Bayesian

Sparse Gaussian processes achieve $O(N)$ inference by replacing the kernel with an appropriate expansion in a fixed basis $\{φ_j\}$ on the input space. Given a compute budget $M \ll N$, practitioners conventionally truncate the basis to its first $M$ entries. Nothing in the formalism, however, prevents one from selecting only those $M$ basis functions that matter for the data at hand. This would avoid spending budget on basis functions where there is no signal, but it requires a criterion for ranking the candidates. We propose three such criteria derived from an information-theoretic view of the basis-function selection problem. Each criterion matches a different state of knowledge at selection time: a no-data state, a no-prior state, and an in-between state. We then study the performance of truncation versus selection strategies on six UCI regression benchmarks across three basis families: Hilbert-space Gaussian processes (HSGP), variational Fourier features (VFF), and variational inducing spherical harmonics (VISH). We observe that the no-data criterion is a safe default, matching or improving on truncation for HSGP, VFF and VISH, with substantial gains for VISH and improvements over a recently developed selection heuristic for that basis family. The data-aware no-prior and in-between criteria provide substantial gains over truncation specifically for HSGP, which is the most broadly used of the three families in practice.

Hierarchical data is ubiquitous in the empirical sciences and is most commonly analyzed with generalized linear mixed-effects models (GLMMs). Bayesian inference for GLMMs yields calibrated uncertainty but requires MCMC; the No-U-Turn Sampler (NUTS) is the gold standard but is slow and must restart from scratch for every new dataset, model and prior. We introduce metabeta, a pretrained neural network for prior-amortized in-context Bayesian inference over GLMMs. Unlike previous neural posterior estimators that fix the prior at training time, metabeta accepts prior families and hyperparameters as inputs at test time, enabling zero-shot generalization. Two set transformers and conditional normalizing flows mirror the posterior's two-level structure (global parameters shared across groups, local parameters per group). The model is trained on millions of realistic simulated datasets spanning continuous, binary, and count outcomes. By default, the flow posterior is refined by Independence Metropolis-Hastings against the unnormalized posterior, so its correctness rests on the sampler rather than the network; this yields tuning-free inference two to three orders of magnitude faster than NUTS. Alternatively, the flow can warm-start NUTS, giving nearly identical inference with substantially increased speed and stability. On controlled benchmarks with ground-truth parameters, metabeta matches NUTS in parameter recovery, calibration and out-of-sample prediction. On out-of-distribution real datasets, its posteriors closely match those of NUTS across all parameter types, and they remain faithful under misspecified likelihoods and priors, out-of-distribution predictors, collinear designs, and data-poor regimes. The model is open-source and open-weights and thus immediately deployable.

Identifying linear time-invariant (LTI) dynamical systems is challenging when trajectories are short, noisy, or high-dimensional. Traditional system identification typically treats each system independently and cannot exploit shared structure across related systems. We propose PBML-LTI, a PAC-Bayesian meta-learning framework for few-shot LTI system identification that learns a transferable prior over task-specific dynamics while preserving task heterogeneity. Each task corresponds to an unknown LTI system, and the meta-learner uses training trajectories to learn a data-dependent prior over transition matrices. For a new system with limited data, PBML-LTI performs Bayesian adaptation under this prior to obtain a task-specific posterior, providing accurate estimates and principled uncertainty quantification. A key challenge is temporal dependence, since LTI trajectories violate the i.i.d. assumptions underlying most PAC-Bayes meta-learning analyses. We address this with a martingale PAC-Bayes analysis for dependent trajectory losses and derive a support-query predictive-risk bound that motivates a fit-KL meta-training objective. The bound clarifies the roles of empirical fit, posterior complexity, and prior quality in few-shot adaptation under sequential dependence. We further derive corollaries for transition-matrix recovery and multi-step trajectory prediction, connecting uncertainty-aware meta-identification with finite-sample guarantees for dependent dynamical data.

This paper presents a real-time semantic world modeling framework specialized for precision agriculture using autonomous robots. The framework combines probabilistic mapping of objects and their semantic attributes, updated through Bayesian inference, with a graph-based Simultaneous Localization and Mapping (SLAM) approach implemented using $g^2o$, a general framework for graph optimization. This integration enables accurate mapping and localization without relying solely on GPS. By leveraging semantic information such as plant type, size, and health, the robot can perform tasks while mapping and localizing itself within a field of crops. The proposed framework was validated through Gazebo simulations and physical experiments on an indoor field with artificial plants using Boston Dynamics' robot dog Spot. A YOLOv8n object detection model was trained to extract object and semantic data from depth camera observations. These simulations and experiments demonstrate that the system can successfully perform real-time mapping of up to at least 400 plants.

We study the Bayes-optimal spherical linear model as the ambient dimension and sample size grow proportionally, under a quantitative Marchenko--Pastur spectral-regularity condition on the design. This condition is satisfied by normalized i.i.d. designs with standardized entries of finite fourth moment, but does not require entrywise independence or impose conditions on the singular vectors. Under this condition, we prove a quantitative all-temperature TAP approximation and characterize the posterior geometry. For the natural finite-aspect-ratio TAP functional, the normalized spherical free energy and the TAP optimum differ by $O_P(p^{-1})$. Each is within $O_P(p^{-1/2})$ of its explicit deterministic equivalent, and this fluctuation scale is sharp. Uniformly over all global TAP maximizers, the normalized squared Euclidean distance to the spherical posterior mean is $O_P(p^{-1})$. We also prove that the posterior mass outside a data-dependent band determined by the ridge estimator has sharp exponential order. More precisely, uniformly over sufficiently small band widths $\varepsilon$, the logarithm of this mass is at most $-cp\varepsilon^2+O_P(1)$. For every fixed geometrically admissible width, a spherical-cap construction gives a matching exponential-order lower bound on this mass. For every deterministic sequence of widths $\varepsilon_p\gg p^{-1/2}$, the corresponding bands capture asymptotically all posterior mass.

We train neural networks on synthesized frames to approach the optimum Bayes estimator for dense emitter localization. The result justifies the future work on training neural networks to achieve high-throughput large-FOV super spatiotemporal resolution SMLM.

The problem of multiobjective optimization under uncertainties is often approached by taking the expectation of each objective. In this work, we propose instead to formulate this as a Bayesian decision problem and to rely on the expected value of the hypervolume, which is to be maximized with respect to a finite set of input points. We show that this can be performed using methods based on gradients in a stochastic optimization framework, provided that care is taken with respect to dominated points. Moreover, in the absence of readily available differentiable code, we propose to use Gaussian Processes as differentiable surrogate models, in order to perform the optimization. An additional contribution in this work are some active learning strategies, through acquisition functions which helps construct a surrogate model well-designed for the multiobjective optimization problem at stake. These strategies are compared on simple analytical problems to assess their performances.

Active subspace methods provide a framework for quantifying predictive uncertainty in high-dimensional models by identifying and performing inference along parameter directions that have the greatest influence on the model output. However, the construction of active subspaces requires storing many full-dimensional model gradients, which becomes prohibitive as model size increases. We address this limitation by proposing Compressed Active Subspaces (CAS), a scalable approach that first maps the model parameters to a compressed space using a structured isometric embedding and then constructs the active subspace within this reduced parameterization. Our approach substantially reduces the memory required for active subspace construction and enables Bayesian inference for large models where standard active subspace methods become impractical. We demonstrate the scalability of CAS on neural networks of increasing size while maintaining predictive performance and robust uncertainty estimates.

Generative models are increasingly central to many de novo discovery pipelines, in which designs are generated at scale and filtered through virtual screens to determine a set of candidates to experimentally validate. While Bayesian optimization (BO) is a natural fit for this setting, as it uses past evaluations to guide future proposals, the computational overhead required for its sequential decision-making becomes a bottleneck when virtual screens are relatively cheap. We make BO practical in this regime by exploiting the unique combination of a linear model constrained to a spherical domain where high-dimensional latents concentrate. We build off recent work justifying the use of linear surrogates, while deriving nearly closed-form solutions to the surrogate modelling and acquisition problems that exploit spherical symmetry. The result is at least a 100x speedup over state-of-the art baselines, with matching or improved performance across molecular and image generation benchmarks. Altogether, our method makes BO a practical drop-in for de novo pipelines where it was previously too slow to consider.

Bayesian Optimization (BO) is widely used for optimizing expensive black-box functions, yet many real-world optimization problems contain substantially richer information than function evaluations alone. Examples include training curves in hyperparameter optimization, expert notes and images in scientific experimentation, and prior knowledge about where optima may lie. We show that large language models (LLMs) can effectively leverage such rich auxiliary information to guide optimization. Motivated by these findings, we develop three methods for incorporating auxiliary information into BO using LLMs. Across hyperparameter optimization benchmarks and a real-world nuclear fusion optimization task, our methods consistently outperform both standard BO and existing LLM-based optimization approaches. Our results demonstrate the effectiveness of LLMs for leveraging rich auxiliary information in BO.

Large language models (LLMs) are increasingly deployed in high-stakes domains such as housing screening. While alignment techniques mitigate explicit racial bias in generated text, they often leave covert attitudinal associations in internal probability distributions untouched. Adapting the matched-guise sociolinguistic paradigm, we examine covert dialect bias in housing-related social judgments across four varieties: Standard American English (SAE), African American Vernacular English (AAVE), Nigerian Standard English (NSE), and Nigerian Pidgin (NP). AAVE reflects the racialized dialect studied in prior covert-bias evaluations, whereas NSE and NP represent Black African, postcolonial varieties absent from this literature. Using 260 meaning-matched sentence quadruples and log-probability scoring over housing-relevant adjectives, we probe ten open-weight LLMs across three contexts varying in social proximity: tenant screening, neighbor acceptance, and roommate selection. Across all ten models, AAVE and NP are consistently associated with more negative adjectives than SAE, with NP penalized most severely. Crucially, each dialect is penalized via distinct stereotype clusters rather than a generic non-standard category. NSE, which carries institutional prestige, displays a context-dependent shift: favored over SAE in formal tenant screening but increasingly penalized as social proximity grows. Our findings reveal that LLMs inherit covert dialect bias along both racial identity and prestige dimensions, echoing documented human housing discrimination and demonstrating its reach across postcolonial English varieties.

Weighted majority votes are central to many successful ensemble methods. PAC-Bayesian theory provides tight generalization guarantees for such models by analyzing the expected risk of stochastic classifiers, while analyzing the risk of deterministic majority votes relies on surrogate bounds. To avoid these surrogates, Zantedeschi et al. ( 2021) introduced guarantees for stochastic majority votes, but the resulting models remain randomized. In this paper, we propose a derandomization framework for stochastic majority votes. To do so, we apply recent advances in disintegrated PAC-Bayesian theory directly to the space of majority vote weight vectors, transforming stochastic guarantees into certificates for a single deterministic majority vote. We derive two families of high-probability generalization bounds, covering both data-independent and data-dependent constructions of the ensemble, which naturally lead to a self-bounding learning algorithm optimizing deterministic majority vote guarantees.

Probabilistic certification of Bayesian neural networks lower-bounds the posterior probability that a model satisfies a verifier-defined safety property. In one-shot federated Bayesian learning, however, the deployed model is obtained by aggregating parameters drawn from client-specific posterior distributions, so local certificates do not directly guarantee safety of the aggregated model. This paper develops a deployment-consistent certification framework by propagating local posterior events through the deployment aggregation rule, with an exact geometric characterization for Federated Averaging (FedAvg). Each client constructs disjoint hyper-rectangular regions in parameter space and computes their probability masses. The server forms Cartesian products of these regions, maps them through the deployment rule, and retains a product event only when its aggregation image is verified to satisfy the safety property. Under independent client posteriors, each product-event probability factorizes into local masses, and summing verified disjoint events yields a lower bound on safety probability of the deployed model. For FedAvg with nonnegative aggregation coefficients, the image of a Cartesian product of axis-aligned hyper-rectangles is exactly a weighted hyper-rectangle, introducing no set over-approximation. We distinguish the proposed transported-event certificate from direct certification under posterior distributions induced by FedAvg and Product-of-Gaussians aggregation. Experiments on MNIST and Fashion-MNIST under label-Dirichlet heterogeneity show that the transported FedAvg certificate ranges from 22.51% to 46.89%, while direct global certificates range from 72.05% to 91.39%. Results show that predictive accuracy and certifiable safety do not necessarily follow the same trend, and that global posterior constructions can exhibit distinct certification behavior across architectures.

Tracking the energy transition requires reliable statistics on renewable deployment. Rooftop photovoltaics (PV) are especially hard to track, owing to their decentralised nature, and the resulting inaccuracies in official statistics are known but not quantified. Remote sensing offers an independent way to identify rooftop PV systems. We introduce a Bayesian framework to estimate the ground-truth rooftop PV capacity from remote sensing detections, turning an imperfect detector into an uncertainty-aware measurement instrument. Applied to France, the corrected detections estimate a capacity of 4.03 GWp [3.96--4.11] (99% credible interval) of rooftop PV below 36 kWp, matching the transmission system operator's connection data within 3.3% nationally, while identifying local under-reports of up to 61% of local capacity. We also document and quantify a significant truncation bias in French rooftop PV open data. Beyond France, the approach paves the way for more reliable estimates of rooftop PV capacity worldwide.

We introduce loss-conditioned state execution, a model-agnostic method that decides whether to execute a world model's fixed feasible proposal or retain the current state. Predictive informativeness alone, however, does not establish whether an update will reduce downstream loss. Occurrence ranking can approach perfection while persistence remains the unique absolute-loss Bayes action. Two transition laws can also share occurrence information and conditional variance yet require opposite absolute-loss decisions. We formalize state movability as the existence of a loss-reducing feasible correction and distinguish it from the benefit of a particular proposal. Our method constructs a loss-specific feasible proposal from a predictive distribution and evaluates its groupwise bounded-loss gain over persistence on independent calibration units. The proposal is executed only in groups with a positive simultaneous lower confidence bound. For fixed proposals and groups with bounded unit losses, we prove that every accepted group has lower expected loss than persistence with high probability when calibration units are i.i.d. draws from the target population. Experiments on public forecasting and action-conditioned dynamics benchmarks show supported updates and a trade-off between certification and coverage. On 28,684 held-out M4 Monthly series, the method executes the proposal for 14.0% of series and achieves bounded loss 0.588, compared with 0.599 for persistence and 0.621 for always executing the proposal. The paired 95% bootstrap intervals for both comparisons lie below zero. In constrained forecasting of six unhealthy-inventory types from JD$\mbox{.}$com, a leading e-retailer in China, strong occurrence-ranking signal coexists with a loss-based preference for persistence, illustrating why event predictability and state execution must be evaluated separately.

Bayesian optimisation (BO) typically relies on a single global Gaussian process (GP) model as its surrogate model. However, GP regression has cubic computational complexity in the number of training data points, limiting its applicability to large-scale optimisation problems. The product-of-experts Gaussian process model with uncertainty calibration (GP-pro-c) mitigates this limitation by combining multiple local GP experts, enabling improved uncertainty quantification, reduced computational cost, and preservation of global correlations. Despite these desirable properties, the use of GP-pro-c in BO has not been thoroughly studied. This paper introduces BO-pro-c, a Bayesian optimisation algorithm that uses GP-pro-c as its surrogate model, and evaluate its performance across a diverse range of BO settings. Experimental results suggest that BO-pro-c maintains competitive optimisation performance while achieving a 0.9% reduction in simple regret and a 39.4% reduction in computational overhead relative to a BO algorithm based on a single global GP model.

While homographies are fundamental to many computer vision tasks, the majority of conventional estimation techniques provide only point estimates without directly quantifying uncertainty introduced by noisy observations. Uncertainty, though, propagates to subsequent processing steps such as camera calibration and 3D reconstruction and is particularly relevant in safety-critical and socially relevant fields including medical imaging, autonomous driving, and defense. We present a fast Bayesian formulation for homography estimation from point correspondences that explicitly incorporates measurement uncertainty and prior knowledge while providing a posterior distribution over the homography parameters. A closed-form solution of the posterior mean of the homography is derived in homogeneous coordinates and supplemented by an iterative Bayesian approach to handle non-linearities. Synthetic experiments demonstrate the applicability to projective transformations and show improved estimation accuracy over DLT under varying noise conditions. Image stitching experiments further demonstrate applicability to real image correspondences while additionally providing uncertainty information.

Large language models are increasingly used as natural-language interfaces to structured data, yet they remain unreliable when answers require consistent evidence conditioning, dependency-aware reasoning, and uncertainty estimation. Bayesian networks provide an explicit probabilistic reasoning layer, but learning useful structures from data remains costly and fragile at scale. We introduce ABSOL, a hybrid LLM-guided Bayesian network structure-learning framework that uses LLMs as bounded semantic guides. Across five discrete BN benchmarks spanning 27 to 1041 nodes, ABSOL is the only evaluated method to produce a viable graph on every benchmark, and achieves the highest Edge F_1 on every benchmark larger than 27 nodes with GPT-5.4. The four LLM augmentations, which contribute complementary semantic evidence to the statistical backbone, improve Edge F_1 over the non-LLM aggregation backbone by +0.23 on average. Complementary post-hoc refinement experiments suggest that these gains depend in part on limiting the LLM's authority over the final structure. Together, these results show that language-derived semantic knowledge can substantially improve scalable probabilistic structure learning when used as bounded guidance within a statistically grounded reasoning pipeline. The code for ABSOL is available at github.com/megagonlabs/absol-bn.

As the population ages, the number of patients with cardiovascular diseases continues to increase, highlighting the need for early detection before progression to severe and irreversible functional decline. Consequently, estimating cardiovascular biomarkers from non-invasive biosignals, such as photoplethysmography (PPG) and arterial pressure wave (APW) signals, has attracted increasing attention. These signals can be measured using wearable and cuff-type devices. Previous studies have used PPG and APW signals to estimate cardiovascular biomarkers. However, these signals exhibit strong similarities in both the temporal and frequency domains and primarily reflect peripheral and arterial pulse waveforms. Therefore, they may provide limited information about cardiac mechanical function. In contrast, the quantitative impact of additional biosignals, such as ballistocardiography (BCG), which reflect the body's minute mechanical responses to cardiac ejection, remains unclear. In this study, we generated synthetic PPG, APW, and BCG signals from a unified whole-body cardiovascular circulation model and evaluated the complementary contribution of BCG to probabilistic cardiovascular biomarker estimation. We estimated posterior distributions of cardiovascular biomarkers using neural posterior estimation and simulation-based inference. The results showed that adding BCG signals significantly improved estimation performance. Furthermore, even in ill-posed cases where PPG and APW alone produced multimodal posterior distributions, adding BCG yielded unimodal posterior distributions. These findings provide fundamental insights into signal selection for estimating cardiovascular dynamics.

In practice, various learning scenarios provide access to auxiliary features exclusively during training. Incorporating such data to enhance model performance gave rise to a paradigm known as Learning Using Privileged Information (LUPI). While this extra information is intended to improve the resulting model, establishing a generalized, cohesive understanding of how privileged information (PI) transfers useful knowledge remains a challenge. Vapnik's original theory and subsequent works offer performance guarantees in certain cases, but these results are inherently per-algorithm and rely on setting-specific proof approaches. Consequently, a more general framework explaining how and when PI transfers useful knowledge is still missing. To bridge this gap, we introduce an algorithm-agnostic, information-theoretic approach based on the PAC-Bayes framework. Rather than asking whether a particular algorithm exploits PI, we ask how much value it could offer: comparing the tightest achievable risk bound with and without PI yields its potential - an upper limit on the extractable gain. We introduce a metric that quantifies this potential directly from empirical training risk, bypassing the need for test-time data access, and validate our findings in both supervised and unsupervised settings. The results demonstrate a robust correspondence between our training-time metric and true test-time performance gains. Ultimately, this work takes a necessary step toward an information-theoretic understanding of LUPI, and quantifying the potential of privileged features before committing to a model.

When multiple LLM agents yield conflicting answers, the decision-making process dictates whether agent diversity improves performance or merely compounds shared errors. Existing collective decision-making methods, including voting, electoral rules, and LLM judges, rely on forward reasoning: they map evidence to labels in one direction. Although these methods can combine diverse forward traces, they still aggregate estimates that share this evidence-to-label factorization and can inherit correlated errors within the forward pool. We therefore construct a reverse posterior for each instance through Bayesian backward reasoning from an explicit likelihood. The forward and reverse posteriors provide differently factorized approximations of the underlying posterior. Because estimates from different factorizations may tend to share the same error less often, we use Jensen-Shannon divergence to rank agents by cross-path consistency. This cross-path consistency signal underlies three strategies: hard selection (MinJS), soft reweighting (FwdJS), and log-linear fusion (LogLin). Evaluated on DDXPlus across five LLM backbones, our proposed strategies show consistent improvements: MinJS outperforms random selection across all backbones, FwdJS generally improves over the strongest baseline, and LogLin achieves the best performance among the evaluated methods, with its largest gains on the subset where the agents disagree. Despite its weaker standalone accuracy, the reverse posterior serves as a more useful anchor than forward-only alternatives, providing complementary information for collective decision-making. When labeled data are available, a lightweight two-stage calibration can further refine the reverse anchor and improve aggregation performance.

Many ML datasets are constructed by running a detector, heuristic, or model over candidate pools; accepted items become labels. Dataset precision is then governed by true-positive prevalence in each pool via Bayes, not solely by detector quality. Using one instrument and period, we hold a detector-defined event dataset plus an independent official index labeling every detected item as real or phantom. One detector, three pools yield phantom rates 81.7%, 9.0%, and 0.0%. Transferring precision from the two high-rate pools to the low-rate pool predicts 0.955 versus measured 0.183, a +422% error; the Bayes expression predicts all three within 3.3%. The detected response curve is an exact convex combination of a true-event and a phantom component (residual 1.1e-16), with phantoms outnumbering true events 473 to 308, so contamination is a second signal with detector-inherited shape, not additive noise. Contamination direction depends on the estimator: on identical windows one statistic is diluted and another inflated because its denominator is also contaminated. A common normalization turns the estimator into a mean of ratios whose expectation need not exist; on the same 335 events it returns 0.40 where the well-defined estimator returns 0.10.

We provide methods for calculating the robustness of the predictions of two types of generative classifiers whose underlying distribution is a Probabilistic Graphical Model (PGM): naive Bayes classifiers and generative forests (a probabilistic extension of random forests). Following the paradigm of robustness quantification, we define the robustness of a prediction as the extent to which the distribution of the classifier can be perturbed without changing this prediction. We consider perturbations obtained by varying the local models of the PGMs within general neighborhoods and focus in particular on epsilon-contamination, total variation distance and chi-squared divergence balls. We test our methods on benchmark datasets, demonstrate that the robustness value of a prediction serves as an indicator for its trustworthiness and compare our approach with other such indicators.

Although probabilistic statements are ubiquitous, foundational disagreements persist about their understanding, as exemplified by debates between Bayesians and frequentists; moreover, it is unclear when and why acting on them actually leads to desirable outcomes. Here, we argue that every probability is the output of a \emph{prediction method}, that is, it depends on both a particular way of constructing abstractions and a way of transforming them into predictions. Through this, we provide a unifying perspective on supposedly different kinds of probabilities and show that even supposedly objective ones are model-dependent. We demonstrate that when a finite calibration criterion is met, one can anticipate the distribution of utilities for a given policy and inform successful decision-making on finite sets of events. Based on the notion of prediction methods, inductive arguments, and the probability calculus, we explain the feasibility of the calibration criterion in many settings. Overall, we develop a coherent perspective on probabilities and their use, connecting key intuitions behind other interpretations along the way.

Amari's contributions to information geometry and machine learning are well known. Here, we revisit Amari's work on Bayesian duality which has not received as much attention. We connect Amari's Bayesian duality to a convex duality of Bayes' rule. Using this connection, we present a generalization of Amari's Bayesian duality and discuss its relevance for modern artificial intelligence.

Sparsity is a powerful structural resource in optimization and statistics. We develop frameworks for leveraging sparsity in sampling problems over the Hamming slice $\mathcal{X}_k^d:=\{\mathbf{x}\in\{\pm 1\}^d:|\{i:\mathbf{x}_i=1\}|=k\}$, in high-dimensional regimes where $k\ll d$ (i.e., where $\mathcal{X}_k^d$ is \emph{highly magnetized}). We use our frameworks to design improved samplers for canonical problems in the study of \emph{Ising models} and \emph{Bayesian sparse linear regression}. Our first main result considers the \emph{Sherrington--Kirkpatrick} (SK) model restricted to fixed-magnetization slices $\mathcal{X}_k^d$. We give a polynomial-time sampler for fixed-magnetization SK models at any inverse temperature $β>0$, under arbitrary external fields, provided that $k\le c_βd$ for an appropriate constant $c_β$. By combining this result with an annealing strategy for estimating normalizing constants, we obtain polynomial-time samplers for the SK model at arbitrarily low temperatures under a sufficiently strong external field of strength $h$. In the large-$β$ limit, our framework permits sampling at field strengths within constant factors of the \emph{Almeida--Thouless line} delineating the replica-symmetric and replica-symmetry-breaking regions ([dAT78]), improving polynomially over the field strength $h(β)$ required by the recent work of [BAR26]. Our second main result concerns the measurement complexity of polynomial-time Bayesian sparse linear regression. Recent work by [KSTZ25] shows how to sample from the canonical \emph{Gaussian spike-and-slab posterior} with expected sparsity $k$, at any signal-to-noise ratio, given $n\gtrsim k^3\log^3 d$ Gaussian measurements. We improve this requirement to $n\gtrsim k^{3/2}\log^2 d+k\log^3 d$, using a common sparsity-aware framework underlying both our results.

In this paper we derive a Probably Approximately Correct (PAC)-Bayesian error bound for partially observed linear time-invariant (LTI) stochastic dynamical systems in state-space form with inputs and sub-Gaussian noise. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. The bound derived in this paper relates the expectation of prediction errors with the prediction error generated by the model on the data used for learning. In addition, we show that it can also be used to derive bounds for the parameter estimation error. In turn, this allows us to provide finite-sample error bounds for the prediction error and parameter estimation error for a wide class of system identification algorithms. Furthermore, as LTI systems are a sub-class of recurrent neural networks (RNNs), these error bounds could be a first step towards PAC-Bayesian bounds for RNNs.

We address the challenge of scalable uncertainty quantification in large-scale scientific applications, where complex state-of-the-art machine learning methods are often computationally infeasible. Our primary contribution is a simple yet effective empirical Bayes method for automatically tuning the hyperparameters of a flexible, heteroscedastic Spectral-normalized Neural Gaussian Process. This approach retains the expressiveness and uncertainty-awareness of semi-Bayesian neural models while significantly reducing the computational burden by integrating hyperparameter learning directly into the training loop. We demonstrate the practical impact of our method on the task of estimating the dynamic aperture in circular particle accelerators, a fundamental problem in high-energy physics colliders and storage rings, using simulation data from the case of the Large Hadron Collider at CERN. Traditional approaches to DA estimation require extensive particle-tracking simulations, which are prohibitively time-consuming and resource-intensive. Our results show that the proposed method achieves competitive predictive performance and well-calibrated uncertainty estimates at much lower computational cost than state-of-the-art approaches. We stress that, beyond this application, the proposed empirical Bayes framework offers a general solution for training heteroscedastic neural models in situations where manual hyperparameter tuning is impractical. Accordingly, we anticipate that this framework can be applied to other domains that encounter comparable computational limitations.

Many scientific graphs attach several variables to each node, so a single scalar edge weight cannot describe direction-dependent interactions. We model each edge by a symmetric positive-definite (SPD) matrix and infer a posterior over matrix-valued graph geometries, which we call the Bayesian matrix-valued graph (BMVG). We ask how these interactions reconfigure across contexts: how large the change is and which multivariate directions strengthen or weaken. The geodesic distance induced by the affine-invariant Riemannian metric (AIRM) quantifies deformation magnitude and generalized eigenvalues resolve its signed directions.Against fused graphical lasso, Bayesian multiple-GGM, and common principal components, BMVG is competitive on global precision recovery while retaining identifiable matrix-valued edge structure and accurately recovering edge-level deformation directions. In controlled known-truth experiments, it resolves structural change with increasing sample size, including orientation changes that leave ordinary eigenvalues unchanged. In one year of Bay Area weather data, the geometry of 12-hour change reconfigures spatial coupling about as much as whole seasons differ. In TCGA-BRCA, estrogen-receptor (ER)-associated reconfiguration concentrates on specific gene-module pairs and persists under graph-scaffold sparsification and removal of subgroup mean differences. These results establish posterior matrix-valued edge geometry as a unified framework for quantifying and interpreting context-dependent multivariate reconfiguration.

The concept of Markov chain Monte Carlo (MCMC) cycles, an analogy to cyclic processes in heat engines, is presented in order to examine Bayesian inference problems. In this effort, we develop adaptive ensemble schedulers that allow the tuning of external parameters of a Bayesian canonical ensemble during an MCMC run, realising the MCMC cycles in practice. We run these cycles on different statistical models. As a fundamental insight, we find (both theoretically and in practice) that such systems can produce a non-zero net work output if and only if the considered model is non-Gaussian. As such, they may serve as a measure of non-Gaussianity in Bayesian inference, which we test on an example from supernova cosmology.

Kernel methods separate data representation from decision-making, but typically require the kernel to be chosen in advance. We show that this kernel can instead be learned by alignment, and develop the resulting framework through the recently introduced Collaborative Learning and Inference (CLaI). We show that Collaborative Learning can be viewed as a kernel alignment process, in which an embedding is trained so that its induced similarity matches a label-derived target kernel. We also prove that Collaborative Inference is equivalent to kernel Bayes classification with Parzen-window density estimation. Motivated by these perspectives, we generalise CLaI by replacing cosine similarity with a learned Mahalanobis distance and extend it to multiclass classification. On CIFAR-10, PathMNIST, and SleepEDF, the Mahalanobis formulation improves accuracy, converges faster, and yields lower calibration error than the cosine-based variant. Auxiliary experiments further support these connections, showing that CLaI produces latent signals of the same form as a Gaussian process, while achieving competitive calibration on sepsis prediction. Together, these results establish a principled learned-kernel framework that unifies representation learning, kernel alignment, and Bayesian classification, and extends naturally to the multiclass setting.

This paper addresses infinite-dimensional Bayesian inference for inverse problem of partial differential equations with model parameters in infinite-dimensional Hilbert space. To effectively incorporate prior information, we propose a novel continuous normalizing flows based infinite-dimensional model. Specifically, by introducing a well-defined neural ordinary differential equation in infinite-dimensional space, a simple reference measure can be transformed into a more complex measure which encodes the prior information. A corresponding theoretical framework is established to ensure the well-posedness of our proposed Bayesian prior in infinite-dimensional space. We also provide training methods of the prior for two distinct data settings, along with two sampling algorithms for the resulting Bayesian posterior. The proposed framework is applied to three representative inverse problems: the simple smooth inverse problem, inverse scattering problem, and the inverse heat conduction problem. Numerical experiments support the theoretical analysis and demonstrate the efficiency of the proposed algorithms.

Gaussian-process Bayesian optimization (GP-BO) excels at black-box optimization of costly functions, e.g., hyperparameter optimization (HPO) and multi-agent system (MAS) design. Convergence-rate guarantees exist for select methods, notably GP upper confidence bound (GP-UCB), but require a fixed kernel. Critically, the kernel encodes how input proximity affects objective value similarity. When raw coordinates poorly match this geometry - as with log-scaled hyperparameters or localized peaks - input warping can greatly improve sample efficiency, yet known GP-UCB proofs require a fixed kernel. We propose Finite-Library Input-Warped Bayesian Optimization (FLIWBO), which selects warps from a finite library of smooth input maps by any history-dependent rule. It adapts the input geometry to accelerate learning while retaining high-probability convergence guarantees under mild hypotheses, with an explicit $\sqrt(N_\varepsilon)$ library-size cost. Controlled diagnostics show that finite-library warping repairs planted geometry mismatches and identify FLIWBO failure cases. Across four repeated benchmarks - warped synthetic objectives, a confidence-fence trap, and Fashion-MNIST HPO - FLIWBO-UCB beats raw-coordinate GP-UCB under misspecified geometry, escapes traps that defeat even oracle-warp expected improvement, and recovers much of the gain from manual log scaling, while leading the tested methods that admit a matching regret guarantee. A 20-dimensional MAS design study further shows feasibility under costly noisy evaluations. Code for experiments is available: https://github.com/edvin-ketabati/bogp-paper-experiments.

Minimum Bayes Risk (MBR) decoding enables high-quality text generation by selecting the hypothesis that maximizes a utility metric over sampled pseudo-references. However, it is highly susceptible to metric overfitting: it can irregularly inflate the chosen utility metric at the direct expense of other unoptimized evaluation metrics. To mitigate this, we introduce SVD-MBR, which frames the pairwise utility matrix as a noisy information signal. By computing a low-rank approximation via Singular Value Decomposition (SVD) and retaining only the top-$k$ components, we effectively decouple true consensus from metric noise. Experiments demonstrate that SVD-MBR successfully regularizes decoding, yielding substantial gains across a range of generalized metrics. Furthermore, we reveal that this denoising is metric-dependent: neural metrics encode a robust low-rank consensus ideal for SVD, whereas surface-level metrics struggle to separate signal from metric noise.

Bayesian methodologies for handling count-valued time series have gained prominence due to their ability to infer interpretable latent structures and to estimate uncertainties. Among these Bayesian models, Poisson-Gamma Dynamical Systems (PGDSs) are proven to be effective in capturing the evolving dynamics underlying observed count sequences. However, the state-of-the-art PGDS still falls short in capturing the transition dynamics that are commonly observed in real-world count time series. To mitigate this limitation, a PGDS with time-varying transition kernel (TV-PGDS), is proposed to allow the underlying transition matrices to evolve over time. Three specifically-designed Dirichlet Markov chains (Dir-Dir, Dir-Gam-Dir, PR-Gam-Dir) are constructed to accommodate heterogeneous structural mutations within these dependencies. Leveraging Dirichlet-Multinomial-Beta data augmentation techniques, a fully-conjugate and efficient Gibbs sampler is developed to perform posterior simulation. Experiments show that, in comparison with related models, the proposed PGDS achieves improved predictive performance due to its capacity to learn time-varying dependency structure captured by the time-evolving transition matrices.

We consider semi-supervised classification from a partially classified sample arising from a two-component Weibull mixture. The feature is observed for all data, whereas some class labels are missing. The probability of a missing label is modelled as a function of classification uncertainty, giving a feature-dependent missing-at-random (MAR) mechanism that shares parameters with the Weibull-mixture classifier. The missing-label indicators can therefore provide information about the classifier in addition to the observed features and available class labels. Under a common Weibull shape, a Bayes' rule has at most one positive decision boundary, which is unique when the rule is nonconstant; under unequal shapes, it can have two. We characterise these decision regions, derive the Fisher information for the classifier after adjustment for nuisance parameters in the missingness model, and obtain a decision-boundary expansion of the expected error rate of the plug-in sample rule relative to the Bayes error. The expansion yields classification-specific asymptotic relative efficiency formulas for the one- and two-boundary cases and shows that a positive-definite increase in Fisher information is sufficient, but not necessary, for a smaller first-order expected error rate. Numerical studies and a semi-synthetic analysis based on hard-drive failure data illustrate potential reductions in expected error rate and improvements in decision-boundary estimation from modelling feature-dependent label missingness.

Item-sensitivity, defined as whether a model's choice depends on the specific input rather than on its own output prior, is widely reported as evidence of task competence. We show this evidence is necessary but not sufficient using a forced-choice signalling task abstracted from the board game Deception: Murder in Hong Kong. In this environment, the reference points against which a coordinate should be judged (a fit-maximising strategy, a posterior-maximising strategy, and uniform random selection) are all computable in closed form. Across seven language models, two model families, a post-training ablation, and three independent scoring rules, every one of 21 model-by-rule cells is reliably item-sensitive. Yet 8 of those 21 cells are not statistically distinguishable from a chooser that ignores the item and selects at random, and 5 score worse than random at describing the target. Item-sensitivity and distance from random correlate at only r = 0.30. We call this consistency without alignment and argue it generalises to any evaluation that relies on item-sensitivity, permutation consistency, or self-consistency without an independent reference for the measured quantity. We further find that a literal-similarity baseline with no pragmatics outperforms most tested language models, that adding a pragmatic layer over two baseline similarity sources moves choosers toward random rather than toward the Bayesian reference, and that a standard labelled multiple-choice format carries no measurable content signal here. All results represent the model side of a pre-registered instrument; a matched human condition is designed and piloted but not yet collected.

Social media platforms increasingly use AI-based credibility indicators to help users judge misinformation. Unlike individual human-AI decision-making, these indicators are embedded in information spread: users see both an AI prediction and earlier judgments shaped by the same AI, and their own judgments may then enter the public history. Yet how to analytically characterize this process remains under-explored. We therefore introduce a social-learning lens for this setting by extending the classical Bayesian cascade model with the AI indicator as a shared public signal. The resulting Gateway condition compares the evidence from the AI prediction with users' private impressions. Through this view, we show that AI changes what public history means. Crowd agreement may reflect accumulated independent human evidence, or repeated dependence on the same AI prediction. This creates a preservation-correction trade-off: stronger reliance on AI can preserve correct predictions, but can also lock in incorrect ones by blocking corrective private impressions. We calibrate the model using human-subject data on news veracity judgments. Although the AI outperforms human users, the average user weights it below her own impression but above several peer judgments, while individual users vary from discounting the AI to relying on it enough to cascade. Simulations show that over-reliance on a weak AI is especially harmful, and that diversifying AI signals across users can better keep the crowd informative. We conclude with implications for understanding human-AI interaction in information spread and designing misinformation interventions.

Reinforcement learning from human feedback (RLHF) has become the dominant paradigm for aligning large language models (LLMs) with human preferences. However, traditional RLHF relies on scalar reward signals that lack interpretability and fail to capture the multifaceted nature of response quality. Rubric-guided reinforcement learning addresses these limitations by introducing structured, interpretable evaluation criteria, or rubrics, as the backbone of reward design, feedback generation, and policy optimization. In this survey, we introduce a Bayesian framework that defines constitutions as prior distributions $P(R)$ over evaluation criteria and rubrics as conditional instantiations $R_x \sim P(R|x)$. Under this unified view, we present a taxonomy of rubric-guided RL along the prior-posterior axis, covering constitutional AI, instance-specific rubrics, process-level supervision, self-evolving rubrics, and their agentic and multimodal extensions. Furthermore, as rubrics are natural-language artifacts, we present a linguistic analysis of how granularity trade-offs, semantic drift, and linguistic reward hacking impact alignment reliability, identifying key open problems for future research.

Hyperparameter selection remains a key challenge in Bayesian optimization (BO) and Bayesian active learning (AL), as model misspecification can lead to suboptimal performance, while more accurate fully Bayesian treatments typically rely on computationally expensive MCMC sampling. This paper proposes a unified framework, KENDO (Kernel ENsemble Disagreement-aware Operator), that integrates Ensemble Gaussian Processes (EGP) with disagreement-aware acquisition strategies. The central idea is to replace hyperparameter sampling with a kernel ensemble and adaptive Bayesian weighting, combined with disagreement-aware acquisition strategies. Within this unified framework, we instantiate KENDO-BO for BO and KENDO-AL for Bayesian AL, demonstrating that both arise from a common self-correcting mechanism with task-specific acquisition objectives. We further extend the approach to multi-objective optimization via random scalarization that preserves the single-optimizer conditioning structure. Thorough numerical tests on synthetic and real-world benchmarks across single-objective optimization, multi-objective optimization, and active learning demonstrate that (i) KENDO-BO achieves competitive or superior optimization performance compared to state-of-the-art methods while reducing computational overhead by up to $5\times$ and (ii) KENDO-AL achieves superior predictive calibration over MCMC-based active learning baselines with up to $27\times$ speedup.

Multi-source evidence fusion under Dempster-Shafer theory faces two persistent challenges: existing conflict measures assess inter-evidence inconsistency and intra-evidence uncertainty independently, yielding incomplete evaluations, and current fusion methods evaluate evidence sources exclusively through instantaneous comparisns without exploiting their long-term reliability across diverse decision contexts. This paper proposes a unified evidence reasoning framework that addresses both limitations. Specifically, a chaos-conflict measurement is introduced to jointly quantify cross-evidence conflict and intra-evidence non-specificity, with five formally proven properties ensuring consistent assessment. A historical experience driven weighting scheme partitions the decision space via spectral clustering and applies regret theory to compute context-specific reliability profiles from past fusion outcomes. These mechanisms feed into a hybrid combination rule that adaptively balances uncertainty preservation against weighted consensus, controlled by the global conflict level, followed by a belief-interval decision strategy that enables robust classification without discarding epistemic uncertainty. Experiments on 16 real-world benchmark datasets demonstrate that the proposed framework achieves an average F1 score of 85.78 and a mean AUC of 93.30, outperforming eight DST-based baselines and three gradient boosting methods. Ablation analysis confirms the contribution of each component we proposed. The framework offers an effective approach for adaptive evidence fusion in multi-source decision making.

Standard probabilistic logic programming frameworks typically rely on grounding logic programs into discrete propositional representations. This operational requirement restricts exact inference to finite domains and discrete probability distributions. In this paper, we introduce Measure-Theoretic Probabilistic Definite Clause Logic (MT-PDCL), a generalized foundational framework that eliminates this finite-domain restriction. By explicitly defining stochastic variables over bounded index domains and equipping the interpretation space with standard Borel $σ$-algebras, MT-PDCL allows logical variables to operate natively over continuous measurable spaces. Building on Continuous Distribution Semantics, MT-PDCL models probabilistic rules as mutually independent causal events. However, rather than aggregating these derivations via finite boolean circuits, declarative entailment is formally defined through exact Lebesgue integration over the continuous measure space. We introduce a continuous immediate consequence operator that unifies the integration of continuous prior distributions with the evaluation of exact continuous observations. We demonstrate that this approach replaces the combinatorial bottleneck of discrete grounding with exact, algebraic, and structurally differentiable inference. While this transition trades discrete combinatorics for the geometric curse of dimensionality, it achieves the expressive power of continuous probabilistic models while preserving the pure declarative syntax of definite clause logic.

McCoy & Griffiths (2025, henceforth M&G) suggest that a Bayesian prior can be distilled into Artificial Neural Networks (ANNs) through Model-Agnostic Meta-Learning (MAML, Finn et al., 2017). They support this empirically by showing that meta-trained networks demonstrate formal language learning abilities comparable to Yang & Piantadosi (2023)'s Bayesian learner, significantly outperforming standard ANNs. We point out that under the standard interpretation of a prior, M&G's procedure does not actually instill one; it merely initializes network weights favorably, leaving the objective function unchanged. We then consider a more permissive interpretation, where the system as a whole can be seen as implementing a Bayesian learner even without an explicit prior in the objective. We show that this interpretation faces nontrivial challenges. Finally, we assess how well MAML approximates the empirical results of Bayesian learning, showing that unlike genuine Bayesian learners, M&G's model overfits and generalizes poorly to unseen data.

The robust treatment of environmental and operational variability (EOV) is an open challenge in population-based structural health monitoring (PBSHM). The difficulty is compounded in the case that the EOV signals are unmeasured. A common approach in conventional SHM is to apply \emph{projection-based} methods that discard subspaces of healthy feature data, reasoning that the EOV signal dominates the variance of the measured features. However, a common pitfall of projection-based approaches is that when damage acts close to the same variance-dominant direction, damage sensitivity is removed along with the EOV. An alternative identifying assumption for the removal of particular unmeasured EOVs is slowness; the latent EOV process is characterised by its long temporal correlation. In this paper, the latent EOV is cast as a state-space Gaussian process, enabling tractable $\mathcal{O}(T)$ inference via a Kalman filter. A robust hierarchical Bayesian identification framework is developed that enables population-level identification of latent EOVs and EOV-free residual features, using a Laplace approximation. The approach is first validated on a single laboratory-scale benchmark structure from the literature, subject to thermal EOVs, demonstrating robust damage detection and EOV recovery. The method is then applied to a simulated nine-turbine offshore wind farm with staggered deployment and damage, where it delivers a substantial true-positive uplift over projection and cointegration-based baselines at matched false-positive rates.

Tool-using LLM agents are commonly trained and evaluated in environments where tool calls succeed reliably, yet deployed tools can fail transiently, persistently, or silently. Robust recovery therefore requires more than repeated retries: an agent may need to retry the same path, switch to an alternative, or recognize that no viable path remains. We present BENCH2ROBUST, a framework that converts failure-free tool-use benchmarks into controlled stochastic environments with scenario-controlled solvability, where episodes explicitly require retrying, switching, or stopping after available paths are exhausted. We use BENCH2ROBUST to study two complementary interventions: structured runtime recovery context through Bayesian Tool Memory (BTM), and curriculum-controlled reinforcement learning. Across 7 models from 4 families and two multi-turn benchmark families, tool failures produce a near-universal robustness gap. On held-out Retail tasks, BTM improves robustness by up to 16.8 percentage points without retraining, while RL learns complementary recovery behavior that remains beneficial without inference-time BTM. Combining the two reaches 40.8-45.5% under injection while preserving failure-free performance. These results suggest that robust tool use benefits from combining environment-specific recovery knowledge with learned recovery behavior.

Analogies are quaternary relations of the form "A is to B as C is to D". Among the various formalizations of analogical reasoning, proportional analogies provide an important axiomatic framework by characterizing valid analogies through a set of postulates. While proportional analogies have been extensively studied over Boolean, symbolic, and real-valued domains, their extension to probability distributions remains largely unexplored. In this paper, we introduce a notion of proportional analogy for probability distributions based on Bayesian updating. Our approach builds upon the idea that two distributions are related whenever one can be transformed into the other through Bayesian updating induced by a suitable set of observations. We investigate this framework for several standard members of the exponential family and discuss how it naturally extends to arbitrary probability distributions through Gaussian mixture approximations.

PAC-Bayes theory provides generalization guarantees by controlling the Kullback--Leibler (KL) divergence between posterior and prior distributions over a chosen hypothesis representation. However, predictive risk depends only on the predictive behavior induced by a hypothesis, not on the particular internal realization that implements that behavior. In over-parameterized systems, many distinct configurations induce identical predictive behavior, yet the classical PAC-Bayes KL divergence does not distinguish uncertainty over predictive behavior from variation among behaviorally equivalent realizations. We show that this distinction induces an exact structural decomposition of classical PAC-Bayes complexity. We formalize behavioral equivalence through a measurable behavior map and use measure disintegration to decompose probability measures on the configuration space into a distribution over predictive behaviors and conditional distributions over behavioral fibers. This yields an exact decomposition of the classical PAC-Bayes KL divergence into a behavior-selection term and a realization-level term given by an expected conditional KL within fibers. We define Z-information as the negative of this realization-level contribution: the exact gap between the KL divergence and the complexity of uncertainty over predictive behavior alone. We further show that the behavior-selection term admits an exact variational characterization: it is the minimum KL divergence among all posteriors inducing the same distribution over predictive behaviors, attained by a canonical fiber-symmetrized representative. Finally, we show that symmetry, behavior-preserving directions, fiber geometry, and invariance under fiber-preserving perturbations arise naturally from the same behavior-map structure. Together, these results identify predictive behavior as the natural object of PAC-Bayes complexity.

The Naive Bayes (NB) classifier remains a standard choice for categorical data, yet its widely used smoothing rules, such as Laplace, Lidstone, Krichevsky-Trofimov, and the $m$-estimate, all prescribe a fixed smoothing strength that ignores feature cardinality, sample size, and class imbalance, inducing a non-vanishing bias on modern high-cardinality tabular data. We propose hierarchical empirical-Bayes Naive Bayes (HEB-NB), in which each class-feature conditional probability is smoothed by a Dirichlet prior whose concentration is learned data-adaptively via Type-II maximum likelihood, enabling principled information sharing across classes while retaining closed-form inference. We further introduce HEB average one-dependence estimators (HEB-AODE), showing that the adaptive smoothing transfers cleanly to structural relaxations of NB. Theoretically, we establish a non-asymptotic $\ell_1$ error bound for HEB-NB matching the empirical-distribution minimax rate plus a vanishing data-adaptive bias, together with a matching Laplace-tight lower bound that yields a finite-sample, risk-level strict separation from Laplace. We further derive a plug-in excess Bayes-risk bound via total-variation tensorization and a population top-1 expected calibration error (ECE) corollary. Empirically, across 31 UCI and OpenML benchmarks, HEB-NB attains the best average Friedman rank on probabilistic metrics, with up to 22.1% log-loss reductions on high-cardinality datasets and consistent improvements of HEB-AODE over vanilla AODE. Combining HEB-NB with mutual-information weighting reduces top-1 ECE by 41%-70%, demonstrating substantial gains in probabilistic accuracy and calibration.

We develop a new approach to Personalized Federated Learning across heterogeneous clients using Nonparametric Empirical Bayes (NPEB). Leveraging the asymptotic normality of local parameter estimates obtained from Empirical Risk Minimization or M-estimation, our method formulates these estimates as noisy observations to estimate an unknown shared prior via Nonparametric Maximum Likelihood. A key challenge in applying NPEB in this setting is that existing approaches assume known fixed variances, which is not true in practice. To address this, we introduce a Variance-Aware Nonparametric Empirical Bayes (VANEB) framework that leverages the parameter-dependent asymptotic variance of local M-estimators. A key technical contribution is a generalized Tweedie's formula for this heteroskedastic setting. We then establish non-asymptotic error rates for density estimation in the average squared Hellinger distance and derive an oracle denoising inequality that provides error bounds for our estimator. While our theoretical guarantees are rooted in the asymptotic regime of M-estimators, we empirically explore heuristic extensions of VANEB to modern federated learning settings involving Deep Neural Networks (DNNs). For DNNs, we propose VANEB-head and VANEB-FT, which personalize the last fully connected layer via an NPEB step using an approximate diagonal variance estimator. We show that our method has strong performance on popular vision datasets MNIST and CIFAR-10, using a convolutional neural network architecture.

For safety-critical software, data from the software's operational past (e.g. a sequence of success and failure events experienced by the software) can provide strong statistical support for reliability claims about the software. However, such data might not describe past software failure events in sufficient detail, and this might leave a reliability assessment (based on this data) unable to account for important features of past software failures. In this paper, by extending conservative Bayesian inference (CBI) techniques used in reliability assessment, we illustrate a principled statistical approach for checking the robustness of reliability claims derived from insufficiently detailed operational data. We demonstrate the extent to which insufficient detail in operational data can undermine software reliability claims in autonomous vehicle (AV) safety assessment scenarios. Reliability claims derived from insufficiently fine-grained data might be dangerously optimistic, despite a concerted effort by an assessor to use such data conservatively during the assessment. While these findings are consistent with previous work on the impact of statistical model fidelity in Bayesian software reliability assessments, our work clarifies why attempts to use low-fidelity data conservatively can be naive, and we give the first conservative estimates of the impact of data fidelity on assessments.

Hamiltonian Monte Carlo (HMC) is a successful generic inference method in probabilistic programming, but in its ordinary formulation it needs gradients and finite-dimensional parameter spaces. In Haskell, lazy evaluation lets probabilistic programs express stochastic processes and other non-parametric Bayesian models over implicit infinite-dimensional spaces. This paper develops new formulations of gradient-based HMC for this infinite-dimensional setting, via lazy evaluation. For automatic differentiation, we provide an analysis based on a new notion of "piecewise analytic under cylindrical analytic partition" (PACAP), to show that even if a program is infinite-dimensional and defined lazily, the gradient of the likelihood function is finitely supported. For the Monte Carlo method itself, we develop several HMC variants and a No-U-Turn Sampler that operate over the infinite-dimensional parameter space but are still productive because of lazy evaluation. Experiments cover Gaussian mixture clustering, random walks, and piecewise-constant regression with Poisson-process changepoints.

Oracle routing measures how much a pool of language models could gain from per-query selection, but the diagnostic has two flaws: testing against a best fixed model selected on the same examples invalidates paired inference, and a full-information oracle sees outcomes no deployable router observes. We separate three estimands (outcome-oracle opportunity, the Bayes-optimal gain from a declared pre-answer signal, and the held-out gain of a learned router) and prove selection-valid confidence intervals that survive choosing the best fixed model or the best member of a router family, a signal-information sandwich, and a $(1-1/e)$ greedy guarantee for building compact pools from submodular complementary coverage. On eight checkpoints from six families over four benchmarks, selection-valid intervals certify a population oracle gap of $9.7$--$30.7$ points on every task, yet the strongest deployable prompt router recovers only $7.5$--$14.4\%$ of it, and the simultaneous interval for the best of eleven tested policies has lower limit zero throughout. The realizable share of oracle opportunity is small and certifiable: strong routers beat the best fixed model, and most of the gap remains.

Preferential Bayesian optimization (PBO) optimizes objectives accessible only through pairwise user comparisons. The standard approach fits a Gaussian process surrogate for observed pairwise comparisons (PairwiseGP) using the Laplace approximation and selects queries with the Expected Utility of Best Option (EUBO) acquisition function. EUBO queries new candidates at each step, producing pairs that share no candidates with previous queries. Each such pair forms an isolated component in the comparison graph, removing one degree of freedom from the likelihood Hessian and making it rank-deficient. This deficiency is structural and cannot be resolved by changing the surrogate modeling approach. Existing approaches to remedy this issue either waste query budget by forcing comparisons to stay connected, or apply uniform regularization that also perturbs directions already well-constrained by the observed comparisons. We propose KappaSharp that enables a diagonal correction to the Hessian to reduce its condition number, with larger corrections where the prior uncertainty is higher. The correction is only applied in the model fitting step, not query selection. An adaptive variant of KappaSharp is also presented that activates the correction only when the surrogate is confident about recent comparisons, avoiding unnecessary corrections when the problem is well-conditioned. On 11 benchmarks (5--20 dimensions), including a 16-dimensional controller tuning problem in plasma medicine, Adaptive KappaSharp outperforms the standard PBO baseline, with up to +10.9% ($p{=}0.003$).

Few-shot learning has emerged as a promising approach for anatomical segmentation when labelled data are scarce. However, different few-shot learning algorithms exhibit complementary strengths and weaknesses, with performance varying across anatomical targets and institutions. Existing few-shot segmentation ensembles, that combine predictions from multiple algorithms, typically employ fixed weighting schemes and therefore cannot adjust model contributions according to the target domain. In this work, we propose a Bayesian adaptively-weighted ensemble framework for segmentation under label scarcity and domain shift. Multiple few-shot segmentation algorithms are first adapted using a small labelled support set. Bayesian optimisation is then used to automatically identify ensemble weights that maximise segmentation performance on a target-domain validation set. The learned weights are subsequently fixed and applied to combine predictions on previously unseen query images from the target domain. The proposed framework is evaluated on the Cross-institution Male Pelvic Structures dataset using held-out anatomical structures and institutions to simulate simultaneous label scarcity and institutional domain shift. Results demonstrate statistically significant improvements over individual few-shot learners, fixed-weight ensembles, training-from-scratch baselines and recent state-of-the-art ensembling approaches. By adapting model contributions to the target anatomy and institutional domain, the proposed framework provides a practical mechanism for deploying segmentation systems to new clinical sites under severe annotation constraints.

This paper proposes the Bayesian Expected Uncertainty Reduction (B-EUR) model, which formalizes the value of trying a candidate design action as its expected reduction of epistemic uncertainty about action--outcome relations. The model addresses one part of the Uncertainty Driven Action (UDA) model's open question concerning how changes in uncertainty perception determine action selection. We examine two environmental properties: generalizability, or how far knowledge from one trial extends to neighboring candidates, and outcome discriminability, or how clearly differences among outcomes can be distinguished. We tested the model through simulations and human experiments using a graph-shape guessing task that isolates learning about action--outcome relations under a limited trial budget. Epistemic value followed an inverted-U-shaped relationship with generalizability and increased with outcome discriminability in the simulations. In the human experiments, the subjective value of trying and enjoyment followed inverted-U-shaped relationships with generalizability, while choice behavior reflected both properties. The B-EUR model provides a computational account of candidate-action evaluation within uncertainty-driven design activity and offers implications for constructing prototype sets, framing design problems, and organizing feedback to support informative exploration.

Calibration requires a predictor to be unbiased after conditioning on its own predictions. Multicalibration asks for this guarantee simultaneously across a collection of groups. Many prediction tasks ask for several related features of the same conditional outcome distribution: variance is defined relative to the mean, skewness relative to both mean and variance, and conditional value at risk relative to a quantile. We study multicalibration for a sequence of $k$ properties in which each property is identifiable once the preceding properties are fixed. This framework includes Bayes pairs but does not require the properties to arise from a single loss. For every fixed $k\ge2$, we establish matching upper and lower sample-complexity bounds up to logarithmic factors under regularity conditions. Even with only polylogarithmically many binary groups, achieving multicalibration error $\varepsilon$ requires $\widetildeΩ(\varepsilon^{-(k+2)})$ samples. Conversely, for any finite group family $\mathcal G$, we give a randomized learner using $O(\varepsilon^{-(k+2)}+\varepsilon^{-2}\log|\mathcal G|)$ samples. Thus the sample complexity is $\widetildeΘ(\varepsilon^{-(k+2)})$ for polynomial-size group families. We instantiate the theory for three canonical examples.

Black-box optimization is a ubiquitous problem in science and engineering, often dealing with expensive objective functions with cheaper lower-fidelity proxies available. Multi-fidelity Bayesian optimization (MF-BO) is a principled approach to this problem, leveraging correlations across different fidelities when querying the objective. However, for many important MF-BO tasks, the true highest-fidelity function is prohibitively expensive to be part of the optimization loop. Nevertheless, practitioners often have gold standard data (observations of the highest-fidelity function) obtained from previous experiments that might provide information for the current task. For instance, in molecular optimization, chemists often pick the top-$k$ candidate molecules using various computer simulations, and later reveal their true objective function values. In this work, we demonstrate the suboptimality of standard MF-BO algorithms in the real-world scenarios above, even under ideal assumptions. Next, we mitigate this problem by incorporating historical high-fidelity data accompanied by task descriptors---which can be explicitly given or extracted from unstructured metadata. We demonstrate the effectiveness of our methods on synthetic functions, as well as real-world problems in chemistry and hyperparameter optimization.

The drift diffusion model (DDM) is a cornerstone of cognitive decision-making research. Although numerous estimation methods exist, researchers continue to seek inference approaches that are both fast and flexible across diverse study designs. Amortized Bayesian inference (ABI) can provide nearly instantaneous inference for complex stochastic models like the DDM, but neural networks trained for one study design cannot generalize to others. In this paper, we propose a divide-and-conquer framework that address this limitation. The core idea is that the DDM's independence assumption allows the full dataset to be decomposed into pairwise shards, each sharing a common structure that a single neural network can learn. Inference is performed on each shard separately and the resulting posteriors are combined via consensus MCMC to approximate the full posterior. Using simulated datasets, we evaluate the accuracy and uncertainty of this method. Our results show that the proposed divide-and-conquer approach achieves accuracy and uncertainty comparable to MCMC while reducing computational cost by several orders of magnitude. This work not only advances DDM estimation but also demonstrates a general strategy for improving the scalability and generalizability of ABI methods across diverse applications.

Combining Bayesian learning and quantitative verification is a powerful toolset for analysing key quantitative properties of software systems, like reliability and response time. However, the accuracy and robustness of verification results strongly depend on the prior knowledge (PK) underlying Bayesian inference. This knowledge reflects original beliefs about the probability of events and typically depends on domain expertise. Using inaccurate or uninformative PK can negatively affect quantitative analysis, yielding incorrect verification results. Our EPIK approach tackles this important challenge by eliciting and embedding PK in quantitative verification equipped with Bayesian estimators. Unlike existing approaches that require PK on formal model transition parameters, EPIK leverages system-level properties that are directly observable and are linked to real-world semantics. EPIK formulates a twofold optimisation problem to derive the distributions of unknown transition parameters and then embeds these distributions to verify new or difficult-to-measure (elusive) properties. The detailed experimental evaluation using multiple variants of real-world case studies and diverse EPIK instantiations shows its effectiveness, flexibility and generality.

The minimum change principle provides an information-theoretic characterization of the Bayes reversal channel in classical probability theory and has recently been proposed as a framework for extending Bayes' rule to quantum information theory. Using quantum relative entropy, we investigate a minimum change principle for the setting of quantum statistical inference. Specifically, we consider a forward process based on a classical-to-quantum preparation channel and a reverse process based on a quantum-to-classical measurement channel. We establish a closed-form characterization of measurements that are optimal for this principle, and this optimal measurement can be found via a dual formulation involving a single unconstrained Hermitian variable. This perspective allows us to recover some notable measurements within the same framework, including pretty good measurements and Fermi-Dirac thermal measurements, and we use it to discover a novel family that we call softmin thermal measurements. We further show that softmin thermal measurements arise as optimal solutions to entropy-regularized semidefinite optimization problems, demonstrating that they play a role for measurements analogous to that of thermal states in statistical mechanics. Finally, we prove an additivity property for the relative-entropy minimum change principle and investigate the performance of Fermi-Dirac thermal measurements for quantum hypothesis testing.