Sarah Rudd, who once ran analytics for Arsenal, made her name applying the tenets of probability theory to movements on the pitch. Even she admits not everything can be solved with data. The role of advanced analytics in sports is a contentious subject. To its defenders, data-driven pragmatism is a natural evolutionary step in the way we play and watch games. For detractors, the approach prioritizes results above all else and drains the soul from a pursuit that should be spontaneous and joyful.

Bayesian Optimal Experimental Design (BOED) provides a rigorous framework for decision-making tasks in which data acquisition is often the critical bottleneck, especially in resource-constrained settings. Traditionally, BOED typically selects designs by maximizing expected information gain (EIG), commonly defined through the Kullback-Leibler (KL) divergence. However, classical evaluation of EIG often involves challenging nested expectations, and even advanced variational methods leave the underlying log-density-ratio objective unchanged. As a result, support mismatch, tail underestimation, and rare-event sensitivity remain intrinsic concerns for KL-based BOED. To address these fundamental bottlenecks, we introduce an IPM-based BOED framework that replaces density-based divergences with integral probability metrics (IPMs), including the Wasserstein distance, Maximum Mean Discrepancy, and Energy Distance, resulting in a highly flexible plug-and-play BOED framework. We establish theoretical guarantees showing that IPM-based utilities provide stronger geometry-aware stability under surrogate-model error and prior misspecification than classical EIG-based utilities. We also validate the proposed framework empirically, demonstrating that IPM-based designs yield highly concentrated credible sets. Furthermore, by extending the same sample-based BOED template in a plug-and-play manner to geometry-aware discrepancies beyond the IPM class, illustrated by a neural optimal transport estimator, we achieve accurate optimal designs in high-dimensional settings where conventional nested Monte Carlo estimators and advanced variational methods fail.

In online clustering problems, there is often a large amount of uncertainty over possible cluster assignments that cannot be resolved until more data are observed. This difficulty is compounded when clusters follow complex distributions, as is the case with text data. Sequential Monte Carlo (SMC) methods give a natural way of representing and updating this uncertainty over time, but have prohibitive memory requirements for large-scale problems. We propose a novel SMC algorithm that decomposes clustering problems into approximately independent subproblems, allowing a more compact representation of the algorithm state. Our approach is motivated by the knowledge base construction problem, and we show that our method is able to accurately and efficiently solve clustering problems in this setting and others where traditional SMC struggles.

This research considers a scalable inference for spatial data modeled through Gaussian intrinsic conditional autoregressive (ICAR) structures. The classical estimation method, restricted maximum likelihood (REML), requires repeated inversion and factorization of large, sparse precision matrices, which makes this computation costly. To sort this problem out, we propose a variational restricted maximum likelihood (VREML) framework that approximates the intractable marginal likelihood using a Gaussian variational distribution. By constructing an evidence lower bound (ELBO) on the restricted likelihood, we derive a computationally efficient coordinate-ascent algorithm for jointly estimating the spatial random effects and variance components. In this article, we theoretically establish the monotone convergence of ELBO and mathematically exhibit that the variational family is exact under Gaussian ICAR settings, which is an indication of nullifying approximation error at the posterior level. We empirically establish the supremacy of our VREML over MLE and INLA.

In many modern applications, a carefully designed primary study provides individual-level data for interpretable modeling, while summary-level external information is available through black-box, efficient, and nonparametric machine-learning predictions. Although summary-level external information has been studied in the data integration literature, there is limited methodology for leveraging external nonparametric machine-learning predictions to improve statistical inference in the primary study. We propose a general empirical-likelihood framework that incorporates external predictions through moment constraints. An advantage of nonparametric machine-learning prediction is that it induces a rich class of valid moment restrictions that remain robust to covariate shift under a mild overlap condition without requiring explicit density-ratio modeling. We focus on multinomial logistic regression as the primary model and address common data-quality issues in external sources, including coarsened outcomes, partially observed covariates, covariate shift, and heterogeneity in generating mechanisms known as concept shift. We establish large-sample properties of the resulting fused estimator, including consistency and asymptotic normality under regularity conditions. Moreover, we provide mild sufficient conditions under which incorporating external predictions delivers a strict efficiency gain relative to the primary-only estimator. Simulation studies and an application to the National Health and Nutrition Examination Survey on multiclass blood-pressure classification.

Gaussian processes (GPs) offer appealing properties but are costly to train at scale. Sparse variational GP (SVGP) approximations reduce cost yet still rely on Cholesky decompositions of kernel matrices, ill-suited to low-precision, massively parallel hardware. While one can construct valid variational bounds that rely only on matrix multiplications (matmuls) via an auxiliary matrix parameter, optimising them with off-the-shelf first-order methods is challenging. We make the inverse-free approach practical by proposing a better-conditioned bound and deriving a matmul-only natural-gradient update for the auxiliary parameter, markedly improving stability and convergence. We further provide simple heuristics, such as step-size schedules and stopping criteria, that make the overall optimisation routine fit seamlessly into existing workflows. Across regression and classification benchmarks, we demonstrate that our method 1) serves as a drop-in replacement in SVGP-based models (e.g., deep GPs), 2) recovers similar performance to traditional methods, and 3) can be faster than baselines when well tuned.

We present AutoStan, a framework in which a command-line interface (CLI) coding agent autonomously builds and iteratively improves Bayesian models written in Stan. The agent operates in a loop, writing a Stan model file, executing MCMC sampling, then deciding whether to keep or revert each change based on two complementary feedback signals: the negative log predictive density (NLPD) on held-out data and the sampler's own diagnostics (divergences, R-hat, effective sample size). We evaluate AutoStan on five datasets with diverse modeling structures. On a synthetic regression dataset with outliers, the agent progresses from naive linear regression to a model with Student-t robustness, nonlinear heteroscedastic structure, and an explicit contamination mixture, matching or outperforming TabPFN, a state-of-the-art black-box method, while remaining fully interpretable. Across four additional experiments, the same mechanism discovers hierarchical partial pooling, varying-slope models with correlated random effects, and a Poisson attack/defense model for soccer. No search algorithm, critic module, or domain-specific instructions are needed. This is, to our knowledge, the first demonstration that a CLI coding agent can autonomously write and iteratively improve Stan code for diverse Bayesian modeling problems.

Model-based design of experiments (MBDOE) is essential for efficient parameter estimation in nonlinear dynamical systems. However, conventional adaptive MBDOE requires costly posterior inference and design optimization between each experimental step, precluding real-time applications. We address this by combining Deep Adaptive Design (DAD), which amortizes sequential design into a neural network policy trained offline, with differentiable mechanistic models. For dynamical systems with known governing equations but uncertain parameters, we extend sequential contrastive training objectives to handle nuisance parameters and propose a transformer-based policy architecture that respects the temporal structure of dynamical systems. We demonstrate the approach on four systems of increasing complexity: a fed-batch bioreactor with Monod kinetics, a Haldane bioreactor with uncertain substrate inhibition, a two-compartment pharmacokinetic model with nuisance clearance parameters, and a DC motor for real-time deployment.

Uncertainty quantification for image data is dominated by complex deep learning methods, yet the field lacks an interpretable, mathematically grounded baseline. We propose Bayesian scattering to fill this gap, serving as a first-step baseline akin to the role of Bayesian linear regression for tabular data. Our method couples the wavelet scattering transform-a deep, non-learned feature extractor-with a simple probabilistic head. Because scattering features are derived from geometric principles rather than learned, they avoid overfitting the training distribution. This helps provide sensible uncertainty estimates even under significant distribution shifts. We validate this on diverse tasks, including medical imaging under institution shift, wealth mapping under country-to-country shift, and Bayesian optimization of molecular properties. Our results suggest that Bayesian scattering is a solid baseline for complex uncertainty quantification methods.

Learned priors based on deep generative models offer data-driven regularization for seismic inversion, but training them requires a dataset of representative subsurface models -- a resource that is inherently scarce in geoscience applications. Since the training objective of most generative models can be cast as maximum likelihood on a finite dataset, any such model risks converging to the empirical distribution -- effectively memorizing the training examples rather than learning the underlying geological distribution. We show that the posterior under such a memorized prior reduces to a reweighted empirical distribution -- i.e., a likelihood-weighted lookup among the stored training examples. For diffusion models specifically, memorization yields a Gaussian mixture prior in closed form, and linearizing the forward operator around each training example gives a Gaussian mixture posterior whose components have widths and shifts governed by the local Jacobian. We validate these predictions on a stylized inverse problem and demonstrate the consequences of memorization through diffusion posterior sampling for full waveform inversion.