Papers

Object-valued time series, including distributions, covariance matrices, networks, and compositions, lack the linear structure required by conventional autocovariance-based definitions of long memory. We develop an intrinsic framework for defining and estimating long memory in metric spaces of negative type. An isometric embedding yields a centered Hilbert-valued process, and the trace of its lag-covariance operator provides a signed, additive measure of temporal dependence. Crucially, this trace equals the difference between the marginal mean pairwise distance and expected lagged distance, so the framework and its estimators use only distances between the original objects. We define the memory parameter through the nonsummable decay of this trace and show that it coincides with the usual parameter in Hilbert-valued settings. Estimation uses Bartlett aggregates of distance-based lag measures. Estimating the common marginal distance from the same dependent sample induces a common-centering bias in these aggregates. We derive its finite-sample form and propose iterated block-difference corrections, log-ratio and multi-bandwidth log-slope estimators, and a localized self-consistency refinement. We establish consistency, document substantial bias reduction in simulations, and find long-memory evidence in foreign-exchange return distributions and U.S. electricity-generation compositions.

LLM judges evaluate candidate language models through pairwise comparisons of their responses. Judge-aware Bradley–Terry models aggregate pairwise comparisons by jointly estimating candidate scores and judge-specific sensitivities (how strongly each judge responds to quality differences), often interpreted as reliability. We show that this reading fails when judges prefer a display position. Even under balanced display orders or swap-and-average, omitting position attenuates fitted judge sensitivity by approximately the factor $1-D^2$ near equal quality, where $D$ is the judge's net first-slot preference between equally good responses. We propose OSCAR, which jointly estimates candidate scores, judge sensitivities and judge-specific position, length and family effects, and provides confidence intervals for pairwise win probabilities that account for dependence within prompts. On held-out data, observed attenuation closely matches the prediction $1-D^2$ (correlation of log ratios 0.99). Across four datasets, position adjustment lowers held-out log loss by 14-23% and changes which judges would be dropped for low sensitivity on three datasets. On MT-Bench, it ranks GPT-4 first instead of claude-v1, matching human votes. Simulations with known parameters show improved recovery of judge sensitivities and candidate rankings, with 94.4-95.2% confidence interval coverage. Low estimated sensitivity alone is therefore not evidence of an unreliable judge.

Black-box machine learning models increasingly deliver strong predictions, but extracting useful information from them, such as a set of important features, remains challenging. Existing model-agnostic methods primarily estimate feature importance or conduct inference on it rather than directly selecting features, whereas many feature selection methods are model-specific or rely on the model-X assumption. We introduce LOCO-guided Adaptive Minipatch Sampling (LAMPS), a model-agnostic ensemble framework that uses any black-box regression algorithm as its base learner to select features important for predicting the response. The base learner need only produce predictions and need not perform feature selection itself. LAMPS operates within a minipatch ensemble framework that subsamples both observations and features, allowing leave-one-covariate-out (LOCO) feature importance scores to be easily computed. It adaptively concentrates minipatch sampling on features with high LOCO scores while maintaining exploration. The resulting sampling probabilities rapidly separate signal from noise features after a few iterations, enabling selection through simple thresholding. We establish that LAMPS achieves exact feature selection in high-dimensional settings, provided that the base predictive models are sufficiently well trained on average. Extensive experiments on synthetic and real data show that LAMPS outperforms state-of-the-art feature selection methods, with particularly strong performance in the presence of correlated features.

Causal Bayesian Optimization (CBO) combines causal inference with Bayesian optimization to enable sample-efficient intervention selection in systems with causal structure. This survey provides a systematic review of CBO through a unified BO-loop perspective, showing how causal assumptions shape intervention search spaces, surrogate models, acquisition functions, and decision policies. We organize existing methods by graph and system-knowledge assumptions, environment, intervention representation, surrogate architecture, and decision rule, and connect CBO to causal bandits, Bayesian experimental design, safe optimization, policy search, and causal abstraction. We also introduce a reproducibility-oriented benchmark spanning hard- and soft-intervention settings, with standardized GAP and a new trajectory-aware Path-Aware GAP (PA-GAP), evaluating seven CBO methods and a non-causal BO baseline across thirteen datasets, three budgets, and two metrics. Results show that no method dominates uniformly: rankings depend on dataset, budget, metric, and how causal information is used, while strong non-causal baselines remain competitive in several settings. Controlled graph-misspecification and omitted-variable stress tests further show that rankings can change substantially when learner-side causal information is perturbed. We conclude by identifying key open challenges, including robustness to causal-assumption violations, scalable unknown-graph optimization, mixed intervention types, realistic cost models, stronger theoretical guarantees, and integration with modern representation learning and causal abstractions.

Large-scale multipurpose cohort studies and biobanks often omit covariates needed for specific downstream analyses. We study target-population inference for generalized linear regression when key covariates are completely absent from the target data but observed in a related source population. Standard missing covariate methods are not directly applicable because they require at least partial observation of the covariates in the target population. We develop a doubly robust transfer learning framework under a sub-population shift assumption, which allows the distribution of the observed outcome and covariates to differ between populations while requiring the conditional distribution of the missing covariates given the observed variables to be shared. Unlike linear regression, nonlinear target estimating equations require parameter-indexed conditional functionals of the missing covariates rather than a fixed collection of low-order conditional moments. Our estimator combines importance-weighted source estimating equations with imputation of these conditional functionals. Under the identifying assumption, the estimator remains consistent when either the density-ratio model or the conditional-covariate model is correctly specified. Under regularity conditions, it is root-$n$ consistent and asymptotically normal, and attains the semiparametric efficiency bound when both nuisance models are correctly specified.

Comparisons of confounding-control methods are informative only when simulated data contain the treatment-covariate relationship those methods address. We assessed whether Plasmode 0.1.0 preserves that relationship, uses the returned exposure to generate outcomes, and reports the effect implied by its generating model. We inspected the source and ran the generator under identical seeds with source rows as supplied, sorted by observed exposure, or using an alignment correction. We ran 200 null-effect replicates per strategy in a synthetic cohort and the SUPPORT/Right Heart Catheterisation cohort. A sampled subject usually received another subject's exposure probability. In the synthetic cohort, source-model AUC was 0.503 before and 0.841 after alignment. Crude risk-difference bias moved from +0.0051 to +0.3037. Other orders preserved or reversed the association. After alignment, deliberately misspecified estimators retained risk-difference biases of 0.10-0.11. Correctly specified estimators were nearly unbiased. With the unmodified generator, all five appeared similarly unbiased. At an injected odds ratio of 2, the conditional log odds ratio appeared on the observed exposure (+0.696; target 0.693), not the returned exposure (+0.017). Misalignment can weaken, preserve, or reverse designed confounding and thereby distort method comparisons. Alignment correction does not repair outcome-exposure decoupling, disagreement between the reported and generating effects, or marginal miscalibration.

We develop a stochastic nested fixed point (SNFP) estimator for random coefficients logit demand models that updates model parameters using stochastic gradients and performs demand inversion one market at a time. Relative to the conventional nested fixed point (NFP) estimator, SNFP substantially reduces memory requirements and computational cost, making estimation feasible in very large datasets. We establish the large-$T$ (number of markets) asymptotic properties of the estimator under regularity conditions. We also characterize the effect of sharing one block of simulation draws across markets and show how to correct for it. Monte Carlo simulations show that the SNFP estimator achieves statistical accuracy comparable to the NFP estimator, and in our benchmark a single online pass estimates a model with 100 million markets in about 5.5 hours. An empirical application using scanner data further demonstrates the practical advantages of SNFP for large-scale demand estimation.

The temporal dependence inherent in time series models poses a fundamental challenge for distributed Bayesian inference, as it precludes embarrassingly parallel algorithms based on naive independence assumptions in the time domain. We propose a frequency-domain framework for scalable Bayesian inference in stationary time series that exploits the asymptotic independence underlying the Whittle likelihood. To exploit parallel computing resources, we develop a distributed fast Fourier transform and integrate it with embarrassingly parallel Markov chain Monte Carlo (MCMC) algorithms within a modern cluster-computing framework. This enables the analysis of time series that exceed the memory capacity of a single computational node or for which computation time is a bottleneck. The proposed methodology is compatible with a broad class of existing divide-and-conquer algorithms for independent data by applying them to frequency-domain rather than time-domain partitions. We establish that the error of the spectral divide-and-conquer MCMC posterior approximation relative to the exact time-domain posterior converges to zero in probability in a shrinking neighbourhood of the full-data Whittle posterior mode. The corresponding convergence rate is also derived. Across several experiments, we demonstrate that our approach provides accurate approximations to the full-data Whittle posterior. The proposed method is shown to outperform the current state-of-the-art time-domain divide-and-conquer methodology, particularly for highly persistent processes. The methodology is further illustrated by fitting a semi-long range model to a long meteorological time series.

We develop exponential family synthetic controls (EFSC), a distributional version of synthetic controls for a panel of datasets. Each cell of the panel corresponds to a dataset drawn from an exponential family whose natural parameters factorize probabilistically across units and times. We estimate the latent factors using black-box variational inference. This replaces the usual weighted-average view of synthetic controls with a flexible probabilistic model that operates on full distributions. We propose causal estimands based on divergences between pre- and post-intervention distributions induced by the posterior of the natural parameters, together with distributional placebo tests to support causal inference and assess the significance of the estimated effects. We validate the proposed framework on synthetic and real data. Across a variety of exponential-family distributions, EFSC accurately recovers causal effects induced by exponential tilts, together with the corresponding divergences between treated and counterfactual distributions. The framework also captures effects induced by structural perturbations of the latent factors and by heavy-tailed noise contamination. Finally, we apply EFSC to study the expansion of Medicaid under the Affordable Care Act (ACA) and its impact on the distribution of health insurance coverage across U.S. states. Code is available at https://github.com/blei-lab/efsc.

Researchers across academia increasingly base significance claims on LLM judge scores and small-sample AI evaluations. Yet without well-calibrated confidence intervals (CIs), hypothesis tests, and judge-bias corrections, such claims are unreliable. We address these issues in several contributions. First, we find that running statistics over raw LLM judge scores leads to inflated false positives: counterintuitively, for many inter-rater agreement metrics, false positive risk peaks at "almost perfect" human-LLM agreement. To help researchers understand how to run statistics over LLM judges responsibly, we present guidance and tooling for the statistical analysis of mixed human-AI judge designs, and implement nine hypothesis tests via prediction-powered inference (PPI), including the first known PPI corrections for four rank-based tests (Wilcoxon signed-rank, Mann-Whitney U, and omnibus variants). To keep PPI++ stable with small human-labeled calibration sets, we introduce bootstrap-adaptive power tuning, which shrinks the estimated weight toward a target estimated from the labeled data, and accounts for that weight's own sampling variance. Second, through Monte Carlo simulations, we derive recommendations for what CI, p-value, and FWER correction methods to use for small-sample AI evaluations (N<100), and warn researchers against bootstrap CIs. We package these recommendations into evalstats, an open-source Python package that selects calibrated methods automatically, and demonstrate it in three scenarios, including one where a real LLM judge validated at "substantial agreement" would have led a researcher to publish a spurious finding. evalstats is publicly available at https://github.com/ianarawjo/evalstats.

This paper presents a probabilistic perspective on large language models (LLMs), developed with the aim of bringing together, in a single self-contained account, tools that are usually treated separately across the literature. LLMs are described through probability measures on the set of sequences of tokens, specified via their autoregressive conditional distributions. Training is formulated as a maximum-likelihood estimation problem, addressed by stochastic gradient methods, while text generation is viewed as the sequential simulation of the resulting stochastic process. The role of the asymmetry of the Kullback–Leibler divergence in text generation is examined in relation with characteristic phenomena such as hallucination and the distinction between statistical plausibility and truth. As a complementary illustration of the same viewpoint, we also discuss diffusion models, built around the score function, which cast generation not as sequential token prediction but as the simulation of a reverse-time stochastic process transforming noise into data both in discrete and continuous time.

The rare type match problem is a challenging situation faced by a forensic statistician who aims at providing the value of a match between some characteristic of a crime stain and the corresponding characteristic of a suspect's stain when this characteristic has not been observed before. Several methods have been designed in the literature to assess likelihood ratios for the rare type match case when evidence consists of a Y-STR profile found on the crime scene matching the Y-STR profile of a designated suspect. We develop a general Bayesian decision-theoretic framework for quantifying the expected cost of alternative approaches using the logarithmic scoring rule. The framework provides a novel formalization of the posterior cross-entropy, which explicitly enhances the contribution of the method-specific strategy used to extract information from the data. We revisit existing decompositions of posterior cross-entropy and introduce a new complementary decomposition that provides a more interpretable characterization of this contribution. This work compares nine different methods by assessing their performance using empirical validation experiments. Its ultimate goal is to provide forensic experts and triers of fact with methodological insight and a pre-experimental guide to these alternative approaches.

Let $\Pi$ denote the independence copula and $M,W$ the upper and lower Fréchet–Hoeffding copulas. Catalano and Lavenant (2025) conjectured that $M$ and $W$ maximize the quadratic Wasserstein distance from $\Pi$ among all bivariate copulas. We prove this conjecture and characterize all equality cases: $\mathcal W_2^2(C,\Pi)\leq 1/10$, with equality if and only if $C\in\{M,W\}$. We also determine explicitly the optimal Monge map from $\Pi$ to $M$. The proof is constructive and combines the optimal transport from independence to the diagonal, a sharp convex-order inequality for 1-Lipschitz functions, the conditional convex order, and a coupling construction based on conditional comonotonicity and the supermodular order. As a consequence, we obtain a normalized Wasserstein-based dependence measure that characterizes independence and attains its maximal value exactly for comonotone and countermonotone dependence.

Robust linear fits can resist response contamination yet remain too dense or unstable for useful global explanations. We propose penalized distillation, which fits a smoothly clipped absolute deviation (SCAD) estimator to a robust initial estimator's empirical fitted surface along a safeguarded coordinate-descent path and evaluates candidate states separately for fidelity, parsimony, perturbation stability, and held-out prediction. The new results attach to the states the algorithm actually computes. Conditional on a fixed uncontaminated design, deterministic bounds transfer response-replacement boundedness from the initial fit to every retained path state. Turning to fixed dimension, we characterize the oracle-support branch by its empirical-Gram projection and influence function, give conditions for covariance-weighted least-squares approximation equivalence, and establish a path-conditional generalized information criterion. By contrast, at large dimension-to-sample ratios the full-coordinate robust fit collapses without warning, and screening restores the construction. Under a sure-screening framework, the robustness bound and the support and selection guarantees transfer to the screened fit. Simulations separate robustness transfer from support recovery, efficiency, and computation across the dimension-to-sample ratio, with p up to 240, and the signal density, which isolates what the sparse stage adds once the screen over-selects. In a duplicate-grouped superconductivity study, the distilled estimator remains predictively stable under prespecified training-response shifts but retains 66.8–68.8 of 81 slopes. Stronger sparsification reduces the model to 12.6–14.0 slopes only at visible fidelity and prediction cost. Distillation therefore preserves predictive stability on these data without substantiating a compact coordinate-level explanation.

This paper establishes time-uniform accuracy guarantees for deterministic square-root ensemble Kalman filters in linear–Gaussian state-space models under perfect-model dynamics. For hyperbolic, detectable systems, we prove that the ensemble means and covariances approximate their Kalman filter counterparts uniformly in time, with high probability and without inflation or resampling. The required ensemble size depends on the effective rank of the initial covariance and the dimension of the unstable subspace, rather than on the ambient state dimension. We also analyze a localized square-root ensemble Kalman filter for weakly coupled spatial systems. We show that the stabilizing Kalman covariance inherits spatial decay from the dynamics and derive accuracy bounds that separate sampling error from localization bias. For the localized filter, the required ensemble size depends on the local intrinsic dimension and only logarithmically on the number of spatial blocks, while the localization bias scales with the interaction strength. Our analysis combines stability and forgetting for possibly singular Riccati recursions, nonasymptotic covariance concentration, and perturbation estimates for localized covariance dynamics.

Density-Ratio Rescoring (DRR) augments a classifier trained at the original class prior with a survey-raking dual score. Raking reweights the majority sample to match minority feature moments within a tolerance. DRR marginally standardizes the dual and base scores and combines them with a fixed weight of one half, using the fitted dual directly for prediction without resampling or refitting the base classifier. Under exact population matching and a correctly specified log-linear tilt model, the dual equals the log density ratio up to an additive constant. A class-separation analysis characterizes the signal strength and correlation conditions under which fusion improves separation under common within-class covariance. On 24 tabular benchmarks, evaluated over 30 trials and five base learners, DRR at the D=128 random-feature setting improves average precision over the standardized base on every dataset, with a mean gain of 0.034. It exceeds the shared-dual raking-and-relabeling resampler on 22 of 24 datasets, with a mean gain of $0.092$, and on all eight one-versus-rest tasks of a shared gene-expression cohort. These results demonstrate the effectiveness of using raking duals as reusable scores for improving rare-class ranking while retaining classifiers trained at the original prior.

For Gaussian location mixtures with identity covariance, we prove that the nonparametric maximum likelihood estimator of the mixing distribution is unique for Lebesgue-almost every dataset, for every sample size and dimension. In particular, uniqueness holds almost surely whenever the joint distribution of the observations is absolutely continuous. The proof uses the finite-support theorem for Gaussian location NPMLEs and almost-everywhere differentiability of the optimal log-likelihood. Wherever this optimal value is differentiable, all NPMLEs have the same density derivatives at the observations. A minimal linear dependence among Gaussian evaluation vectors then rules out distinct solutions. An appendix extends the result to arbitrary fixed, known, observation-specific positive-definite covariances.

Randomized perturbations can reveal downstream responses without identifying which causal relationships are direct. Conditioning on intermediate responses may induce associations through unmeasured common causes, even when the source is randomized. We introduce Proxy-Adjusted Balanced Masking (PABM), a nonparametric framework for recovering directed acyclic graphs from targeted interventions, response data, and recorded proxies. PABM uses valid target-specific interventions to identify candidate ancestors. It then compares conditional target distributions with and without a candidate response and, when supported by the design, its intervention variable, adjusting for other ancestors and proxies. Each comparison uses matched observations, adjustment variables, and fitting procedures. Identification requires every included nonparent comparison to have zero conditional gain and each parent to have positive gain in at least one comparison. We establish population identification, finite-sample recovery conditions that accommodate incomplete adjustment and estimation error, and familywise error control under valid held-out p-values. Continuous-response simulations show favorable graph recovery relative to specified comparator pipelines; K562-calibrated count simulations reveal a selection–ranking tradeoff. Proxy ablations assess sensitivity to recorded information. An analysis of K562 Perturb-seq data illustrates descriptive candidate-network construction; proxy adequacy for unmeasured biological variation remains unresolved.

The target trial framework is increasingly used to define the causal estimand for an observational analysis. However, designing time-to-event analyses that align with the causal estimand is challenging in pregnancy because the timescale on which outcomes are often defined, "gestational age", differs from the timescale on which outcomes are measured, "time since study entry". We argue that neither timescale is "best" but that each aligns with a different estimand. In this article, we use the target trial framework to define the causal estimands corresponding to each timescale. We consider a (hypothetical) target trial comparing randomization to 17-alpha hydroxyprogesterone caproate (17-OHPC) versus placebo between 16-20 weeks of gestation to prevent delivery before 37 weeks' gestation, whether by miscarriage, stillbirth, or live birth. We define the estimand of time-to-event analyses on the "time since study entry" timescale as the effect of randomization to 17-OHPC versus placebo between 16-20 weeks of gestation on delivery before 37 weeks'. In contrast, the causal estimand of time-to-event analyses using the "gestational age" timescale corresponds to the effect had, counter to fact, all participants been randomized at one gestational age (e.g., 16 weeks). We review specific challenges to estimation and possible solutions for each timescale.

Mixed Graphical Models (MGMs) provide a flexible framework for structure learning from heterogeneous data by treating sets of both continuous and discrete. Bayesian inference for MGMs remains challenging due to the combinatorial complexity of graph space exploration and posterior computation. In this paper, we model discrete components through latent Gaussian variables and consider two likelihood specifications: a copula-based ranked likelihood, yielding the copula-MGM, and a probit formulation based on cut-off points, yielding the probit-MGM. We then propose the Mixed Graph WWA, a class of MCMC methods for posterior simulation in Bayesian MGMs. Building upon the WWA algorithm, we develop two specialized algorithms: copula-WWA for copula-MGMs and probit-WWA for probit-MGMs. Both methods exploit the latent Gaussian representations to perform posterior inference through a Gibbs sampling scheme that alternates between latent-variable augmentation and graph-structure updates. Through extensive simulation studies we demonstrate that the proposed methods achieve graph recovery accuracy comparable to or better than existing approaches, including copula-BD MCMC and probit-BD MCMC based on the Birth-Death MCMC methodology, while maintaining efficient posterior exploration and favorable effective sample size per unit computational time. We further illustrate the practical utility of our approach through an application to the PAM$50$ breast cancer gene expression dataset, where the inferred MGMs reveal meaningful dependencies between gene expression profiles and cancer subtypes. These results highlight the effectiveness of the Mixed Graph WWA method as a scalable and principled tool for Bayesian structure learning in MGMs.

The high-frequency spectral roll-off of continuous-time ARMA (CARMA) models can magnify the effect of narrow-band disturbances when aliasing is weak, making standard maximum-likelihood Whittle estimation sensitive to affected ordinates. We show that a logarithmic transformation $r_k = \log \rho_k$ converts the Whittle scale problem into a Gumbel location problem, connecting robust spectral estimation to the classical minimax theory of Huber and Rieder. A piecewise centering correction—closed-form for small $b$, implicit closed-form for the practitioner range—preserves Fisher consistency for any clipping level without numerical optimisation. Clipping the Gumbel score symmetrically and transforming back yields a two-sided clipped Gumbel score (the standard Rieder–Hampel bounded-influence form) whose normalised influence curve is proved locally asymptotically minimax under shrinking gross-error contamination. The efficiency loss is quantified by a single scalar $K(b)$: at $b = 1.5$, only $16\%$ nominal asymptotic variance overhead. In the stated AR(2) Monte Carlo design, the sample-size trends are compatible with the asymptotic rate, and at $b=1.5$ the bias reductions are about $40\%$, $90\%$, and $96\%$ for $a_1$, $a_2$, and $\lambda$, respectively.

The Generalized Naive Bayes (GNB) model was introduced for discrete and categorical random variables as an extension of classic Naive Bayes. We now accommodate the GNB framework to continuous explanatory variables. A central result of the paper is that structure learning of the GNB depends only on the pair copulas of the bi-variate marginals. We proved that the GNB structure can be assigned to the basis of a matroid, therefore we give greedy algorithms for finding the optimal GNB structure on the training data, in sense of minimizing Kullback-Leibler divergence. Three cases are considered: joint Gaussian distribution, then a more flexible model where we suppose the dependence structure to be described by a Gaussian copula with arbitrary marginals, and an even more flexible case where the joint continuous probability distribution is arbitrary, i.e. copula and marginal distributions are arbitrary. A method for model reduction, based on the newly introduced concept of GNB forest is given. We close the paper by comparing the newly introduced GNB classification results to other classical "glass-box" algorithms on real datasets.

Dictionary learning seeks to recover an unknown dictionary $A$ from observations ${\bf y}_i = A{\bf x}_i$ with sparse coefficient vectors ${\bf x}_i$. We introduce the Iterative Atom Refinement (IAR) algorithm, a simple procedure for recovering individual dictionary atoms. Starting from a random direction, IAR repeatedly selects the observations most strongly correlated with the current iterate and updates the direction by averaging the selected data. Our main contribution is a rigorous convergence theory of IAR. Using high-dimensional probabilistic estimates and a novel monotonicity principle for atom-selection probabilities, we show that a small initial advantage of one atom is amplified until that atom is isolated. Under our model assumptions, IAR identifies a generating atom after only three refinement steps. Numerical experiments support the theory and show that the resulting dynamics accurately capture the behavior observed in dictionary refinement.

Early-phase combination oncology trials increasingly seek the optimal biological dose combination (OBDC) rather than a maximum tolerated dose combination, in line with the FDA's Project Optimus initiative. Existing Bayesian OBDC designs span rule-based, model-assisted, and model-based paradigms, but no unified framework exists for choosing among them, and none supports borrowing information across indications sharing a combination regimen. We propose a corrected three-by-two taxonomy of OBDC designs by mechanism and objective, a rule-based design (Ji3+3-Comb) closing a documented gap in transparent, model-free combination dose-finding, and a Bayesian Hierarchical Utility-based Cross-indication (BHUC) design that borrows information across indications via a robust mixture prior while discounting non-exchangeable information. We derive the shared utility function as the Bayes-optimal decision under a linear clinical loss, and prove that BHUC's mixture-prior posterior weight automatically vanishes as two indications' true rates become discordant, formalizing its robustness to non-exchangeable borrowing, and complement this asymptotic guarantee with an exact, sample-size-free ceiling on borrowed influence that holds even before any own-indication data accrue. A roadmap links trial features to design choice, and a 5000-replication simulation study, sensitivity analyses, and a case study built from a published phase Ib trial show that model-assisted designs offer the most robust safety-efficacy trade-off, while BHUC improves correct selection over independent per-indication designs even under discordance. We conclude with concrete, protocol-actionable recommendations for pharmaceutical and trial biostatisticians, and directions for future methodological and computational development.

Modern conditional generative models face significant challenges when learning complex covariate dependencies. While sufficient dimension reduction (SDR) provides a principled approach to compress these dependencies, traditional SDR frameworks were not formulated for conditional generation. To bridge this gap, we propose Belted Engression, a unified and architecturally parameter-efficient framework for generative distributional regression. Our approach establishes an end-to-end compress-then-generate paradigm driven by sufficient representation learning, embedding a structural bottleneck into the generative architecture. Theoretically, we prove that the standard SDR condition is equivalent to a law-preserving generative factorization, which is achieved at the global optimum of the population Belted Engression objective. Furthermore, by uncovering a localized Bernstein-type control for the energy-score loss, we establish finite-sample convergence rates that are sharper than those of existing results. We also prove that this belted architecture is strictly smaller, operating with an asymptotically vanishing parameter count relative to the unstructured baseline. Extensive simulations and real-world applications demonstrate that Belted Engression achieves superior distributional prediction and SDR recovery with fewer trainable parameters.

Estimating regression models with nonlinear predictor transformations is challenging when data are missing, because nonlinearity typically renders the conditional distribution of the missing values intractable. Previous methods require specific nonlinear forms, such as polynomials or interactions, or rely on approximations that can induce bias. We propose a stochastic EM algorithm that uses sampling-importance resampling (SIR-StEM) to handle missing data under arbitrary nonlinear transformations of predictors, either substantively motivated or incidental, such as spline basis expansions. Unlike approaches that require linearity or closed-form conditionals, SIR-StEM only requires evaluating the complete-data likelihood up to proportionality, making it applicable across a broad class of nonlinear regression models. We construct an algorithm that makes use of missing data pattern information for computational efficiency, and establish asymptotic normality of the estimator. We demonstrate the method in two simulation studies, one with parametric nonlinear transformations and another with a spline basis expansion based on real behavioral measures. Results show that SIR-StEM yields low bias and near-nominal confidence interval coverage, outperforming other common approaches. We conclude with limitations and directions for future research.

Accurate risk prediction is crucial for clinical-decision making, intervention planning, treatment and transplant allocation. However, longitudinal clinical data are often irregular and subject to censoring. We propose PAR2COX, a joint framework that integrates PARAFAC2 decomposition with Cox proportional hazards model, using patient-specific latent factors as covariates in the likelihood. The proposed alternating optimization framework jointly estimates phenotypes and survival parameters, enabling survival-guided representation learning. PAR2COX accommodates both historical patients with observed outcomes and current patients whose outcomes remain unknown. Numerical experiments and a case study based on MIMIC-IV data demonstrate improved risk stratification compared with existing approaches, highlighting the value of survival-informed phenotype learning.

Probabilistic forecasting requires accurate central predictions and calibrated uncertainty estimates. This paper presents TEMPER, the Temporal Encoder-Masked Probabilistic Ensemble Regressor, a univariate time-series forecasting algorithm that combines a temporal autoencoder, a differentiable masked neural decision forest, continuous ranked probability score (CRPS) training, and Gaussian-mixture post-processing. The R implementation is built on torch for R and returns horizon-wise density, distribution, quantile, and sampler functions. We evaluate TEMPER on three deterministic synthetic level series with trend, periodic, regime-switching, nonlinear-threshold, and heteroskedastic components. Across 96 rolling-origin forecasts at horizons t + 1, t + 5, t + 20, and t + 60, TEMPER obtains 2.824% mean CRPS normalized by origin level, 3.635% median absolute error, and 68.8% empirical 90% interval coverage after training with a 300-epoch cap and early-stopping patience of 100. A naive persistence bootstrap has the best aggregate CRPS, 2.763%, while TEMPER has the best median absolute error and the best CRPS at t+1 and t+5. The ablation study uses matched series-origin-horizon cells, horizon-wise CRPS deltas, endpoint sensitivity summaries, and a calibration-specific interval study. Relaxing the learned mask improves average CRPS by 0.472 percentage points on the ablation subset, mainly through long-horizon gains. A twofold interval inflation improves held-out coverage from 54.2% to 91.7% and gives the best 90% interval score among tested calibration rules. The results identify calibration, horizon-specific tuning, and component selection as the central research priorities.

Let \(\{X_n:n\ge0\}\) be a controlled branching process with controls \(\phi_n\), and let \(N_n=\phi_n(X_n)\) be its progenitor count. We study large deviations of \(X_{n+1}/N_n\), \(N_n/X_n\), and \(X_{n+1}/X_n\), on their natural positive-denominator events, in supercritical and critical regimes. We treat generation-wide random slopes and asymptotically deterministic first-order slopes. For asymptotically affine random-slope controls, an affine recursion and a random environmental product yield a sharp Bahadur–Rao–Petrov upper-tail asymptotic. A positive Cramér tilt identifies the prefactor via a nontrivial normalized limit, and conditional on the upper-deviation event the three ratios concentrate around their environmental centers. With individual-sum controls, the process is an exact branching process in a random environment with immigration (BPREI). Motivated by this connection, we establish sharp upper large-deviation asymptotics for BPREI and transfer its positive-transform and harmonic-moment profiles to the controlled process. In the nondecreasing supercritical case, harmonic moments exhibit an exact environmental–boundary–persistence trichotomy. In a centered finite-variance critical class, finite-horizon-minimum methods yield exact \(n^{-1/2}\)-scale limits for all positive harmonic orders and for the three ratio-deviation probabilities. For deterministic first-order slopes, the environmental-product mechanism disappears. Matched lower and upper probability-generating-function envelopes transfer auxiliary exact-recursion profiles to controls without an exact branching recursion, producing distinct supercritical and critical harmonic-moment trichotomies. Thus the location of first-order randomness determines the lower-tail mechanism, while immigration may change a sharp constant, a phase boundary, or the decay exponent.

In this paper, we propose a model-free feature screening framework tailored for high-dimensional and heterogeneous datasets, based on a novel distributionally robust dependence measure termed Copula Divergence. The proposed screening method, named CD-Screen, addresses critical limitations of existing feature screening methods, such as restrictive modeling assumptions and sensitivity to heterogeneous feature distributions. CD-Screen ranks features according to their Copula Divergence without relying on a specific regression model or distributional assumptions. Additionally, we introduce CD-FDR, a data-driven procedure to control false discoveries, ensuring accurate and efficient feature selection. Theoretical analyses establish the sure screening and rank consistency properties of CD-Screen, along with asymptotic control of the false discovery rate by CD-FDR. Extensive simulation studies demonstrate the superior performance of our methods compared to traditional screening approaches across diverse scenarios. Furthermore, a real data analysis of the relationship between stock returns and inflation in the United States {illustrates the practical use of our method and provides descriptive evidence on} sector-specific responses to economic changes.