Papers

A multivariate transformation of the unit cube with component transformations that are piecewise continuously differentiable and uniform distribution preserving (udp) is considered. A stochastic inverse transformation is defined using randomization to overcome the non-injective nature of the udp transformations. The inverse transformation preserves the uniform margins of a random vector distributed according to a copula and yields different copulas for different randomizations. A copula density transformation result for the multivariate stochastic inverse is proved and illustrated in the bivariate case.

In partially exploratory factor analysis (PEFA), the loading structure and factor numbers are weakly specified. The regularized variational approximation for partially confirmatory factor analysis (PCFA VA) recovers this structure via Bayesian variable selection, using spike and slab priors to assign inclusion probabilities to unspecified loadings. This research introduces a post selection assessment framework for this approach. We convert converged solutions into covariance models using either hard selection (thresholding probabilities into a sparse pattern) or soft selection (retaining them as weights for effective parameter counts). We derive the resulting degrees of freedom, absolute fit diagnostics (RMSEA, SRMR, CFI, TLI), and relative criteria (AIC, BIC, ELBO). To determine factor numbers, we propose a scale free gain rule with a sustained drop guard. Simulations show absolute indices successfully track loading recovery and flag under factoring. While raw criteria over factor, our gain rule accurately recovers true dimensionality, with the ELBO variant proving most robust. Finally, a 100 item PID 5 example demonstrates that our model fits better than a confirmatory 25 facet model and concordantly recovers major structures across disjoint specifications.

Newly developed items must ordinarily be field tested before their psychometric properties are known, creating a cold start problem for item calibration. Predicting item parameters from features is a long standing measurement problem dating back to the Linear Logistic Test Model; modern text embeddings now automate the design matrices traditionally specified by hand. We propose an evaluation framework combining regularized regression on item text embeddings, repeated cross validated R squared reported with its resampling standard deviation, and two performance upper bounds: a reliability ceiling derived from parameter standard errors, and a design ceiling derived from simulation based power calibration. Applying this framework to a mathematics item bank (EEDI) and a medical licensure benchmark (BEA 2024), we find that item difficulty is highly predictable from text (repeated cross validated R squared = 0.53, or about 57% of its reliability ceiling), whereas discrimination and pseudo guessing appear less predictable. However, evaluating these results against our ceilings reveals that this apparent hierarchy stems from target reliability rather than text signal strength: text uniformly recovers 57 to 63% of the reliable variance across difficulty targets, whereas the 3PL pseudo guessing parameter has a reliability ceiling near zero, making it an unviable target at current precision. On BEA, embedding based regression matches leaderboard RMSE despite explaining almost no variance, highlighting the critical need for scale free metrics and explicit ceilings in benchmarking. Finally, we show that a single train and test split can inflate apparent accuracy by 0.1 to 0.15 in R squared, underscoring the necessity of repeated cross validation for calibration support applications and future benchmark construction.

Purpose. Confounding adjustment in health-care database studies screens large proxy libraries where events per variable are low, straining standard propensity score (PS) methods, and many regularized variable-selection strategies exist (outcome-adaptive LASSO [OAL], group LASSO/GLiDeR, highly adaptive LASSO [HAL]). Yet few comparisons have varied exposure prevalence within such a selection menu, and none pairs it with doubly robust estimation, compute accounting, and a null-RD truth anchor. Methods. We conducted a plasmode simulation anchored on National Health and Nutrition Examination Survey data (2013-2018; 25 investigator-specified covariates, 142 prescription-derived proxies), comparing ten pipelines combining these strategies with inverse probability of treatment weighting (IPTW) and targeted maximum likelihood estimation (TMLE). Three scenarios were evaluated under a known null (true risk difference, RD = 0): frequent, rare-exposure, and rare-outcome. We report bias, standard error (SE), relative error, 95% coverage, and runtime. Results. HAL (G-Computation) had near-zero bias but highly concentrated estimates, giving near-unity coverage and large relative error (106-186%). OAL (IPTW), GLiDeR, and HAL (TMLE) were best calibrated, whereas the regularized-LASSO TMLE pipelines under-covered modestly (91-93%) in the rare scenarios. Under rare exposure, LASSO-IPTW had the largest bias and inflated SE and over-covered (conservatively), problems that TMLE removed. On real data, methods agreed (RD approximately 0.07-0.085). Runtimes spanned <1 s to >16 h. Conclusions. Under a null benchmark, pairing outcome-aware selection (OAL, GLiDeR) or doubly robust estimation (TMLE) with regularized models best balanced bias, calibration, and robustness to rarity. The rare-exposure arm exposed the largest gaps; method choice should weigh the prioritized metric against compute.

Spectral positional encodings (PEs) for directed graphs face two obstacles: magnetic Laplacians require an $O(n^3)$ Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures. We propose learnable spectral PEs of the form $h_\theta(A_q)\,R$, where $A_q$ is a normalized magnetic operator, $h_\theta$ a learnable scalar spectral response, and $R$ a block of random probes. Because the PE is a matrix function of the operator, it is gauge-invariant by construction. We compute it in a Hermitian block Krylov subspace from sparse matrix–vector products only, prove that $k = O(\log(1/\varepsilon))$ block steps suffice uniformly over heat–resolvent response families, and give a covering-number argument for why low-dimensional structured families generalize where free per-eigenvalue weights overfit. On a directed SBM whose symmetrization is uninformative by construction, direction-blind PEs stay at chance while magnetic Krylov PEs converge to the exact-eigendecomposition oracle as the depth grows. The same probes yield gauge-invariant pairwise features with $1/\sqrt{s}$ Monte-Carlo error, and the undirected $q{=}0$ case improves heterophilous benchmarks over no-PE and polynomial baselines.

We study a random compositional model for the growth of affine regions in deep piecewise-linear networks. The model is generated by i.i.d.\ perturbations of the symmetric height-one tent map, and the main observable is the number \(N_n\) of affine pieces after \(n\) layers. We prove the existence of a submultiplicative pressure for \(N_n\), yielding exponential upper bounds for both tails of \(n^{-1}\log N_n\). The same argument applies to abstract submultiplicative complexity observables and gives higher-dimensional extensions for convex-polytopal affine-cover counts and worst-line affine-piece counts. Since the true branch count has no matching supermultiplicative inequality, lower bounds require a separate certified construction. We introduce a finite-state defect process that records branches whose future splitting can be guaranteed, and use bridge words to obtain constructive upper-tail lower bounds. In a uniformly favorable small-noise regime, this process is governed by a companion matrix whose Perron root tends to \(2\), implying eventual exclusion of lower tails below \(\log 2-\xi\).

A common method for the representation and analysis of time-series data is the hidden Markov model (HMM), where each observation is associated with a hidden state that evolves over time. However, many real-world systems are influenced by multiple independent factors, which are more naturally represented by factorial hidden Markov models (fHMM), where several hidden Markov chains jointly generate the observed data. Although an fHMM provides a richer and more realistic representation of many real-world systems, it can be reformulated as an equivalent HMM, but with a significantly larger state-space, leading to a severe increase in computational cost. In particular, the forward filtering algorithm, which is central to evaluation, decoding, and estimation tasks, becomes prohibitively expensive even for small systems. This work focuses on developing scalable methods for time-series analysis using tensor algebra to exploit the multidimensional structure of fHMM directly, without constructing intermediate HMM representations. Our novel filtering approach significantly improves computational performance and enables the efficient analysis of large systems and datasets, extending the scope of fHMM and providing a practical framework for data intensive applications.

In this article, we investigate the cumulative residual extropy associated with linear consecutive k-out-of-n:G systems, which play an important role in reliability theory and engineering applications. We first derive explicit expressions for the proposed measure and examine the behavior of cumulative residual extropy under a variety of stochastic orderings. In addition, several bounds and meaningful results of the characterization are established. Moreover, we also introduced the dynamic version of the cumulative residual extropy and explored the relationship between the proposed dynamic version and the mean residual life function. Fromaninferentialperspective, wedevelopanonparametricestimationprocedure for the cumulative residual extropy and establish the corresponding consistency properties of the estimator. The finite-sample performance of the proposed estimator is further investigated through extensive Monte Carlo simulation studies under different parametric settings and validated through a real dataset.

Cryptocurrencies have experienced repeated large-scale losses from protocol exploits and exchange breaches, exposing insurers and investors to severe operational risks. This paper develops an equilibrium pricing framework for catastrophe bonds tailored to the cryptocurrency ecosystem. We introduce a double-trigger structure that jointly captures short-term catastrophic shocks and longer-term systemic deterioration. To model the multi-risk environment, we incorporate dual dependence, combining dependence across triggers with multivariate dependence among financial risk factors through vine copulas. Beyond expected prices, we characterize the full distribution of discounted cash flows and return rates, enabling risk-sensitive metrics such as Value-at-Risk and Tail Value-at-Risk. Furthermore, we propose an on-chain settlement architecture where calibrated payout functions are embedded directly into smart contracts. This design eliminates basis risk associated with settlement delays and minimizes the agency costs inherent in traditional intermediation. Our results demonstrate that multi-trigger crypto CAT bonds offer a statistically robust and economically efficient vehicle for transferring systemic digital asset risks to capital markets.

The problem of signal detection under an unknown background can be framed as one of inferring the weight of a mixture model with one misspecified component. Banerjee and Algeri (2026) show that, for this problem, the conservativeness of the inference is entirely determined by one single parameter, called the compensator. They demonstrate that, when the data are independent and identically distributed, an inferential approach based on the compensator circumvents the need to estimate the density of the misspecified component and the associated challenges. The main purpose of this manuscript is to broaden the scope of such an approach and extend it to the case in which, as is often encountered in modern experiments in physics and astronomy, the data consist of Poisson counts observed over a large number of bins.

This paper studies transfer learning for linear discriminant analysis in high-dimensional two-class classification. We consider one target domain and several source domains, where the mean difference in each domain is decomposed into a deterministic common component and a domain-specific random deviation. The common component represents a shared classification signal across domains, while the random deviation captures domain-specific heterogeneity. Under spiked covariance models, we derive deterministic limits for the target-domain Gaussian-calibrated error of weighted transfer classifiers under both homogeneous and heterogeneous covariance settings. These limits quantify the effects of the shared signal, domain-specific variation, dimension-to-sample-size ratios, and spike structures on transfer performance. They further lead to oracle transfer weights and consistent data-driven plug-in estimators. We also characterize the intercept bias induced by unbalanced target-domain class sample sizes and provide an asymptotically optimal correction.

Reinforcement learning (RL) is increasingly grounded in tools from probability, optimization, and operator theory. This survey organizes the mathematical structures that underpin the design and analysis of modern algorithms in RL. We begin from Markov decision processes (MDPs) and the Bellman operators, emphasizing contraction mappings, monotonicity, and fixed-point theory that yield convergence guarantees and rates for value and policy iteration, and temporal-difference schemes. We then develop the optimization perspective: stochastic approximation and martingale methods, convex duality and the role of regularization linking mirror/proximal methods. Function approximation is treated through linear and non-linear settings, covering stabilization, error decomposition, and sample-complexity via concentration inequalities for dependent data and mixing processes. We further cover off-policy evaluation/learning, constrained RL and constrained MDPs (CMDPs). Throughout we unify algorithmic templates under common operator and variational lenses, highlighting both finite-sample bounds and asymptotic results. Our presentation is intended to provide a unified mathematical entry point for researchers in probability, optimization, and statistics interested in reinforcement learning.

Cardiovascular mortality is shaped by interacting demographic, environmental, and structural processes operating across multiple spatial scales. Conventional epidemiologic analyses often rely on aggregate summaries or single-model formulations that obscure hierarchical variation and contextual heterogeneity. We present a reproducible multilevel statistical inference framework integrating Normal (age-adjusted), Poisson (count-based), and population-offset Poisson models to quantify cardiovascular mortality across nested geographic units while separating demographic effects from structural variation. The framework was applied to county-level mortality data from Ohio and Pennsylvania (1999-2020) using MLwiN hierarchical models for seven cardiovascular disease (CVD) subtypes. Fixed effects included year, sex, race, PM2.5, and O3, while county-level random intercepts captured spatial heterogeneity. Complete model equations are provided in the Supplementary Material. The framework reveals complementary perspectives on cardiovascular risk unavailable from a single model. Age-adjusted mortality declined more rapidly in Pennsylvania than Ohio, whereas Poisson models identified post-2010 stagnation or reversal for several CVD subtypes. Black populations experienced elevated mortality risks, males exhibited higher mortality than females, and PM2.5 showed stronger associations with ischemic and hypertensive mortality in Pennsylvania. Population-offset models reduced unexplained variance while preserving county-level structural disparities. Beyond cardiovascular epidemiology, this work introduces a generalizable hierarchical statistical framework for structurally nested health systems. The methodology provides a scalable foundation for disease surveillance, environmental health assessment, health equity research, reproducible statistical analysis, and AI-assisted scientific inference.

We present a framework for causal inference in behavioral health case studies – and observational N=1 settings more generally – under unmeasured confounding. The framework rests on a class of causal estimands, termed $\Omega$ estimands, defined as contrasts of functions of an outcome variable's support rather than its distribution. Because such estimands do not depend on how probability is distributed over supports, they are insensitive to the confounding that limits other methods. We prove that, in a structural causal model, the observational and interventional supports of an outcome coincide under a single assumption – positivity – without any requirement that confounders be known, measured, or adjusted for. Two optional conditions extend the framework: one licensing a client's recalled baseline as a stand-in for sparsely measured baseline periods, and one connecting support contrasts to conventional mean contrasts through an expected-average identity. We adopt a subjectivist (de Finetti) interpretation of probability and situate the framework within mandates for measurement-based care. A case study of cognitive behavioral therapy for anxiety illustrates an elementary approach a provider can use.

High-stakes computerized adaptive tests (CATs) must continually calibrate new items in their item bank. When an item is new, few responses are available, so item parameter estimates – and thus test scores – are poor. Item features and explanatory item response theory (IRT) models mitigate this by folding item content into calibration. Neural IRT models, whose item parameters are neural-net outputs, are powerful, but tuning hyperparameters and architectures in real time while a CAT is scoring is impractical and threatens validity. We propose a pre-launch step that fits a neural net to produce low-dimensional item embeddings, so the production system can use a simple linear explanatory IRT model on top of them. We use a neural parameterization of the 3-parameter logistic (3PL) model in which a feature network maps each item's content features to a representation $h_j = z(x_j) \in \mathbb{R}^d$, from which the discrimination and difficulty $(a,b)$ follow generalized linear forms; the guessing parameter $c$ is fixed to a global constant to avoid identifiability issues. The feature network and latent abilities $\theta$ are fit jointly via Monte Carlo Expectation-Maximization (MCEM), with no separate ability-estimation or pre-calibration stage. Using an item-split protocol that holds out entire items to simulate feature-only evaluation, we apply this to two Duolingo English Test practice task types – yes/no vocabulary and vocabulary-in-context – searching over feature sets, architectures, and dimensions $d$. A shallow two-layer ReLU network with $d=6$ and hand-engineered scalar features matches or beats larger architectures on held-out items for both. This is a first step toward a compact, content-derived item embedding for the Scalable Parametric Item Calibration Engine (SPICE), the fully Bayesian engine at the core of the S2A3 adaptive-testing system.

We consider the problem of finding stationary points for stochastic convex optimization problems. Rather than surrogates to stationarity, such as a proximity-to-stationarity guarantee or small gradient of the Moreau envelope, we ask for a stronger notion: that the subdifferential of the objective actually contains a small element. This criterion is non-trivial, because subdifferentials of convex functions fail to converge uniformly, even in arbitrarily small neighborhoods of the optimum. Our convergence guarantees rely on dimension theory to decompose the graph of the subdifferential of a convex function, showing how stochastic sampling preserves "pieces" of these graphs, and allowing effective application of proximal-point-like methods.

Best-arm identification is a canonical model for data-driven decision-making, but in many applications each reward observation is costly. Motivated by the growing availability of cheap predictions from machine learning and large language models, we study fixed-confidence best-arm identification in which each costly reward pull is paired with a cheap but correlated proxy score. The marginal mean of the proxy can be estimated offline and is treated as known, whereas its correlation $\rho$ with the reward, which governs how much the proxy helps, is unknown and must be learned online in pair with real rewards. We show that a control-variate adjustment turns this model into a heteroscedastic identification problem whose oracle sample complexity improves by residual variance $1-\rho^2$. The central difficulty is that the correlation must be learned from the same costly samples that identification consumes online, and that a plug-in estimate of the residual variance is anti-conservative and can compromise correctness. We propose PROBE (PRoxy OLS for Best-arm Exploration), a phase-elimination algorithm that directly maintains an upper certificate on the residual variance with an ordinary least squares fit, whose exact chi-square law keeps the certificate valid regardless of the unknown correlation. We prove that PROBE is $\delta$-PAC and attains the known-correlation oracle sample complexity up to a constant multiplicative factor and a constant additive calibration cost. The guarantee extends to the $(\epsilon,\delta)$-PAC setting under minimal changes to the algorithm. Numerical experiments on synthetic instances and on an auto-loan pricing replay with large language model and tabular proxies confirm that the sample savings of PROBE scale with the strength of the reward-proxy correlation, exactly as the theory predicts.

We prove a Gaussian comparison inequality for closed convex sets with reference probability at least 1/2. For centered Gaussian vectors whose covariance matrices are ordered in the Loewner sense, the smaller covariance law assigns at least as much probability as the larger covariance law to every closed convex set with measure at least 1/2 under the larger covariance. More generally, we show that convolving a nondegenerate Gaussian law with any independent centrally symmetric distribution cannot increase the probability of a closed convex set when the convolved probability is at least 1/2. This provides a one-sided analogue of Anderson's Theorem for Gaussian measures. As a statistical application, the result justifies one-sided and order-restricted inference using conservative covariance estimators at significance levels below 1/2 for test statistics whose acceptance regions are closed and convex but not necessarily symmetric.

We consider observables $X$ whose realizations in a sampled dataset are restricted, for example by measurement resolution, to a finite set of distinguishable categories within their possibly infinite theoretical domain. Given the probabilities of observable categories, we study the coarse-grained probability mass of families of possible datasets generated through a forward sampling process. The combinatorial construction induces an intrinsically discrete $p$-value defined directly from the sampling process rather than through additional probabilistic structure on observables. Specifying the sample means of $d+k$ arbitrary functions $g_\alpha(X)$ defines a linear family of datasets whose probability mass is obtained as a weighted sum over integer lattice points contained within the associated polyhedron. To overcome the intractable large-$N$ combinatorics, we derive via saddle-point techniques a density approximating these probability masses in the continuum limit of forward sampling within the multinomial universality class. As a demonstration, we consider conditional sampling, where $d$ structural means are fixed while $k$ means vary over admissible datasets. The information geometry emerging from the saddle-point density, together with the spherical symmetry arising at large $N$ from the intrinsic $p$-value construction, enables efficient computation of the $p$-value in the Laplace approximation via the $\chi^2_k$ distribution. The resulting statistic is given by the semi-analytic expression $2N$ times the Kullback-Leibler divergence between the information projections associated with the corresponding structural and observed linear families. These projections can be computed efficiently via standard numerical routines converging for sufficiently well-behaved sample means.

Small-area estimation (SAE) enables researchers and policymakers to identify spatial disparities in health outcomes, but survey-based SAE products carry an inherent lag. Gold-standard estimates such as CDC PLACES are released roughly two years after the underlying survey data are collected, limiting their use for time-sensitive decision-making. This study evaluates the potential for machine learning (ML) to serve as a surrogate, learning the relationship between frequently updated area-level predictors and existing SAE outputs to generate timely, comparable estimates in years when SAE from surveys are unavailable or delayed. We evaluate several global and geographically weighted ML models for county-level SAE of ten chronic conditions across the US: COPD, asthma, heart disease, arthritis, cancer, depression, diabetes, high blood pressure, high cholesterol, and stroke. Our findings suggest that geographically weighted ML frameworks like geographically weighted random forest and geographically weighted regression offer scalable and open data surrogates for rapidly generating SAE and supporting data driven decision making.

External and synthetic control arms (ECAs) are entering psychiatric drug development, but the field lacks a benchmark that evaluates the properties regulators care about: not only how accurately a method reconstructs untreated trajectories, but whether its uncertainty is calibrated, whether it is robust to the informative observation times common in mental-health records (sicker patients are seen more often), and what false-positive rate it induces in go/no-go trial decisions. Real psychiatric trial data (e.g. STAR-D and registry cohorts) require credentialed access and lack ground-truth counterfactuals, so, following established semi-synthetic benchmarks in causal inference (IHDP, ACIC, and the PK-PD tumor-growth simulator), we release Psych-ECA, a fully reproducible generator of longitudinal symptom trajectories for depression (PHQ-9), anxiety (HAM-A), and psychosis (PANSS) with known counterfactual control arms, informative visits, and validated-scale measurement noise. We benchmark eight estimators spanning carry-forward, pooled real-world-data averages, nearest-neighbour matching, linear mixed models, gradient boosting, and the Scribe trajectory-bridge method. Three findings emerge. First, trajectory and flexible machine learning methods achieve the best counterfactual accuracy (about 2.3 PHQ-9 RMSE), outperforming cross-sectional baselines. Second, only Scribe is both accurate and calibrated, achieving 93-96% empirical coverage of nominal 90% prediction intervals, compared with 87-88% for gradient boosting and 62-75% for uncalibrated SDE models. Third, inverse-intensity correction reduces bias under informative sampling, while Scribe's calibrated intervals are the only trajectory method that maintains nominal false-positive rates as informativeness increases. We release all code, data-generation scripts, and random seeds to enable fully reproducible evaluation.

Sepsis is a leading cause of mortality, yet optimal treatment policies remain contested. Existing reinforcement learning (RL) approaches learn fixed strategies for sepsis treatment, limiting adaptability to changing clinical objectives during inference. We propose EHRMPC, a framework that decouples learning patient dynamics from optimizing treatment by training a patient digital twin in the form of a generative electronic health record (EHR) model. The digital twin predicts clinical trajectories under interventions and enables model predictive control (MPC) to optimize treatments via inference-time planning over simulations. We evaluate EHR-MPC on a multicenter ICU sepsis cohort spanning 8 hospitals in the Mass General Brigham health system using both off-policy importance sampling and on-policy simulation-based evaluation. Relative to RL baselines, EHR-MPC achieves comparable off-policy performance and improved simulation performance. Unlike RL, this work frames sepsis treatment optimization as inference-time control over learned patient dynamics, establishing a general framework for decision making with generative clinical models.

Sparse precision matrix estimation provides an interpretable and computationally efficient framework for modeling conditional dependencies in high-dimensional, low-sample-size data. A recurring challenge is appropriately selecting the regularization parameter that controls estimator sparsity and strikes a balance between underfitting and overfitting. We propose a closed-form, matrix-valued regularization parameter derived from the sampling distribution of the first-order optimality conditions of the $\ell_1$-regularized Gaussian maximum-likelihood estimator. By prescribing the probability that each nonzero entry of the estimator satisfies its optimality condition under resampling, we eliminate the need for cross-validation. The resulting regularization parameter is shown to attain asymptotic scaling properties that, under standard conditions, provide consistency and sparsistency of the estimator. On synthetic Gaussian and non-Gaussian datasets, as well as real-world gene microarray and neuroimaging applications, the proposed approach achieves estimation accuracy comparable to cross-validation, delivers superior support recovery, and reduces runtime by several orders of magnitude.

Diffusion models offer a powerful framework for sampling from complex probability densities by learning to reverse a noising process. A common approach involves solving for the time-reversed stochastic differential equation (SDE), which requires the score function of the evolving sample distribution. The logarithm of this distribution's density is governed by a Hamilton-Jacobi-Bellman (HJB) type partial differential equation (PDE). However, current methods for solving this PDE, such as PINNs or trajectory-based techniques, often suffer from long training times and significant sensitivity to hyperparameter tuning. In this work, we introduce a novel and efficient solver for the underlying HJB equation based on the functional tensor train (FTT) format. The FTT representation leverages latent low-rank structures to efficiently approximate high-dimensional functions, enabling both model compression and rapid computation. By integrating this efficient representation with a backward-in-time iterative scheme derived from backward stochastic differential equations (BSDEs), we develop a fast, robust and accurate sampling method. Our approach overcomes primary bottlenecks of existing techniques, enabling high-fidelity sampling from challenging target distributions with improved efficiency.

We study exact support recovery by best subset selection in linear regression under a fixed design. For a known support size $s$ and ambient dimension $d$, we derive a non-asymptotic upper bound on the probability that best subset selection fails to recover the true support. The bound is expressed through a deterministic subset-separability parameter, which measures how well the true support can be distinguished from competing supports after projection. The result holds for all sample sizes $n$ exceeding a certain sufficient threshold which we state explicitly in terms of the signal-to-noise ratio, the subset separability of the realized design, and a logarithmic factor of order $\ln s + \ln(d - s)$. In contrast to random-design analyses, no full log-combinatorial term over the candidate support class appears. We discuss how such terms may reappear when the design is random and the separability parameter must be controlled uniformly over many competing subsets. The fixed-design formulation and the proof strategy also indicate settings in which the effective complexity of best subset selection may be reduced, for instance, under structured designs or restricted candidate subset classes.

We investigate when multivariate kriging improves on separate prediction, using separable multi-output Gaussian processes as the principal theoretical benchmark. We introduce directed coverage, directed proximity and borrowing potential indices to describe design complementarity beyond exact overlap without response values. Under common radial kernels, we derive finite-design bounds linking geometry to prediction gains with known covariance parameters and information for estimating cross-output dependence. An exact risk decomposition under correct specification separates oracle gains from differences in estimation penalties; we give a mean-square expansion under explicit conditions. We distinguish parameter sharing from response coupling: auxiliary data can improve prediction through shared-scale estimation even when direct borrowing is negligible. This distinction guides a calibrated screening-and-validation procedure comparing separate marginal models, a shared-scale independent model and a coupled model, retaining the shared-scale option when coupling is excluded. In simulations with sparse target data and auxiliary observations retained during validation, the procedure reduces mean latent squared-error risk by 6.66% relative to separate fitting. After calibration, screening reduces runtime by 14.01% relative to direct validation of the same candidates, with similar mean risk. Controlled experiments show that covariance misspecification can reverse fitted gains, while queueing and monitoring applications demonstrate the importance of auxiliary coverage and the prediction task. These findings support task-matched validation and separate assessments of predictive accuracy, uncertainty calibration and computational cost.

We show fast mixing of Glauber dynamics for certain quadratic Gibbs measures with large external fields. The main ingredient is an overlap condition that allows us to control correlation matrices uniformly over all pinnings, by controlling norms of small submatrices of the interaction matrix. Using stochastic localization, we then obtain a lower bound on the spectral gap and, consequently, polynomial-time mixing of Glauber dynamics. As a direct application, we consider the Sherrington-Kirkpatrick model, whose interaction matrix is a scaled GOE matrix. For this model, we show that for any fixed finite inverse temperature $\beta$, there exists a strength of external field $\theta$, not depending on the size of the system, for which Glauber dynamics mixes in polynomial time (with high probability on the draw of the interaction matrix).

Drought monitoring across heterogeneous climate regions requires explicit modeling of drought persistence rather than relying solely on index classification. We develop a Bayesian spatial, duration-augmented framework for drought persistence that captures both temporal memory and spatial dependence. Because persistence estimates depend on the underlying drought index, we also assess the influence of (i) reference evapotranspiration (RET) formulations and (ii) the choice of probability distribution used to standardize the climatic water balance. The framework is applied to a high-resolution (0.25$^\circ \times$ 0.25$^\circ$) dataset over 730 grid cells across Italy. The choice of RET formulation (Penman-Monteith, Hargraves, Thornthwaite) has only a marginal influence on the overall drought signal: spatially averaged SPEI series at the 3-, 6-, and 12-month scales are identical across methods. By contrast, the proposed bulk-and-tails (BATs) distribution substantially improves index estimation, consistently outperforming commonly used alternative models across all RET methods and timescales and achieving normality acceptance rates of 98.4%–99.9% compared to 37%–92% for the generalized extreme value (GEV) distribution and 0%–30% for others. The spatial duration-augmented model further outperforms a nonspatial counterpart, producing spatially continuous, uncertainty-aware recovery and survival fields. Results reveal a strong south-north gradient in temporal memory: southern peninsular and island regions (Sicily, Sardinia) exhibit pronounced persistence and duration-dependent recovery hazards, whereas northern and Alpine regions show near-zero duration dependence. Drought characterization is more sensitive to the distributional and persistence assumptions than to the RET formulation, and duration-insensitive classifications systematically understate extended drought risk in southern Italy.

High dimensional biomedical data often exhibit nonlinear, heterogeneous, and manifold driven structures that challenge global parametric and tree-based models. We propose MAPLE (mapper-based Adaptive Prediction via Local Estimation), a localized prediction framework grounded in topological data analysis. The method is formulated as a nonparametric estimator of conditional class probabilities that adapts to the intrinsic geometry of the predictor space. Neighborhoods are defined through connectivity in a data-adaptive Mapper graph, enabling localized averaging within graph induced regions that capture complex structures such as branching and multi-scale heterogeneity. We introduce a statistically principled, data driven procedure for cover selection based on a bias-variance trade off, yielding optimal asymptotic scaling for interval widths and overlaps. The framework accommodates binary, nominal, and ordinal outcomes and incorporates a permutation-based variable importance measure to quantify covariate contributions in prediction. We establish theoretical guarantees, including pointwise consistency and Bayes risk consistency under standard regularity conditions. Simulations show that MAPLE consistently outperforms or matches multinomial regression, ordinal regression, and random forest, with the largest gains observed under heterogeneous and high-noise settings. Applications to Parkinson's disease progression (PPMI) and glioma classification (TCGA RNA sequencing) demonstrate strong predictive accuracy and interpretable, topology-aware summaries of underlying data structure.

Repeated directional responses may vary between individuals in both overall orientation and sensitivity to explanatory cues. We study homogeneous angular regression with correlated Gaussian cue coefficients and a circular random intercept, separating observation concentration from resultant length. Constructive identification conditions recover the joint coefficient law, intercept concentration and observation-error parameters, including unknown uniform contamination. The argument combines repeated contrasts, an arc-support restriction and joint Fourier deconvolution, and extends to exogenous continuous designs. Integration in coefficient coordinates yields smooth population scores even when resultant cancellation prevents a regular latent mode. A fixed-proposal adaptive importance algorithm evaluates the marginal likelihood and its derivatives. In 8,000 independent simulated datasets, 7,998 fits pass numerical checks, and 12,799 of 12,800 local profile intervals are available. Correlation profile-set coverage nevertheless ranges from 91.9% to 94.8% at the nominal 95% level. Greater precision and numerical availability therefore do not ensure calibrated inference. An exploratory extension to 150 Canadian weather stations is evaluated on 259,807 three-hour forecasts from a new 2026 period after fitting on 2023-2025 data. With circular station rotations in all matched models, independent station slopes give modest predictive gains over slopes shared within regions, in all five regions. Estimating their correlations adds no useful predictive gain; this does not assess parameter uncertainty, and predictive calibration remains imperfect. The application uses a noncancelling mean and Laplace integration, so it illustrates the model class rather than testing the proposed algorithm.