Evaluating large language models (LLMs) on open-ended tasks without ground-truth labels is increasingly done via the LLM-as-a-judge paradigm. A critical but under-modeled issue is that judge LLMs differ substantially in reliability; treating all judges equally can yield biased leaderboards and misleading uncertainty estimates. More data can make evaluation more confidently wrong under misspecified aggregation. We propose a judge-aware ranking framework that extends the Bradley-Terry-Luce model by introducing judge-specific discrimination parameters, jointly estimating latent model quality and judge reliability from pairwise comparisons without reference labels. We establish identifiability up to natural normalizations and prove consistency and asymptotic normality of the maximum likelihood estimator, enabling confidence intervals for score differences and rank comparisons. Across multiple public benchmarks and a newly collected dataset, our method improves agreement with human preferences, achieves higher data efficiency than unweighted baselines, and produces calibrated uncertainty quantification for LLM rankings.

Classifier-guided diffusion models generate conditional samples by augmenting the reverse-time score with the gradient of a learned classifier, yet it remains unclear whether standard classifier training procedures yield effective diffusion guidance. We address this gap by showing that, under mild smoothness assumptions on the classifiers, controlling the cross-entropy error at each diffusion step also controls the error of the resulting guidance vectors: classifiers achieving conditional KL divergence $\varepsilon^2$ from the ground-truth conditional label probabilities induce guidance vectors with mean squared error $\widetilde{O}(d \varepsilon )$. Our result yields an upper bound on the sampling error under classifier guidance and bears resemblance to a reverse log-Sobolev-type inequality. Moreover, we show that the classifier smoothness assumption is essential, by constructing simple counterexamples demonstrating that, without it, control of the guidance vector can fail for almost all distributions. To our knowledge, our work establishes the first quantitative link between classifier training and guidance alignment, yielding both a theoretical foundation for classifier guidance and principled guidelines for classifier selection.

Learning causal structures from observational data remains a fundamental yet computationally intensive task, particularly in high-dimensional settings where existing methods face challenges such as the super-exponential growth of the search space and increasing computational demands. To address this, we introduce VISTA (Voting-based Integration of Subgraph Topologies for Acyclicity), a modular framework that decomposes the global causal structure learning problem into local subgraphs based on Markov Blankets. The global integration is achieved through a weighted voting mechanism that penalizes low-support edges via exponential decay, filters unreliable ones with an adaptive threshold, and ensures acyclicity using a Feedback Arc Set (FAS) algorithm. The framework is model-agnostic, imposing no assumptions on the inductive biases of base learners, is compatible with arbitrary data settings without requiring specific structural forms, and fully supports parallelization. We also theoretically establish finite-sample error bounds for VISTA, and prove its asymptotic consistency under mild conditions. Extensive experiments on both synthetic and real datasets consistently demonstrate the effectiveness of VISTA, yielding notable improvements in both accuracy and efficiency over a wide range of base learners.

Transformers often appear to perform Bayesian reasoning in context, but verifying this rigorously has been impossible: natural data lack analytic posteriors, and large models conflate reasoning with memorization. We address this by constructing \emph{Bayesian wind tunnels} -- controlled environments where the true posterior is known in closed form and memorization is provably impossible. In these settings, small transformers reproduce Bayesian posteriors with $10^{-3}$-$10^{-4}$ bit accuracy, while capacity-matched MLPs fail by orders of magnitude, establishing a clear architectural separation. Across two tasks -- bijection elimination and Hidden Markov Model (HMM) state tracking -- we find that transformers implement Bayesian inference through a consistent geometric mechanism: residual streams serve as the belief substrate, feed-forward networks perform the posterior update, and attention provides content-addressable routing. Geometric diagnostics reveal orthogonal key bases, progressive query-key alignment, and a low-dimensional value manifold parameterized by posterior entropy. During training this manifold unfurls while attention patterns remain stable, a \emph{frame-precision dissociation} predicted by recent gradient analyses. Taken together, these results demonstrate that hierarchical attention realizes Bayesian inference by geometric design, explaining both the necessity of attention and the failure of flat architectures. Bayesian wind tunnels provide a foundation for mechanistically connecting small, verifiable systems to reasoning phenomena observed in large language models.

Standard Bayesian Optimization (BO) assumes uniform smoothness across the search space an assumption violated in multi-regime problems such as molecular conformation search through distinct energy basins or drug discovery across heterogeneous molecular scaffolds. A single GP either oversmooths sharp transitions or hallucinates noise in smooth regions, yielding miscalibrated uncertainty. We propose RAMBO, a Dirichlet Process Mixture of Gaussian Processes that automatically discovers latent regimes during optimization, each modeled by an independent GP with locally-optimized hyperparameters. We derive collapsed Gibbs sampling that analytically marginalizes latent functions for efficient inference, and introduce adaptive concentration parameter scheduling for coarse-to-fine regime discovery. Our acquisition functions decompose uncertainty into intra-regime and inter-regime components. Experiments on synthetic benchmarks and real-world applications, including molecular conformer optimization, virtual screening for drug discovery, and fusion reactor design, demonstrate consistent improvements over state-of-the-art baselines on multi-regime objectives.

We study model-agnostic post-hoc calibration methods intended to improve probabilistic predictions in supervised binary classification on real i.i.d. tabular data, with particular emphasis on conformal and Venn-based approaches that provide distribution-free validity guarantees under exchangeability. We benchmark 21 widely used classifiers, including linear models, SVMs, tree ensembles (CatBoost, XGBoost, LightGBM), and modern tabular neural and foundation models, on binary tasks from the TabArena-v0.1 suite using randomized, stratified five-fold cross-validation with a held-out test fold. Five calibrators; Isotonic regression, Platt scaling, Beta calibration, Venn-Abers predictors, and Pearsonify are trained on a separate calibration split and applied to test predictions. Calibration is evaluated using proper scoring rules (log-loss and Brier score) and diagnostic measures (Spiegelhalter's Z, ECE, and ECI), alongside discrimination (AUC-ROC) and standard classification metrics. Across tasks and architectures, Venn-Abers predictors achieve the largest average reductions in log-loss, followed closely by Beta calibration, while Platt scaling exhibits weaker and less consistent effects. Beta calibration improves log-loss most frequently across tasks, whereas Venn-Abers displays fewer instances of extreme degradation and slightly more instances of extreme improvement. Importantly, we find that commonly used calibration procedures, most notably Platt scaling and isotonic regression, can systematically degrade proper scoring performance for strong modern tabular models. Overall classification performance is often preserved, but calibration effects vary substantially across datasets and architectures, and no method dominates uniformly. In expectation, all methods except Pearsonify slightly increase accuracy, but the effect is marginal, with the largest expected gain about 0.008%.

In this paper, we consider the problem of parametric empirical Bayes estimation of an i.i.d. prior in high-dimensional Bayesian linear regression, with random design. We obtain the asymptotic distribution of the variational Empirical Bayes (vEB) estimator, which approximately maximizes a variational lower bound of the intractable marginal likelihood. We characterize a sharp phase transition behavior for the vEB estimator -- namely that it is information theoretically optimal (in terms of limiting variance) up to $p=o(n^{2/3})$ while it suffers from a sub-optimal convergence rate in higher dimensions. In the first regime, i.e., when $p=o(n^{2/3})$, we show how the estimated prior can be calibrated to enable valid coordinate-wise and delocalized inference, both under the \emph{empirical Bayes posterior} and the oracle posterior. In the second regime, we propose a debiasing technique as a way to improve the performance of the vEB estimator beyond $p=o(n^{2/3})$. Extensive numerical experiments corroborate our theoretical findings.

Bayesian optimisation is a sample efficient method for finding a global optimum of expensive black-box objective functions. Historic datasets from related problems can be exploited to help improve performance of Bayesian optimisation by adapting transfer learning methods to various components of the Bayesian optimisation pipeline. In this study we perform an empirical analysis of various ensemble-based transfer learning Bayesian optimisation methods and pipeline components. We expand on previous work in the literature by contributing some specific pipeline components, and three new real-time transfer learning Bayesian optimisation benchmarks. In particular we propose to use a weighting strategy for ensemble surrogate model predictions based on regularised regression with weights constrained to be positive, and a related component for handling the case when transfer learning is not improving Bayesian optimisation performance. We find that in general, two components that help improve transfer learning Bayesian optimisation performance are warm start initialisation and constraining weights used with ensemble surrogate model to be positive.

Hierarchical Bayesian models are increasingly used in large, inhomogeneous complex network dynamical systems by modeling parameters as draws from a hyperparameter-governed distribution. However, theoretical guarantees for these estimates as the system size grows have been lacking. A critical concern is that hyperparameter estimation may diverge for larger networks, undermining the model's reliability. Formulating the system's evolution in a measure transport perspective, we propose a theoretical framework for estimating hyperparameters with mean-type observations, which are prevalent in many scientific applications. Our primary contribution is a nonasymptotic bound for the deviation of estimate of hyperparameters in inhomogeneous complex network dynamical systems with respect to network population size, which is established for a general family of optimization algorithms within a fixed observation duration. While we firstly establish a consistency result for systems with independent nodes, our main result extends this guarantee to the more challenging and realistic setting of weakly-dependent nodes. We validate our theoretical findings with numerical experiments on two representative models: a Susceptible-Infected-Susceptible model and a Spiking Neuronal Network model. In both cases, the results confirm that the estimation error decreases as the network population size increases, aligning with our theoretical guarantees. This research proposes the foundational theory to ensure that hierarchical Bayesian methods are statistically consistent for large-scale inhomogeneous systems, filling a gap in this area of theoretical research and justifying their application in practice.

Reinforcement learning (RL) has achieved remarkable success in real-world decision-making across diverse domains, including gaming, robotics, online advertising, public health, and natural language processing. Despite these advances, a substantial gap remains between RL research and its deployment in many practical settings. Two recurring challenges often underlie this gap. First, many settings offer limited opportunity for the agent to interact extensively with the target environment due to practical constraints. Second, many target environments often undergo substantial changes, requiring redesign and redeployment of RL systems (e.g., advancements in science and technology that change the landscape of healthcare delivery). Addressing these challenges and bridging the gap between basic research and application requires theory and methodology that directly inform the design, implementation, and continual improvement of RL systems in real-world settings. In this paper, we frame the application of RL in practice as a three-component process: (i) online learning and optimization during deployment, (ii) post- or between-deployment offline analyses, and (iii) repeated cycles of deployment and redeployment to continually improve the RL system. We provide a narrative review of recent advances in statistical RL that address these components, including methods for maximizing data utility for between-deployment inference, enhancing sample efficiency for online learning within-deployment, and designing sequences of deployments for continual improvement. We also outline future research directions in statistical RL that are use-inspired -- aiming for impactful application of RL in practice.