Papers

Causal effect estimation from observational data requires careful adjustment for confounding. Classical estimators such as inverse probability weighting and augmented inverse probability weighting can perform well under favorable model specification but may become unstable in complex settings. Machine-learning and representation-learning methods provide greater flexibility, but joint optimization may allow outcome information to alter treatment representations and compromise the intended causal structure. We propose MOCA (Modular One-way Causal Attention), a transformer-based framework that separates treatment and outcome modeling through a modular architecture. This design preserves directional information flow while retaining the flexibility of transformer architectures. Using a Markov-kernel formulation, we show that MOCA learns an autonomous treatment representation and is predictively KL-optimal within its Gaussian model class. Under correct specification, MOCA recovers the true average treatment effect, while additional two-way feedback does not further reduce average treatment effect estimation error. We also propose a conformal inference procedure for individual treatment effects. Across multiple simulation scenarios, MOCA achieved competitive or improved average treatment effect estimation compared with IPW, AIPW, the X-learner, TARNet, DragonNet, BART, Causal Forest, and Do-PFN. Ablation studies supported the contributions of the proposed architectural components. We further evaluated MOCA on the Infant Health and Development Program benchmark and the observational Dehejia-Wahba dataset. Overall, modular attention with one-way information flow provides an effective and interpretable framework for causal inference using modern deep-learning models.

Evaluating generative AI models is increasingly resource-intensive due to slow inference, expensive raters, and a rapidly growing landscape of models and benchmarks. We propose ProEval, a proactive evaluation framework that leverages transfer learning to efficiently estimate performance and identify failure cases. ProEval employs pre-trained Gaussian Processes (GPs) as surrogates for the performance score function, mapping model inputs to metrics such as the severity of errors or safety violations. By framing performance estimation as Bayesian quadrature (BQ) and failure discovery as superlevel set sampling, we develop uncertainty-aware decision strategies that actively select or synthesize highly informative inputs for testing. Theoretically, we prove that our pre-trained GP-based BQ estimator is unbiased and bounded. Empirically, extensive experiments on reasoning, safety alignment, and classification benchmarks demonstrate that ProEval is significantly more efficient than competitive baselines. It requires 8-65x fewer samples to achieve estimates within 1% of the ground truth, while simultaneously revealing more diverse failure cases under a stricter evaluation budget.

Pooling genome-wide association studies of multiple related traits can substantially increase power for detecting genetic variants with pleiotropic effects. ASSET, which exhaustively searches all subsets of studies for association signals, has been widely used to detect modest effects and improve interpretability. Under a normality assumption, ASSET computes p-values via an analytic approximation that accounts for multiple testing. However, this approximation has been evaluated only in limited scenarios and for p-values no smaller than $10^{-3}$. A systematic assessment in the extreme tail is therefore needed, yet naïve Monte Carlo methods would require prohibitively many simulations. We develop a computationally efficient importance-sampling (IS) algorithm that provides accurate ASSET p-value estimates for both independent and overlapping studies, achieving substantial efficiency gains over naïve Monte Carlo, particularly for very small p-values. Using IS, we show that ASSET's analytic approximation is highly accurate across nearly the entire p-value range when normality holds. In contrast, when normality is violated (due to small sample sizes, low-frequency variants, or non-normal traits), ASSET p-values can be inflated or deflated by orders of magnitude, whereas our IS approach remains accurate. We illustrate the method through applications to single-cell eQTL mapping using peripheral blood mononuclear cells from the OneK1K cohort and lung cells from a Korean population.

Generative approaches to clustering provide information on geometric properties of clusters, whereas discriminative approaches provide boundaries between clusters. Ideas from both approaches are incorporated to present a fully unsupervised, probabilistic, and discriminative clustering method via a regularized mutual information objective function, wherein a mixture of mixtures of Gaussian and uniform distributions is used for formulation of the conditional model. Overfitting is avoided by the introduction of a regularizing term and a cluster merge step, similar to those applied in reversible jump Markov chain Monte Carlo methods used in Bayesian clustering. Consequently, the turtle shell method – a fully unsupervised clustering method capable of estimating non-linear boundary lines, automatically selecting the number of components, and capturing intuitive clusters in the presence of data abnormalities such as noise and/or irregular cluster shapes – is introduced. We test this method on various simulated and real datasets commonly explored in clustering research, and extend the analysis to datasets arising from flow cytometry experiments and image analysis.

We argue that current definitions of machine unlearning are underspecified for second-order optimizers. We compare first-order and second-order learners for their ability to handle the data deletion task with varying degrees of eigendecomposition to mimic the loss model memory. While both first and second-order methods realign with the ideal counterfactul in terms of performance and gradient, the second-order optimizer shows significant volatility in the optimizer state. This indicates residual information, supposedly deleted, that isn't detectable by first-order analysis. Various eigendecay treatments show that stability and information loss is regained only under controlled state pertubation where geometric information (or memory) is erased.

Fay-Herriot (FH) models with variance smoothing typically use chi-squared sampling distributions for the design variance estimators. This choice is only valid under strong assumptions on the population and the sampling design, and the choice of sampling distribution is understudied for complex survey designs such as the stratified two-stage clustering design used by the Demographic and Health Surveys (DHS). DHS conducts surveys in low- and middle-income countries and result in low sample sizes for unplanned domains of interest. Thus, accounting for the uncertainty in the estimated design variances is important. We derive two sampling distributions under the DHS design, a simple and a more complex, while clearly specifying and discussing the required superpopulation and design assumptions. In a simulation study, we compare the two sampling distributions to the empirical sampling distributions, and the resulting FH models with variance smoothing to the standard FH model. We find that the standard model exhibits undercoverage, while the variance smoothing models produce better credible intervals according to proper scoring rules. Interestingly, the simple sampling distribution, which is easiest to implement, performs equally as well as the more complex sampling distribution. We illustrate the proposed models by estimating height-for-age z-scores using the 2022 Kenya DHS.

Two-stage recommender systems first choose a candidate generator and then rank items within the generated set. Because the generator decides which items are available to the ranker, changing the generator changes both the policy value and the data support used to estimate that value. This creates an offline selection problem that standard single-stage objectives do not capture: a policy may look good under a retrieval score or a raw off-policy value estimate, but still be unreliable if it depends on weakly supported generator-item pairs. We propose CASP (Coupled Action-Set Pessimism), a support-aware offline selector for finite libraries of two-stage recommender policies. CASP combines doubly robust value estimation with a support-burden penalty. We show that stagewise rules that ignore downstream continuation value can be arbitrarily suboptimal, and we derive population, finite-class, and reconstructed-propensity guarantees for conservative selection. In simulations and a reconstructed MovieLens 1M application, CASP selects lower-burden policies when estimated value and support credibility are in tension.

Not only does mobile health technology enable researchers to track changes in multiple longitudinal outcomes of interest and to record the occurrence of health-related events over time, but it also allows for the delivery of repeated low-cost treatments directly to individuals in real time. We present a model-based approach for estimating the effect of repeatedly delivered treatments in a micro-randomized trial (MRT) via an extension of a joint longitudinal-survival model. We discuss different ways that these repeated treatment effects can be incorporated into the joint model; these different model specifications correspond to different mechanisms by which treatment is assumed to impact the longitudinal and event processes. Taking a Bayesian approach to inference, we model the association between repeated treatments, multiple longitudinally measured outcomes, and recurrent events. We also demonstrate how to calculate information criteria for model selection and present goodness-of-fit plots for assessing survival submodel calibration. We then illustrate the performance of our method via simulations and analysis of data collected in an MRT of substance use.

This study investigates perceptions and use of generative artificial intelligence (GenAI) tools among students and faculty in statistics and data science at a historically Black college or university. Survey data from 119 valid student responses and 14 faculty responses were used to examine familiarity, usage patterns, perceived benefits, awareness of limitations, and instructional support needs. Students reported substantial use of GenAI, with ChatGPT as the dominant tool, primarily for coding assistance and writing support. Although student perceptions of AI in data science workflows and careers were generally positive, confidence in interpreting AI-generated outputs was limited, and concerns about accuracy, reliability, and over-reliance were common. Faculty also viewed GenAI favorably, but self-rated proficiency and the frequency of classroom integration remained limited. Comparisons across student subgroups suggested that familiarity with GenAI and awareness of its limitations varied more by academic level than by gender. These findings highlight a gap between AI adoption and AI literacy and underscore the need for structured training, validation practices, and clearer institutional guidance for responsible AI integration in data science education.

In this paper, we consider the problem of testing independence in high-dimensional settings with missing data. Building upon a recently proposed Kendall-based statistic, we introduce two new modifications specifically designed to accommodate incomplete observations. The proposed methods are studied from both theoretical and empirical perspectives. A comprehensive simulation study illustrates the robustness and applicability of the new approaches. The findings contribute to the development of nonparametric methods for analyzing high-dimensional and incomplete data structures.

Trust Region Bayesian Optimization (TuRBO) is an effective strategy for alleviating the curse of dimensionality in high-dimensional black-box optimization. However, inappropriate lengthscale design can cause the local Gaussian process (GP) model within the trust region to degenerate, leading to suboptimal performance in high dimensions. In this work, we show that TuRBO's local GP may remain either excessively complex or overly simple as the dimension $D$ and trust region side length $L$ vary. To address this issue, we propose a straightforward variant, AdaScale-TuRBO, which scales the GP lengthscale with both the problem dimension and trust region size, thereby preserving kernel geometry and maintaining consistent prior complexity. Empirically, we show that AdaScale-TuRBO can robustly outperform standard TuRBO and other popular high-dimensional BO methods on synthetic benchmarks and real-world trajectory planning tasks.

Agreement coefficients provide a fundamental framework for quantifying the concordance between two or more measurement methods applied to the same continuous variable. Unlike correlation, which measures the strength of a linear relationship, agreement focuses on assessing whether measurements are numerically similar, capturing both precision and accuracy. This review provides a comprehensive overview of the primary statistical approaches for assessing agreement between continuous variables. Such a synthesis is timely, as it has been 15-20 years since the last major review in the field. Beginning with the seminal contributions of Bland and Altman (1986) and Lin (1989), the paper discusses extensions of their methods to robust, multivariate, and repeated-measures settings, as well as recent developments like the probability of agreement and measures based on alternative distance functions measures. Special attention is given to probabilistic and spatial generalizations, including frameworks designed for geostatistical and areal data, which have become increasingly relevant in modern applications such as image analysis and environmental statistics. Through illustrative examples and comparative discussions, this review highlights the evolution, connections, and limitations of existing agreement measures, identifying open challenges and directions for future research.

History-dependent sampling can reduce long-run Monte Carlo variance by discouraging redundant revisits, but existing schemes typically encode history through empirical measure on finite state spaces, which is infeasible in high-dimensional discrete configuration spaces or ill-posed in continuous domains. We propose Score-Repellent Monte Carlo (SRMC) framework that summarizes trajectory history by a running average of score evaluations in $\mathbb{R}^d$, where $d$ is the dimension of the score and state representation. This history is converted into a surrogate target through an exponential score tilt, indexed with $\alpha$ that represents the strength of repellence in controlling the magnitude of the history-based repulsion. The surrogate family is normalization-free in the standard MCMC sense, yielding a generic wrapper: at each iteration, any base kernel targeting $\pi$ can instead be run on the current surrogate $\pi_{\theta_n}$ while the history is updated online. We analyze the coupled evolution of the history recursion and Monte Carlo estimators using stochastic approximation with controlled Markovian noise, establishing almost sure convergence and a joint central limit theorem. We further identify regimes in which the asymptotic covariance decreases as $\alpha$ increases, with scaling $O(1/\alpha)$, extending the near-zero-variance effect of finite-state history-dependent samplers to general state spaces with constant memory. Experiments on continuous targets and discrete energy-based models demonstrate improved estimator variance and mode coverage, while retaining $O(d)$ memory usage and modest per-iteration overhead.

We focus on Borel measures that have a globally subanalytic density function. We prove, given such a measure $\mu$ on a set $A$ and a globally subanalytic mapping $\Phi:A\to \Omega$, with $\Omega$ bounded open subset of $\mathbb{R}^n$, a Sobolev embedding theorem for the Sobolev space $W^{k,p}_{\Phi_*\mu}(\Omega)$ of the push-forward measure $\Phi_*\mu$. We derive an embedding of $W^{k,p}_{\Phi_*\mu}(\Omega)$ into the space of inner Lipschitz functions and give an application to kernel theory.

Statistical modeling of popular music presents a unique challenge due to the complexity of song structures, which cannot be easily analyzed using conventional statistical tools. However, recent advances in data science have shown that converting non-standard data objects into real vector-valued embeddings enables meaningful statistical analysis. In this work, we demonstrate an approach based on logistic principal component analysis to construct embeddings from global song features, allowing for standard multivariate analysis. We apply this method to a corpus of Lennon and McCartney songs from 1962-1966, using embeddings derived from chords, melodic notes, chord and pitch transitions, and melodic contours. Our analysis explores how these song embeddings cluster by Beatles album, how songwriting styles evolved over time, and whether Lennon and McCartney's compositions exhibited convergence or divergence. This embedding-based approach offers a powerful framework for statistically examining musical structure and stylistic development in popular music.

Objective: Triaxial accelerometers (TAAs) are widely used in home care medicine. This study investigates whether TAA signals recorded at the fingertip encode respiratory information, particularly instantaneous respiratory rate (IRR) and respiratory effort, during sleep. Method: We propose an antiderivative-based nonlinear transformation to convert TAA signals into a respiratory surrogate, termed TAA-resp. To quantify the embedded respiration-induced oscillation, a modern time-frequency analysis tool is applied to derive an index, referred to as the respiratory motion index (RMI). The proposed TAA-resp and RMI are validated on two datasets comprising 39 Asians and 10 white or African Americans with full-night recordings from simultaneous polysomnography (PSG) and a fingertip TAA measurement. Criteria for labeling TAA-resp signal quality as good, moderate, or poor are established, and two experts' independent annotations with consensus are obtained. Result: On average, TAA-resp encodes high-quality respiratory information in over 21.25%$\pm$16.11% of full-night recordings, reaching up to 58.33% in some cases. TAA-resp shows stronger correlation with thoracic and abdominal motion than with airflow, indicating predominant capture of respiratory effort. High-quality TAA-resp segments offer an accurate IRR estimate with root mean square error $0.027\pm 0.022$ Hz. RMI is higher for high-quality segments and lower for poor-quality segments, and its distribution aligns with physiology, with higher values during REM, N2, and N3 sleep and in the absence of apnea or hypopnea events. In leave-one-subject-out cross-validation, RMI predicts quality labels with 0.84 sensitivity and 0.88 specificity. Conclusion: Fingertip-mounted TAAs encode meaningful respiratory information intermittently, and our method recycles underutilized respiratory information.

Classical design theory treats the type of an object as a given: the designer decides in advance that this will be a cup, then optimizes its parameters. This paper argues that object type is not a presupposition but an inference, something that can be determined from physical data and functional requirements jointly. We call this problem requirement-steered interface type inference and show that it is inexpressible within existing design frameworks. This paper makes two contributions that are jointly necessary and individually incomplete. The first is the problem itself, which classical design cannot pose because it presupposes the very thing our problem seeks to determine. The second is C-DMBD, a constrained extension of the Dynamic Markov Blanket Detection algorithm, which makes requirement-steered inference computationally tractable. Drawing on the free-energy principle and active inference, established frameworks in theoretical neuroscience and Bayesian mechanics, we model a product's surface as a Markov blanket: the minimal boundary through which all causal exchange between object and environment must pass. Different blanket structures correspond to different object types; different parameterizations of the same structure correspond to different functional modes of the same type. This paper is a proof of concept and a theoretical proposal. It reframes design as inference rather than optimization, and as a relation between generative models rather than a specification of parameters.

Exponential families form the backbone of modern statistics and machine learning, but textbooks seldom derive them from first principles in an accessible way. Although minimal sufficiency and the principle of maximum entropy, originating in physics, provide core motivation, they are often presented as technical and requiring advanced prerequisites. Here, a short, self-contained derivation of exponential families based on maximum entropy is presented that is straightforward to carry out, requires only a modest background in information entropy, and avoids technicalities like constrained optimisation. Two propositions are demonstrated in this fashion: i) exponential families with a general base maximise information entropy with respect to that base subject to fixed expectations of canonical statistics, and ii) exponential families with a uniform base maximise standard information entropy under the same constraints. Maximum entropy therefore provides a principled foundation for exponential families with minimal prerequisites, highlighting the value of teaching entropy in statistics courses.

These lecture notes introduce the statistical analysis of continuous-time generative models built from Markov dynamics. We begin with the stochastic-calculus foundations of score-based diffusion models, including time reversal, score matching, and sampling from learned scores. We then present the broader framework of generator matching, which describes flows, diffusions, jump processes, and discrete generative models through their infinitesimal generators. We then focus on finite-sample guarantees. We explain how errors in the learned drift or generator propagate to the final generated distribution, why stability and regularity properties are essential, and how time-adaptive neural network classes can achieve optimal Wasserstein rates for smooth target distributions. Overall, the notes aim to connect modern generative modeling algorithms with the probabilistic, analytic, and statistical tools needed to understand their worst-case performance.

This study quantifies the association between air pollution and mortality in Ontario, Canada. Exposure-response relationships in air pollution epidemiology are complex due to three features: time-lagged associations, non-linear associations, and multiple pollutants. To address the first two features, two distinct classes of distributed lag non-linear model (DLNM) have been proposed, but extending them to multiple exposures and selecting an appropriate model remain challenging. We propose a unified framework for multiple exposure DLNMs, integrating model specification, estimation, selection and stacking. The framework applies to four different model structures: two additive and two proposed single-index DLNMs, all applicable to general outcome types, including the mortality counts in the motivating application. We develop an estimation approach that applies to all four models. Choosing among the candidate DLNMs is challenging a priori, and we derive an AIC to select among them. As an alternative to selecting a single model, we also extend a model stacking approach to combine inferences across the four DLNMs and propose an implementation scalable to our dataset with 106,346 observations. In the motivating analysis, the four DLNMs yield different estimates, and the proposed stacking approach identifies significant associations between respiratory mortality and a mixture of PM2.5, O3 and NO2.

We study the sharp bounds on $\mathbb{E}[X_1\cdots X_d]$ when the univariate marginal distributions are known, but the dependence structure between them is unspecified. Maximizing products over non-negative variables is straightforward via the comonotonic coupling, but the problem is more subtle when the marginals can take both positive and negative values. Specifically, two negative realizations can be matched to yield a positive product, whereas a single negative realization necessarily yields a negative product. We decompose the problem into a magnitude part and a sign part, and show that universal upper and lower bounds for the product expectation follow from comonotonic coupling of the absolute values and properly chosen sign vectors. Under a mild regularity assumption, we give necessary and sufficient conditions for these universal bounds to be attainable. For the upper bound, the marginal sign-bias vector must belong to the even-parity polytope, while for the lower, the corresponding condition involves the odd-parity polytope. We construct the extremal couplings via measurable selections on the parity polytope whenever these conditions hold. We study the case of identical marginals in more detail and provide examples of nonsymmetric extremal couplings that attain the universal bounds. We explicitly construct the extremal copulas in three dimensions, and use a recursive parity decomposition to obtain higher-dimensional extremal copulas from the trivariate ones.

Predicting customers' long-term revenue from sparse and irregular transaction data is central to marketing resource allocation in non-contractual settings, yet existing approaches face a trade-off. Traditional probabilistic customer base models deliver robust long-horizon forecasts by imposing strong structural assumptions, while flexible machine-learning models often require substantial training data and careful tuning. We propose a variational-autoencoder-based model that preserves the process-based likelihood of established attrition-transaction-spend models conditional on customer heterogeneity, but replaces the restrictive parametric mixing distribution with a flexible latent representation learned by encoder-decoder networks. The resulting approach (i) provides a single model for customer attrition, transactions and spending, (ii) remains reliable when contextual covariates are unavailable, and (iii) flexibly incorporates rich covariates and nonlinear effects when they are available. This design balances structural stability with the flexibility needed to capture complex purchase dynamics. Across multiple real-world datasets and prediction horizons, the proposed model improves upon the latest benchmarks. Businesses benefit directly, as a better assessment of customers' future revenues improves the efficiency of campaign targeting. For research, this work provides guidance on how to embed domain-specific models into the variational autoencoder framework, enabling flexible representation learning while retaining an econometrically meaningful process structure.

The Gaussian mixture model is widely used in unsupervised learning, owing to its simplicity and interpretability. However, a fundamental limitation of the classical Gaussian mixture model is that it forces each observation to belong to exactly one component. In many practical applications, such as genetics, social network analysis, and text mining, an observation may naturally belong to multiple components or exhibit partial membership in several latent components. To overcome this limitation, we propose the mixed membership sub-Gaussian model, which extends the classical Gaussian mixture framework by allowing each observation to belong to multiple components. This model inherits the interpretability of the classical Gaussian mixture model while offering greater flexibility for capturing complex overlapping structures. We develop an efficient spectral algorithm to estimate the mixed membership of each individual observation, and under mild separation conditions on the component centres, we prove that the estimation error of the per-individual membership vector can be made arbitrarily small with high probability. To our knowledge, this is the first work to provide a computationally efficient estimator with such a vanishing-error guarantee for a mixed-membership extension of the Gaussian mixture model. Extensive experimental studies demonstrate that our method outperforms existing approaches that ignore mixed memberships.

As the demand to integrate Artificial Intelligence into high-stakes environments continues to grow, explaining the reasoning behind neural-network predictions has shifted from a theoretical curiosity to a strict operational requirement. Our work is motivated by the explanations of autoregressive neural predictions on dynamic physical fields, as in weather forecasting. Gradient-based feature attribution methods are widely used to explain the predictions on such data, in particular due to their scalability to high-dimensional inputs. It is also interesting to remark that gradient-based techniques such as SmoothGrad are now standard on images to robustify the explanations using pointwise averages of the attribution maps obtained from several noised inputs. Our goal is to efficiently adapt this aggregation strategy to dynamic physical fields. To do so, our first contribution is to identify a fundamental failure mode when averaging perturbed attribution maps on dynamic physical fields: stochastic input perturbations do not induce stationary amplitude noise in attribution maps, but instead cause a geometric displacement of the attributions. Consequently, pointwise averaging blurs these spatially misaligned features. To tackle this issue, we introduce WassersteinGrad, which extracts a geometric consensus of perturbed attribution maps by computing their entropic Wasserstein barycenter. The results, obtained on regional weather data and a meteorologist-validated neural model, demonstrate promising explainability properties of WassersteinGrad over gradient-based baselines across both single-step and autoregressive forecasting settings.

Problem definition: Data-driven models in machine learning have enabled efficient management of production systems. However, a majority of machine learning models are devoted to modeling the mean response or average pattern, which is inappropriate for studying abnormal extreme events that are often of primary interest in aircraft manufacturing. Since extreme events from heavy-tailed distributions give rise to prohibitive expenditures in system management, sophisticated extreme models are urgently needed to analyze complex extreme risks. Engineering applications of extreme models usually focus on individual extreme events, which is insufficient for complex systems with correlations. Methodology/results: We introduce an extreme spatial model for multi-output response control systems that efficiently captures the dynamics using a bilinear function on two spatial domains for control variables and measurement locations. Marginal parameter modeling and extremal dependence have been investigated. In addition, an efficient graph-assisted composite likelihood estimation and corresponding computational algorithms are developed to cope with high-dimensional outputs. The application to composite aircraft production shows that the proposed model enables comprehensive analyses with superior predictive performance on extreme events compared to canonical methods. Managerial implications: Our method shows how to use an extreme spatial model for predicting extreme events and managing extreme risks in complex production systems such as aircraft. This can help achieve better quality management and operation safety in aircraft production systems and beyond.

We introduce two federated learning frameworks for the classical SPDnet model operating on symmetric positive definite (SPD) matrices with Stiefel-constrained parameters. Unlike standard Euclidean averaging, which violates orthogonality, our approach preserves geometric structure through two efficient aggregation strategies: ProjAvg, projecting arithmetic means onto the Stiefel manifold, and RLAvg, approximating tangent-space averaging via retractions and liftings. Both methods are computationally efficient, independent of the optimizer, and enable scalable federated learning for signal processing applications whose features are SPD matrices. Simulations on EEG motor imagery benchmarks show that FedSPDnet outperforms federated EEGnet in F1 score and robustness to federation and partial participation, while using fewer parameters per communication round.

The Pareto distribution plays a crucial role in various disciplines, necessitating robust goodness-of-fit tests for its validation. This article introduces a novel tests based on Stein's characterization and the Laplace transform, offering a fresh perspective on model assessment. We establish the asymptotic properties of the proposed test and evaluate its empirical performance against existing methods in terms of size and power. Our findings demonstrate that the new test often outperforms or performs comparably to established tests. In addition, real data applications illustrate its practical utility.

We study barycenters of filtered Gaussian processes in adapted Wasserstein space. The adapted Wasserstein distance refines classical optimal transport by requiring transport plans to respect the temporal flow of information, making it the natural metric for stochastic systems with filtration constraints, as in stochastic control, mathematical finance, and sequential decision problems. We prove that the unrestricted barycenter problem for weighted Fréchet means of filtered Gaussian inputs admits a solution with Gaussian underlying law, representable as an enlarged filtered Gaussian process but not necessarily as an ordinary one. The problem decomposes into finitely many classical Bures–Wasserstein barycenter problems for the covariance contributions of the successive innovations. We then treat the restricted problem, in which the barycenter is required to be an ordinary filtered Gaussian process, giving a rank and common-noise criterion for when the two problems agree, sufficient conditions for uniqueness, and first order optimality and regularity results. Under a martingale constraint we obtain an explicit solution via martingale projection and Bures–Wasserstein barycenters of the Gaussian increments. Beyond their intrinsic theoretical interest, our results provide a principled way to build representative models from collections of Gaussian stochastic systems, with applications to stochastic optimization, robust finance, and sequential statistical analysis.

We introduce Robust Bayesian Sequential Borrowing (RBSB), a framework for extrapolating evidence across adjacent subgroups in multi-population clinical programmes where studies are conducted in sequence and populations are ordered by clinical proximity. Conventional approaches weight all historical sources uniformly or exclude distant populations entirely, failing to reflect the natural gradient of similarity in such programmes. RBSB encodes the programme order through path-dependent borrowing via robust mixture priors that combine an informative component with a unit-information component to guard against prior-data conflict. Posterior weights, derived in closed form from marginal likelihood ratios, provide transparent dynamic attenuation when heterogeneity arises between sequential populations. The framework supports prospective evaluation of Bayesian Type I error, power, and extends naturally to assurance at both the study and programme level. Simulation studies demonstrate superior false-positive control relative to full pooling, while preserving substantial efficiency gains over standalone analyses. A case study of the START trial illustrates the approach across adult, adolescent, and paediatric populations. RBSB offers a practical, regulator-aligned method for disciplined evidence borrowing that exploits temporal and biological proximity while preventing implausible extrapolation across distant populations.

The Super Learner (SL) is a widely used ensemble method that combines point predictions from a library of learners based on their predictive performance. Interval predictions are of considerable practical interest because they allow uncertainty in predictions produced by an individual learner or an ensemble to be quantified. Several methods have been proposed for constructing interval predictions based on the SL, however, these approaches are typically justified using asymptotic arguments or rely on computationally intensive procedures such as the bootstrap. Conformal prediction (CP) is a machine learning framework for constructing prediction intervals with finite-sample and asymptotic coverage guarantees under mild conditions. We propose coupling CP with the SL through a natural construction that mirrors the original SL framework, using individual learner weights and combining learner-specific conformity scores via a weighted majority vote. We characterize the properties of the resulting SL-based prediction intervals for continuous outcomes. We cover settings under exchangeability, potential violations of exchangeability, and data-generating mechanisms exhibiting heteroscedasticity, sparsity, and other forms of distributional heterogeneity. A comprehensive simulation study shows that the conformalized SL achieves valid finite-sample coverage with competitive performance relative to the true data-generating mechanism. A central contribution of this work is an application to predicting creatinine levels using socio-demographic, biometric, and laboratory measurements. This example demonstrates the benefits of an ensemble with carefully selected learners designed to capture key aspects of complex regression functions, including non-linear effects, interactions, sparsity, heteroscedasticity, and robustness to outliers.