework for adaptive topological simplification and probabilistic modeling. Experiments on synthetic and neuroimaging PDs demonstrate that the proposed framework can preserve dominant topological structure while reducing diagram complexity.

Model evaluations may fix all tests before observing any responses or select later tests using earlier responses. We study this choice in a conditional-query model on a finite outcome space $\mathcal{X}$ with $|\mathcal{X}|=N$. We first ask which pairs of distribution classes can be reliably distinguished. We then ask how many additional queries are required to match an adaptive tester when all queried events must be fixed in advance. We show that learnability holds if and only if the two classes have positive separation in their pairwise conditional probabilities. When this separation is zero, the optimal worst-case error is exactly $1/2$ at every finite query budget. For any $T$-query adaptive policy and any $\rho \in (0,1)$, we construct a randomized non-adaptive procedure using $O(N^2(T + \log(1/\rho)))$ pair queries chosen before any response is observed. Its simulated transcript is within $\rho$ in total variation of the adaptive transcript, uniformly over all distributions in the model. We also construct a matching family with constant adaptive query complexity and $\Omega_\varepsilon(N^2)$ non-adaptive query complexity. Consequently, the worst-case fixed-error adaptivity gap is $\Theta_\varepsilon(N^2)$. Thus interaction can reduce the required number of tests by a quadratic factor, but the apparent exponential branching of an interactive evaluation does not yield an exponential query advantage.

In overparameterised classification, training data can be linearly separable even when the underlying distribution is not. In this setting, gradient descent (GD) on the logistic loss diverges in norm while converging in direction to a max-margin interpolating classifier, whose implicit bias can be statistically suboptimal. In this work, we show that early stopping can overcome this suboptimality: in a Gaussian mixture model with label-flipping noise, GD stopped at an appropriate oracle time achieves minimax-optimal excess zero-one risk for covariance spectra with fast and continuous decay, including polynomial and exponential spectral decays. Our analysis combines a sharp upper bound for the early-stopped iterate with a matching statistical lower bound over arbitrary classifiers, yielding optimal rates that are validated by experiments. A central technical contribution is a new calibration result that converts excess logistic risk into excess zero-one risk; it handles the model misspecification induced by the label-flipping noise, and removes the square-root rate in standard bounds. We also establish a lower bound for linear interpolators, showing that interpolation can require exponentially more samples than early stopping to achieve the same excess risk.

A unified inferential framework is developed to address the stress-strength reliability of multicomponent systems under progressive Type II censoring. The maximum likelihood estimate of reliability is obtained using an expectation-maximization algorithm, followed by the determination of the corresponding Fisher information matrix and confidence intervals based on the missing-value principle. To facilitate a comparative inferential assessment, maximum product spacing estimates are also developed. By employing both informative and non-informative prior models, a comprehensive analysis is conducted within a Bayesian framework, and suitable summaries are obtained using the Markov chain Monte Carlo algorithm. The performance of all the estimators is analyzed through an extensive simulation study. Finally, a practical application of the proposed methodology is presented using a reliability data set. Furthermore, we determine optimal progressive censoring strategies using three different optimality measures and discuss their usefulness in reliability studies.

Conformal prediction endows arbitrary black-box predictors with finite-sample, distribution-free marginal coverage, yet marginal validity can hide severe covariate-specific miscalibration, while exact distribution-free conditional coverage is finite-sample unattainable. Randomly localized conformal prediction (RLCP) mitigates this gap by calibrating near the test point while preserving marginal coverage. Existing theory, however, lacks finite-sample guarantees for the realized localized set that jointly control conditional validity and oracle efficiency. We provide such guarantees. For any fixed score, under Hölder regularity of the conditional score CDF and standard density and kernel assumptions, we prove high-probability bounds, uniform over a realized localization neighbourhood, for the conditional-coverage gap and the length error relative to the oracle. The bounds decompose into an $O(h^\beta)$ localization bias and a calibration term decreasing with calibration size, clarifying the bandwidth bias-variance tradeoff and when RLCP tracks the oracle. We also analyze data-split learned scores: when the score targets a pivotal score, as in conformalized quantile regression, uniform local guarantees decompose into fixed-score calibration and uniform score-estimation errors, showing that improved learning sharpens localized guarantees.

Probabilistic Regression Trees (PRTrees) are a smooth and consistent alternative to classical regression trees, producing continuous predictions through probabilistic split assignments. This paper extends the PRTree framework to accommodate missing predictor values directly during tree construction, eliminating the need for prior imputation. Three strategies are proposed, each exploiting the available information differently: a uniform-probability approach, a partial-observation approach, and a dimension-reduced smoothing approach. These modifications are defined to preserve the fundamental probabilistic properties of the original methodology, including probability conservation and marginal compatibility, under arbitrary patterns of missing covariate values. The proposed methods are evaluated on several real-world datasets exhibiting different levels of missingness and are compared with classical regression trees. The results show that the effectiveness of probabilistic tree construction depends strongly on the treatment of missing observations. Across the considered datasets, the fill strategy emerged as the dominant modeling component, often exerting a larger influence on predictive performance than either the smoothing distribution or the proxy-selection criterion. In datasets where a substantial proportion of observations contained missing predictor values, the proposed methods frequently outperformed CART, while maintaining the interpretability and flexibility of tree-based models.

A framework is developed for the inference of dynamics described by a generalized system of ordinary differential equations. A stochastic gradient method is coined that infers dynamics from observed marginal probability density functions using the joint probability density function of the observable and latent variables. Diffusion and other irreversible processes observed in a low-dimensional state can be recast as deterministic, reversible flows in a sufficiently augmented state space, where the joint density satisfies the hyperbolic Liouville equation. The marginal distribution observed is the projection of these hyperbolic dynamics onto the observed coordinates, with the latent components carrying the randomness and memory. This reframing allows inference for irreversible or stochastic dynamics into the recovery of a deterministic Ordinary Differential Equation (ODE) from marginal observations. Instead of solving the high-dimensional Liouville equation for the joint density, the algorithm exploits its characteristic representation. Particles sampled from the initial distribution are transported along characteristic lines. The Eulerian sensitivity with respect to parameters is obtained by sensitivity propagation along the characteristic lines, with a crossed U-statistic producing an unbiased gradient estimator, which enables stochastic gradient descent. Four experiments validate the method: recovery of a three-mode linear system observed through the marginal of a single mode; a nonlinear Gompertz growth model with a hidden mode; a bistable system whose hidden mode turns a unimodal marginal bimodal; and Stokes–Oseen drag law recovery for particles in a cellular flow. Convergence behavior is analyzed across these settings.

Conditional expectation operators (CEOs) and their associated conditional mean embeddings (CMEs) play a central role across applied mathematics and machine learning, appearing in nonparametric regression, Bayesian inverse problems, and Koopman operator theory. A fundamental question is when a CEO maps a function space on $\mathcal{Y}$ into a prescribed function space on $\mathcal{X}$, particularly a reproducing kernel Hilbert space (RKHS). We show that such mapping properties are characterized by the regularity of the Radon–Nikodym density of the conditional law, and establish a simple, verifiable sufficient condition under which the CEO is bounded and Hilbert–Schmidt. For RKHSs norm-equivalent to Sobolev spaces, this condition reduces to Sobolev regularity of the conditional density. The result yields a direct route to validate CME representations and error bounds for Galerkin-type and CME-based estimators. We verify the regularity condition in three settings: nonparametric regression, Bayesian inverse problems, and Koopman operator theory for stochastic dynamical systems. We show in each case that classical regularity results on the underlying probabilistic model imply the required mapping properties. The resulting framework offers a unified perspective on conditional expectation operators across probability, operator theory, kernel methods, and stochastic dynamics.

The extended-onion and C-vine constructions of Lewandowski, Kurowicka and Joe (2009) are standard methods for sampling from the LKJ_n(eta) distribution on correlation matrices. We show that both arise from the simpler row-normalized Bartlett construction associated with the restricted-Wishart representation of Wang, Wu and Chu (2018), which exposes redundancies hidden in standard gamma-based implementations of the classical constructions. Two exact row-wise couplings establish this: the squared norm of the Gaussian vector supplying the onion's direction has exactly the Gamma law required for one component of the Beta radius, and the same vector, with one chi-squared variate, generates the entire row of mutually independent C-vine partial correlations with their required symmetric-Beta laws. We also show that, relative to flat off-diagonal measure, the LKJ family maximizes entropy at fixed expected log-determinant; the dual natural parameter is eta-1. Under gamma-ratio accounting, normalized Bartlett, the onion, and the conventional symmetric-Beta C-vine require n-1, 2(n-1), and n(n-1) Gamma-equivalent calls. Controlled benchmarks confirm a low-dimensional advantage over the onion implementation and a persistent advantage over the C-vine implementations examined; direct Bartlett normalization also avoids subtractive complements, moving the small-eta zero-diagonal threshold from machine-epsilon scale toward the subnormal range. The sampler is valid for every real eta > 0 and requires only standard normal and chi-squared variates.

A sample of parametric Bayesian inference applications from astronomy, cosmology and particle physics is studied, augmented by mock data sets and toy problems. The parameter spaces of these parametric physical models and their posterior distributions from analysing specific data are characterized by (1) the number of model parameters, (2) whether the posterior shape is similar to a Gaussian, (3) whether the posterior has light or heavy tails, (4) how small the posterior is compared to the prior, i.e., how informative the data are, (5) whether some parameters remain unconstrained while others are highly constrained, (6) whether the posterior has multiple, disconnected modes, and (7) whether the inference undergoes phase transitions. These axis define a parameter space of inference problems. We characterize each of the inference problems and observe that inference in astrophysics spans the entire parameter space, from low to high dimensionality, mono- to multi-modal, and a variety of complex distributions that range from uninformative to highly informative. Furthermore, the computational cost of the physical models can range from milliseconds to dozens of seconds. The collated sample of inference problems is proposed as a standard test bed for new samplers. For reproducibility and ease of use, a Docker compute image is provided.

Major depressive disorder (MDD) risk reflects both stable inherited liability and dynamic behavioral patterns, yet these dimensions are rarely examined together using long-term objective data in real-world settings. Here, we integrated genomic, electronic health record (EHR), and longitudinal Fitbit wearable data from 3,030 adults of genetically inferred European ancestry in the All of Us Research Program, 284 of whom developed EHR-recorded incident MDD after a 180-day baseline period. Time-varying Cox models examined associations of MDD polygenic risk scores (PRS), monthly wearable-derived physical activity and sleep features, and interactions between wearable features and MDD PRS with incident MDD. Higher MDD PRS, lower daily steps, lower light and vigorous physical activity, lower sleep efficiency, and greater sleep duration variability were associated with higher risk of EHR-recorded incident MDD. The associations of sedentary time and sleep duration variability with incident MDD differed across MDD PRS levels, with stronger risk associations among participants with higher MDD PRS. PRS-stratified hazard ratio curves further indicated that comparable estimated risk corresponded to more favorable behavioral levels (such as higher daily step counts and more stable sleep) among participants with higher MDD PRS than among those with lower MDD PRS. Sequentially integrating MDD PRS, baseline wearable features, monthly wearable features, and selected interactions between wearable features and MDD PRS increased model discrimination, with the C-index increasing from 0.637 to 0.705. These findings support the complementary value of inherited liability and longitudinal real-world behavioral monitoring for incident MDD risk characterization and may inform future work on genetically informed digital phenotyping for personalized risk monitoring and prevention.

Analyzing the heterogeneity of treatment effects is crucial in personalized medicine to identify which patients will benefit from specific treatments. The performance of a conditional average treatment effects model to guide treatment decisions can be assessed in different ways, with an important one being the model's ability to effectively discriminate between individuals who benefit from the treatment and those who do not. While many methods and algorithms have been proposed to develop conditional average treatment effects models and individualized treatment rules, little is known about the discriminative ability that can be achieved according to the population's underlying distribution of treatment effects. In this work, we computed the discrimination that can be achieved under oracle CATE for a panel of 20 distributions with varying average treatment effects and levels of heterogeneity. The assessment included the following discrimination metrics: the c-statistic for benefit, the concentration of benefit, and the population average prescriptive effect (PAPE). Results showed that the three metrics employed in this study did not require the same levels of treatment effect heterogeneity to lead to high discrimination results. Notably, achieving high c-statistic for benefit and PAPE values required greater heterogeneity than obtaining high concentration of benefit values. The three metrics considered behave very differently across the distributions. For instance, the concentration of benefit can indicate perfect discrimination in settings with negligible treatment-effects heterogeneity.

Among $2^{nR}$ length-$n$ random sequences, one sequence is an outlier whose index is random. Under hypothesis $\mathcal{H}_0$, the components of the outlier are independent and identically distributed (IID) according to $Q_0$, whereas under hypothesis $\mathcal{H}_1$ they are IID according to $Q_1$. The remaining $(2^{nR}-1)$ sequences are mutually independent, independent of the outlier, and IID according to $P$ under both hypotheses. Based on the observation of all $2^{nR}$ sequences, one wishes to decide between $\mathcal{H}_0$ and $\mathcal{H}_1$. Under the constraint that the decision error probability under $\mathcal{H}_0$ must be bounded away from $1$, we determine the fastest exponential decay rate of the decision error probability under $\mathcal{H}_1$.

The experience sampling method (ESM) is a longitudinal research design where participants report their thoughts, emotional states and behaviours multiple times a day. Our work is motivated by such data collected by the GrowIt! app, which was released to investigate daily emotions among adolescents during the COVID-19 pandemic. Current procedures to analyse ESM data face various challenges. While standard statistical techniques may not scale well to a high-dimensional setting, machine learning procedures can give biased results due to selection bias introduced by missingness. In our motivating dataset, adolescents dropped out due to previous strong feelings of negative emotions. Hence, the implied missing data are of the missing-at-random type that standard machine learning procedures cannot accommodate. We develop a novel neural network architecture that generalises mixed effects models to deep learning to overcome these challenges. It allows semi-parametric and flexible modelling of data's mean and correlation structure through fixed and random effects. For estimation, we use an adaptation of variational auto-encoders and a Bayesian data augmentation algorithm. Through this approach, the model can accommodate longitudinal outcomes following generic distributions, scale well to high-dimensional settings and provide valid inference when data are missing-at-random. We applied the Deep Generalised Mixed Model to the GrowIt! study and various simulations. The results show potential for the Deep Generalised Mixed Model, yet suboptimal performance due to model instability.

A polynomial structural regression model is studied, where the covariate is observed with a mixture of the classical and Berkson measurement errors. Both variances of the classical and Berkson errors, as well as some of their higher moments are assumed to be known. Without normality assumptions, consistent estimators of model parameters are constructed using the Corrected Score method, and conditions for their asymptotic normality are given. Under mild conditions, we found pairs of asymptotically independent estimators. A simulation study illustrates the results.