Bayesian

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.

Reliable semi-dense matching is essential for modern geometric vision systems. Designed under a coarse-to-fine paradigm, it achieves an optimal balance between performance and computational cost. However, existing methods often struggle to provide well-quantified uncertainties, where catastrophic coarse-assignment failures are ignored, leading to truncated error distributions and severely misjudged geometric estimations. In this paper, we propose a lightweight, post-hoc overall uncertainty estimation framework that introduces a two-component calibrated Laplace mixture model with only 9 learnable parameters. The objective is to explicitly capture both the sharp local refinement noise and the broader tail of coarse-assignment failures. We introduce the Coarse-success posterior Refit (CoRe) method, a geometric refitting module that utilizes the posterior probability of coarse-assignment success as soft correspondence weights. Extensive experiments show that our method consistently improves downstream geometric accuracy across various pretrained-only matchers and robust estimators with minimal computational overhead. Our code is available at https://github.com/khoavpt/Probabilistic-matching.

We consider the problem of sampling compositional and discrete objects from a given unnormalized posterior distribution. Notably, recent studies have shown that this problem can be efficiently solved by learning a deterministic Markov Decision Process (MDP) that progressively builds each object in proportion to the posterior. In this work, however, we demonstrate that the Markovian assumption can both hamper signal propagation during training and catastrophically reduce the learned sampler's expressivity due to state aliasing. To address these issues, we propose lifting the MDP with a learnable latent dynamical system that allows the underlying policy to depend on the entire past trajectory---and not only on the current state. In view of this, we refer to the resulting method as path-dependent discrete amortized inference. Importantly, we provably extend existing learning algorithms for discrete amortized samplers to our setting. In experiments on standard benchmark problems, we also show that our approach often leads to faster learning convergence and improved state space exploration relatively to prior techniques.

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.

Ordinal regression, also called ordinal classification, is classification of ordinal data, in which the underlying target variable is categorical and considered to have a natural ordinal relation. Previous works have indicated that, in many real-world ordinal data, the conditional probability distribution (CPD) of the target variable given a value of the explanatory variable would be unimodal in a large domain of the explanatory variable and close to be unimodal even in a remaining domain. Therefore, unimodality-promoting regularized learning (UPRL), which promotes a predicted CPD closer to be unimodal with the aim of decreasing a prediction variance without inducing much bias for ordinal data of the unimodality, is promising to improve the prediction performance especially with small-size training data. In this study, we show that previous UPRL methods promote a predicted CPD to not only become closer to be unimodal but also have a larger scale (in other words, be smoother or less-confident). Therefore, we develop a novel method that more strictly reflects the idea of UPRL and evades a scale-related bias, and verify through experimental comparison that the unimodality-promotion indeed contributes to improve the prediction performance. Additionally, while our proposed UPRL method could perform better for smaller-scale data or with larger-size training data compared to a previous UPRL method, our analysis explains this experimental observation in terms of the presence or absence of an unexpected scale-related bias.

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% ($p{=}0.003$).

Probabilistic long-term time-series forecasting commonly relies on trained models. Training-free conformal methods typically construct intervals around a pre-existing point forecaster and do not natively represent a complete predictive distribution; sequential variants additionally suffer from increasingly delayed feedback at long horizons. We propose KReF, a training-free retrieval framework that treats retrieved historical futures as a querylocal empirical predictive distribution. After robust preprocessing, KReF embeds each lookback using handcrafted statistics or frozen random Fourier features and retrieves similar historical lookback-future pairs. Their similarity weights directly define predictive masses, quantiles, CRPS, and a weighted-mean point forecast. KReF further uses the observed query lookback to construct a probability-integral-transform map and applies validation-selected expansion and shrinkage rates to adapt interval boundaries. Across six LTSF benchmarks and four horizons, KReF obtains the lowest CRPS in all 12 dataset-embedding settings and the lowest IS90 in 9 settings. Without gradient-based fitting, its point forecasts also match or surpass trained baselines on two of six datasets. An archive-oracle analysis further reveals substantial headroom under finer horizon- and channel-wise routing. These results establish retrieval as a useful and underexplored inductive bias for LTSF.

Tabular Foundation Models (TFMs) are currently the best approach to tabular prediction problems. They are constructed as transformers that approximate the Bayesian posterior predictive distribution based on a pre-training prior. These univariate predictors can be converted into multivariate ones autoregressively by sampling one target and adding it to the features. However, the faithfulness of the resulting joint has not been investigated. Furthermore, TFMs cannot be evaluated against the posterior itself, at least not on real-world datasets, because the ground-truth distribution is unknown. We therefore propose asking a different question: could a model's predictions result from any joint distribution? To answer this question, we pose two requirements that any such model must satisfy. The first is marginalization consistency, which demands that marginalized conditionals are equal to directly predicted marginals. The second is factorization consistency, which demands that different factorization orders result in equal joint distributions. Every TFM that we evaluate violates both of these requirements for both classification and regression across all datasets.

Reliable uncertainty estimates are critical in safety-sensitive applications, where understanding the sources of predictive uncertainty is essential. This often requires disentangling epistemic uncertainty from aleatoric uncertainty, yet these uncertainty types are not defined consistently across the literature, making it difficult to assess whether a method produces accurate uncertainty estimates. Evaluation is further complicated by the fact that ground-truth epistemic uncertainty is typically unavailable. Existing benchmarks therefore mostly rely on proxy tasks such as out-of-distribution detection, which do not provide complete ground-truth uncertainty targets and offer limited insight into the structure and quality of uncertainty estimates. We propose a unified definition of uncertainty as pointwise posterior risk, the expected loss of a predictor under the distribution of plausible ground-truth functions given the data. This view combines Bayesian uncertainty over functions with estimator-dependent deviations from the posterior mean, capturing effects such as misspecification and optimization error. This formulation constitutes the foundation of a theory-backed benchmark that enables direct computation of oracle epistemic and aleatoric uncertainty using semi-synthetic datasets with real covariates and known generative processes. By avoiding proxy evaluations, the benchmark enables fine-grained analysis of uncertainty estimates. Empirically, we find that accurate prediction does not guarantee reliable uncertainty disentanglement. The benchmark reveals practically useful differences between methods, identifying approaches with meaningful alignment to oracle uncertainty targets while exposing sensitivity to datasets and modeling choices.

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.

Data assimilation (DA) uses Bayesian inference to update the state of a numerical forecast model with observed data. In this study, we propose a fundamentally different, unified approach to atmospheric data assimilation. We use latent video flow-matching to sample temporally consistent trajectories from a prior trained using ERA5 reanalysis (69 variables over an 8-day window). We also use posterior sampling to assimilate real observation sources, such as those from the NOAA Integrated Global Radiosonde Archive and the Integrated Surface Database. Because the prior generates a continuous trajectory, it naturally propagates information between observed and unobserved frames. Therefore, we can perform various DA tasks, such as filtering and smoothing, simply by changing the observed frames. Moreover, we generate full-state ensemble forecasts directly from sparse observations, achieving performance competitive with state-of-the-art observation-to-forecast models.

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.

Automatic radiology report generation (RRG) aims to simulate the workflow of radiologists, assisting them in clinical diagnosis. However, existing methods often fall short in utilizing all information relevant to the examination, as is typically done in clinical practice. Although some works attempt to incorporate multi-view images and historical data, these additional inputs may sometimes lead to avoidable diagnostic errors on the contrary. To address these challenges, we introduce a decision-making stage after report generation for the first time and propose a Posterior Diagnostic Decision framework (PDD-RRG) to integrate potentially conflicting diagnoses. Specifically, we create various subsets of input data and utilize an existing RRG model to generate reports from different perspectives. Then the Bayesian posterior probability and the learned thresholds for each clinical observation are calculated to obtain an aggregated diagnostic conclusion, which is subsequently used to refine the generated report. Experiments on MIMIC-CXR demonstrate that our proposed PDD-RRG can effectively enhance the clinical efficacy of existing RRG models without any retraining.

With AI systems gaining more access to individuals' information, it is important to protect privacy when reporting statistical answers. Equally important is to privatize the reporting of uncertainty in such answers. To this end, we adopt a Bayesian likelihood-free framework and make simulation from the posterior private. In particular, we propose a new private instantiation of the Bayesian bootstrap using a blocking strategy. Rather than assigning idiosyncratic random weights to each individual, we randomly group individuals and assign a single weight to each group. By concealing individuals' contributions within a group, we fortify differential privacy gates. We harness amortized inference that decouples private learning from posterior sampling. A push-forward map from observation weights to posterior samples is learned privately by adding calibrated noise during training. Subsequent posterior draws require no additional privacy and computation budget. We call the resulting method the Private Generative Bayesian Bootstrap (PGBB). We establish a differential privacy guarantee, analyze convergence to the non-private blocked-bootstrap target, and quantify the discrepancy between the ordinary and blocked Bayesian-bootstrap posteriors. In addition, we derive data-free tuning of the block Dirichlet concentration parameter that restores posterior dispersion asymptotically. We also show a single fit of PGBB can support a family of loss-based decision rules simultaneously without additional privacy cost. In simulations and in applications to U.S. Census returns to schooling and U.S. natality birthweight quantiles, PGBB gives competitive private uncertainty quantification and improves over private Bayesian alternatives that require a specified data-generating model in common settings.

Competitive analysis is central to the study of online algorithms, but upper bounds are often highly problem-specific. We develop a more unifying methodology via the minimax viewpoint. Guided by Yao's principle, we reduce worst-case competitive analysis to Bayesian online design under an arbitrary correlated prior over arrival sequences. For such a prior, let $X^*$ be the hindsight-optimal fractional solution for the realized instance, and let $X^{(t)}=\mathbb E[X^*\mid \mathcal F_t]$ be its posterior process. Our guiding rule is posterior matching: at each time $t$, choose the feasible online action that tracks the current posterior $X^{(t)}$ as closely as the online constraints permit. We show that this single principle yields optimal or near-optimal guarantees for several classical online fractional problems, including set cover, load balancing, matching and more general resource-allocation problems, recovering or improving state-of-the-art bounds in these settings with norm/concave objectives. Via known rounding reductions, it also yields randomized integral guarantees for weighted paging, MTS on star metrics, and ski-rental. At a technical level, our analysis reduces competitive guarantees to key probabilistic inequalities for the vector martingales generated by the posterior of the offline optimum. The resulting framework gives a reusable route from Bayesian online design under arbitrary correlated priors to information-theoretic worst-case competitive guarantees.

Machine-learning benchmarks often pair a label that aggregates a long temporal horizon with input observed through one or a few short windows. Their apparent performance ceiling may therefore be an acquisition-protocol ceiling rather than a model-capacity ceiling. We study labels of the form $Θ_{g,T}=T^{-1}\int_0^T g\{Z(t)\}\,\mathrm{d}t$ when the latent Gaussian process contains both a stable individual trait and a correlated within-individual state. An exact protocol-conditioned Bayes-risk identity provides a common tool. First, we decompose label variance into an $O(1)$ trait component and an $O(T^{-1})$ state component, explaining why a snapshot can retain cross-sectional predictability while poorly tracking within-person change. Second, we derive task-dependent effective temporal spans: mean labels depend on the ordinary correlation time, whereas occupation-time labels depend on an entire spectrum of higher-order correlation times. Third, state-driven occupation-label variance is maximal when the stable trait lies at the threshold; window efficiency decays much more slowly away from that boundary. Under an equal segment budget, exact risks and Monte Carlo experiments show that repeated segments at one time rapidly saturate, whereas temporally dispersed observations continue to increase state explainability. The trait ceiling uses quantities available from ordinary test-retest data; only the state ceiling requires short-lag temporal calibration. The results distinguish architectural limits from protocol limits and show that the label, rather than duration or segment count alone, defines the relevant timescale.

Whether large language models perform genuine algorithmic reasoning or mere pattern completion is hard to test, because most benchmarks lack a ground truth for correct inductive inference. We introduce F-ICL, an in-context-learning benchmark that supplies one exactly. Using the Turing-complete machine F, complement-symmetrised into sF to remove output-polarity bias, we exhaustively enumerate all 1.5 billion programs of length $L\le13$ and compute the Bayes-optimal posterior in closed form under a bounded universal (Levin--Solomonoff) prior; models are scored by how closely their served distributions approach it at matched evidence. Each task is paired with its bitwise complement, on which the optimum scores identically, so an original-twin gap isolates the model's inductive bias. Across 105 serving configurations spanning 37 open models (0.8B--675B) and frontier systems from four laboratories, models answer up to 92% of queries correctly, yet 45 of 46 models yield distributions farther from the optimum than a keystroke reference, and their behaviour is bracketed by low-order prefix statistics fitted only on visible evidence. That reference is itself an algorithmic mixture, induced by a print-only machine with no loops, so the panel's implied measure sits closer to a loop-free mixture than to the loop-bearing optimum, independently of the reference machine. Updating is also non-monotone, which no prior explains: a Bayes-rational solved set can only grow in this realisable, noiseless setting, yet added examples produce $6{,}545$ solved-to-unsolved transitions against $13{,}702$ gains. The gap is not predicted by accuracy (Spearman $ρ=-0.19$, $p=0.21$), does not close with scale or across frontier generations in the serving modes that expose distributions, and is widened by instruction and reasoning post-training. F-ICL is released as an open, reproducible benchmark and toolkit.

Vision-language models return structured chest-radiograph findings through interfaces exposing no confidence score, so a receiving institution cannot read off how far to trust an individual judgment. Whether agreement with an institution's reference standard transfers across sites, findings, prediction directions and question formats is largely unmeasured. We evaluated three generative vision-language models on three institutional chest-radiograph corpora and six findings under two elicitation protocols, comprising more than 345,000 finding-level predictions, and estimated finding-by-direction reference agreement at a receiving institution from a small budget of local labels. Estimation strategies were then stress-tested under repeated strict institution-held-out evaluation. Under evaluation excluding the receiving institution from development entirely, adaptive selection among the seven estimators that design admits did not improve on simple fixed alternatives: it achieved a mean Brier score of 0.1083, against 0.0853 for always using a Beta-Binomial empirical-Bayes estimator and 0.0855 for a target-only logistic model. Those two differ by 0.0003, less than this family's own sensitivity to a change of solver version, and each leads in about half the settings, so no default can be recommended. Their advantage over estimators pooling across institutions was concentrated at one site and not confirmatory once clustered by institution, and a plug-in empirical-Bayes posterior-predictive count interval at a nominal 95% level covered 87.0%, less at the hardest institution. Reference agreement therefore has to be re-evaluated per site and per interface; these results concern agreement with institutional labels, not clinical correctness.

Latent Flow Models have revolutionized compressed-space image synthesis, yet their application to high-fidelity inverse problems remains bottlenecked. In this paper, we trace this dilemma to a fundamental geometric limitation of pre-trained autoencoders, which we term \emph{First-Order Manifold Blindness}. Severe decoder compression (e.g., retaining only $\sim\!2\%$ of the original degrees of freedom) produces a rank-deficient Jacobian, rendering high-frequency measurement residuals in its orthogonal complement invisible to latent gradients even when the decoder can represent the target image. To overcome this bottleneck, we propose Hybrid-Domain Posterior Sampling (HDPS), a decoupled inference framework that disentangles physical measurement consistency from semantic prior modeling. HDPS diverges into the pixel space, leveraging Langevin dynamics to absorb precise orthogonal measurement gradients, and subsequently projects these structural corrections back onto the generative manifold. An optimization-based latent alignment is introduced to filter pixel-space artifacts while avoiding the semantic drift of direct encoding. Extensive experiments on diverse inverse problems demonstrate that HDPS establishes a new state-of-the-art, successfully recovering the high-frequency structural precision that latent-only solvers inherently discard. The code is available at \href{https://github.com/74587887/HDPS}{https://github.com/74587887/HDPS}.

In Non-Intrusive Load Monitoring (NILM), adaptive segmentation of electricity consumption time series is critical for appliance recognition. However, prevailing methods face challenges including heuristic parameter tuning, boundary sensitivity, and metric saturation. This paper proposes BayesSeg, a unified framework integrating time-series segmentation, multidimensional evaluation, and automatic parameter optimization. The segmentation layer employs a dual steady-state criterion based on the tail value and mean of preceding subsequences, combined with a sequential extraction and complement-set parsing strategy, to achieve precise unsupervised partitioning of steady-state and transition-state segments. The evaluation layer maps segmentation results to binary state sequences and formulates a composite metric integrating an event-level F1 score (event_F1) with Normalized Mutual Information (NMI). The event_F1 quantifies switching-event precision and recall via tolerance matching, while NMI captures global structural consistency, jointly overcoming the boundary sensitivity and limited discriminability of point-wise metrics. In the optimization layer, the composite score serves as the objective function for Bayesian optimization, which constructs a TPE surrogate model for efficient global parameter-space exploration. Experiments on the SustDataED2 dataset demonstrate that Bayesian optimization requires only ~100 objective evaluations to locate a parameter region within 0.35% deviation of the exhaustive grid-search optimum. The framework achieves a weighted composite score of 0.7149 and an event_F1 of 0.9340 while reducing optimization latency from ~5300 seconds to under 1 second, a speedup exceeding 5700x. BayesSeg automates segmentation configuration and provides a scalable, efficient solution for time-series analysis in NILM and related domains.

Predictive coding offers a powerful theory of cortical computation, but corresponding scalable algorithmic implementations for artificial intelligence have remained elusive. This paper introduces the Bayesian reflex, a computational framework that directly instantiates predictive coding through three pillars: belief maintenance via hierarchical generative models, sequential Bayesian updating via prediction-error minimization, and uncertainty-driven action via active inference. We show that recent breakthroughs---ellipsoidal decomposition for exact $i.i.d.$ sampling, recursive Gaussian processes for deep hierarchical inference, and derivative-aware Bayesian optimization---provide the missing algorithmic ingredients. The resulting framework enables mathematically principled, scalable, and brain-inspired continual learning, perception, and decision-making. We illustrate its versatility through applications ranging from climate model evaluation to prime number discovery, offering a blueprint for truly adaptive artificial intelligence.

Bayesian optimization (BO) has become the standard tool for sample-efficient optimization and owes its efficiency to uncertainty-aware search driven by generic statistical priors. Richer domain priors can improve BO in principle, but encoding them through tailored kernels or problem structure is difficult and rarely done in practice. LLMs can help sidestep this difficulty by making informal priors from natural language, code, and documentation directly available to the optimizer. However, existing LLM-based BO methods either insert the LLM into a fixed role (surrogate, acquisition proxy, or configuration interface) or hand it broad control, sacrificing the systematic exploration that makes BO reliable. We introduce agentic Bayesian optimization: a paradigm in which an LLM agent is the central decision maker in the BO loop while a Bayesian backend provides the uncertainty-aware optimization substrate. The agent configures the problem, queries the backend, selects and commits evaluations, and can revise the optimization strategy during the run by tightening bounds, switching acquisition functions, proposing targeted evaluations, or even reframing the problem following new instructions or observed evidence. We instantiate this idea in Sara, a surrogate-augmented autoresearch agent, and lenz, a modular BoTorch-based backend that the agent can inspect and modify through a structured interface. Across synthetic and real-world benchmarks, Sara preserves the reliability of state-of-the-art BO without prior knowledge, outperforms LLM-based baselines, and uses natural-language priors to improve beyond standard BO. We further demonstrate the practical value of agentic BO in dynamic settings, where Sara reconfigures the full optimization problem on the fly as requirements change, a capability not previously available in standard BO.

Bayesian Optimization (BO) is widely adopted for data-efficient optimization in scientific and engineering applications, yet its computational cost is rarely evaluated alongside optimization performance. Here we present a systematic, compute-aware study of BO that evaluates surrogate models along two axes: optimization quality and computational frugality. Across eight benchmark functions and nine real-world datasets spanning materials science, mechanics, robotics, chemistry, and machine learning, we benchmark four surrogate models: Gaussian Processes, Random Forests, NGBoost, and Bayesian Adaptive Spline Surfaces. We show that Gaussian Process-based BO consistently incurs the highest time and memory overhead without delivering superior optimization or sample efficiency. In contrast, scalable alternatives achieve equal or better performance at a fraction of the computational cost. Motivated by these findings, we introduce a surrogate-recommendation framework that predicts the most suitable BO surrogate from inexpensive dataset characteristics. Together, these results establish FruBO as a reproducible, compute-aware baseline for Bayesian Optimization and provide practical guidance for surrogate selection under limited computational and experimental budgets.

Counterfactual explanations (CEs) enhance the interpretability of machine learning models by identifying the smallest change to an input required to obtain a desired output. Although CEs are conventionally formulated as a distance-minimization problem, the theoretical basis of this formulation has received limited attention. We show that a distance-minimization-based CE is mathematically equivalent to the maximum a posteriori (MAP) estimate of a Gibbs posterior within the generalized Bayes framework, specifically when a distance-based prior is used. We call this formulation the Distance-Prior Generalized Bayes CE (DP-GBCE). Building on this posterior perspective, we introduce two decision rules beyond MAP within a unified framework: a Bayes decision that minimizes expected decision loss and CVaR-CE, a risk-averse decision rule. We also propose an extension that uses Bayesian model weights to mix the posterior distributions of multiple models, thereby accounting for model multiplicity, where several models have comparable predictive performance. Finally, we define metrics for evaluating both individual CEs and the posterior distribution as a whole, and use experiments on simulated data and Google Trends data to quantify the trade-offs among the decision rules.

Computational cognitive modeling seeks to infer latent cognitive mechanisms underlying observed behavior. Bayesian inverse planning provides a principled framework for such inference, but its success depends critically on the experimental environment. Existing approaches typically treat environments as fixed, leaving open the question of which cognitive experiments are most informative for cognition parameter inference. We formulate the design of cognitive planning experiments as a Bayesian Experimental Design (BED) problem, treating the experimental environment as the design variable. We establish an exact Monte Carlo BED benchmark and introduce an amortized Bayesian experimental design framework for efficient posterior inference and design evaluation. Experiments on the Mouselab-MDP process-tracing paradigm show that amortized BED closely matches the environment rankings of exact Monte Carlo BED while substantially reducing computational cost. We further show that no single environment is uniformly optimal across cognitive inference objectives, revealing trade-offs between expected information gain, posterior recoverability, and information efficiency. These results provide a principled framework for designing informative cognitive experiments for Bayesian parameter inference.

Longitudinal causal studies often record histories as irregular functional fragments: laboratory values, physiologic signals, sensor streams, and image-derived summaries measured at unequal and informative times. Standard doubly robust estimators usually require scalar summaries, whereas sequence learners optimize prediction losses that need not stabilize the efficient influence function. We propose Doubly Robust Functional Representation Learning (DR-FRL), a cross-fitted workflow that turns irregular histories into estimand-targeted states for observed-history regimes. Functional and temporal encoders map point clouds and prior histories into states; nuisance heads estimate outcome, treatment, and censoring functions; and EIF-targeted validation, calibration, overlap, tail, and ablation diagnostics assess whether the state supports the estimating equation. If the selected state preserves the nuisance information needed by the EIF, representation error enters the same second-order product remainder as ordinary nuisance error, and the mean estimator is asymptotically linear under explicit rate, overlap, calibration, and stability conditions. Catoni aggregation is treated separately as a bounded-influence point estimator, not a replacement for Wald inference. Simulations show gains when functional confounding is high-dimensional, measurement is informative, support is weak, or pseudo-outcomes are heavy-tailed. A VitalDB audit shows that DR-FRL can use irregular laboratory point clouds and deliver a useful negative finding: for this ICU-disposition endpoint, scalar laboratory summaries already carry much endpoint-relevant information.

This thesis studies policy learning in interactive systems where an agent observes a context, selects an action from a very large set, and receives partial feedback. The main framework is contextual bandits, with two paradigms: on-policy learning, where the agent interacts sequentially with the environment and minimizes regret, and off-policy learning, where it learns from logged data collected by a logging policy. In large action spaces, both settings face major challenges: inefficient exploration, sparse data coverage, high-variance importance weights, extrapolation bias, and difficult optimization landscapes. The first part develops structured Bayesian methods for on-policy learning. We introduce meTS, a mixed-effect extension of Thompson sampling, and dTS, which leverages diffusion-inspired priors to model dependencies between actions. These methods share information across actions and yield regret guarantees depending on an effective number of actions. The second part addresses off-policy learning. We propose sDM, a structured direct method based on latent variables, show that optimization error can dominate estimation error in large action spaces, and introduce concave, efficiently optimizable policy-weighted log-likelihood objectives. Finally, we develop differentiable pessimistic methods based on exponential smoothing and PAC-Bayesian bounds to control the bias-variance trade-off of regularized importance-sampling estimators.

PET provides functional images relying on two-photon coincidences from positron-electron annihilation. In human tissue, about 40% of annihilations are preceded by Ps formation, of which o-Ps component partially decays into three photons, with the remainder annihilating via pick-off or spin-exchange into two photons. This three-photon channel carries additional information about the surrounding micro-environment, including the three-to-two-photon yield ratio as a potential diagnostic marker. We propose the TRIO algorithm, a novel three-photon event-by-event image reconstruction algorithm formulated as a Bayesian maximum a posteriori inference problem. TRIO unifies time-based trilateration, energy-based reconstruction and, for the first time, a physics-informed prior derived from the QED description of Ps decay within a single probabilistic framework. In contrast to positronium lifetime imaging, which requires a prompt photon and is therefore restricted to specific radionuclides, TRIO relies solely on the three photons and is fully compatible with standard radionuclides such as 18F. Monte Carlo simulation modelled after the Siemens Biograph Quadra scanner demonstrates a mean position error of 1.62~cm, improving by approximately a factor of two over the time-based trilateration (3.05 cm) and by about an order of magnitude over energy-based reconstruction alone (18 cm). More importantly, the proposed Bayesian approach is compatible with existing TOF-PET scanners that can register three-photon annihilation coincidences.

Wavelength selection is one of the important preprocessing methods in near-infrared spectroscopy to improve prediction accuracy and interpretability of spectral data. We formulate wavelength-region selection for sugar content estimation as a binary black-box optimization problem and propose a method based on Bayesian optimization. The proposed method constructs a sparse quadratic surrogate model and sequentially extracts interested wavelength regions by Thompson sampling. Minimizing an acquisition function is performed as a quadratic unconstrained binary optimization problem by simulated or quantum annealing. Experiments show that the proposed method improves the prediction accuracy of partial least squares regression and yields more consistent wavelength regions than genetic-algorithm-based selection and simulated annealing. Under one-bit local perturbations, the selected wavelength regions show minimal fluctuations in root mean square errors between observed and predicted values of a validation set. This local stability suggests that our method converges to a smoother error landscape and avoids isolated overfitted solutions. These results indicate that combinatorial Bayesian optimization is a useful framework for robust feature selection in spectroscopic prediction tasks.

Evaluating large generative models across benchmarks is time-consuming and computationally expensive. This drives the need for methods that can estimate full benchmark performance by evaluating models on only a subset of items, known as a coreset. Current literature mostly requires the practitioner to input a coreset size. However, when reliable performance estimation takes priority over efficiency, an evaluation method should also be capable of automatically determining a coreset size that reflects this priority. We introduce BayesAME, a sequential Bayesian framework specifically targeting automatic determination of the coreset size. BayesAME models performance as a random variable by defining a latent ability for each group of items sharing the same historical model performances, with a joint prior distribution encoding the belief that the target model behaves similarly to these historical models. The posterior distribution over these abilities is used to derive performance estimators, quantify performance uncertainty, and select items to add to the coreset via an information-gain criterion. The coreset is iteratively augmented until the performance estimate fluctuation and the performance uncertainty fall below their respective user-defined thresholds. We propose a multi-target extension that captures performance correlations across multiple target models to further reduce the coreset size. Through extensive experiments across diverse benchmarks, we demonstrate that BayesAME consistently outperforms sequential adaptations of existing methods. Crucially, our comprehensive analysis addresses recent skepticism in the literature, establishing that non-random coreset selection is advantageous over random selection. Finally, we highlight that leveraging continuous response log-likelihoods over traditional binary scores significantly enhances estimation accuracy.

YOLO-Pose models provide efficient keypoint localization, but do not quantify the associated spatial uncertainty. We introduce a lightweight post-hoc probabilistic extension that augments a trained YOLO-Pose model with calibrated bivariate predictive distributions over keypoint locations, centered at the model's original predictions. Concretely, we train additional probabilistic heads with an importance-weighted negative log-likelihood to predict an input-dependent $2\times2$ dispersion matrix for each keypoint, followed by Gaussian calibration for broad downstream compatibility or Student-$t$ calibration for distributional fidelity. Complementing this, we propose an evaluation protocol that combines a suite of distributional calibration diagnostics with average keypoint precision (AKP), a keypoint-level extension of the COCO AP protocol for assessing reliability rankings. Experiments on COCO show that the learned uncertainty estimates enable effective keypoint-level reliability ranking, Student-$t$ calibration best captures the empirical residual distribution, and uncertainty-based pruning removes unreliable keypoints. A central application-level demonstration is vision-based aircraft landing, where calibrated covariances for runway keypoints support uncertainty-aware aircraft position estimation and downstream sensor fusion.

In this work, we present a simulation-based parameter estimation framework for a model defined by a computational simulation of a physical system. We specifically outline an estimation framework consisting of two closely-integrated steps that facilitate an overall end-to-end parameter estimation scheme. The first step involves utilizing an embedded normalizing flow which is used to transform the unknown complex distribution of the residual information into a simple base distribution corresponding to the transformed residual information. In the second step, an empirical-likelihood estimator, under moment restrictions, is utilized for imposing an indirect constrain on the base distribution, where such an instantiated task reasonably allows us to treat the transformed residual information as random variables arising from discretely distribution population with each transformed data point as a single-cell from a set of finite-cell contingencies. Moreover, we use first-order gradient methods for updating the estimated parameter values of the model defined by the computational simulation and the corresponding parametrized embedded normalizing flow, that call for all gradient-related information by leveraging implicitly differentiations of the empirical-likelihood function, which is constructed from the implied empirical probabilities under moment restrictions. Here, it is worth mentioning that the problem formulation presented in this work, which highlights an information-theoretic interpretation, allows to present a computational framework for algorithmic implementations. Finally, as a-by-product, the inverse of the parametrized embedded normalizing flow, w.r.t. the estimated parameter values, serves as a surrogate model for the computational simulation model, which provides useful information for quantifying model discrepancies and sensitivity analysis.

Here we introduce ELECTRIC (Evidential Learning-Enhanced CT Reconstruction via Iterative Correction), a physics-guided Bayesian formulation. An evidential neural network provides an image proposal and an error-predictive epistemic-uncertainty surrogate. The latter is converted into an adaptive precision field and inserted into a Poisson-weighted MAP update. The resulting image-evidence-precision-reconstruction loop treats prior confidence as a learned state variable of iterative reconstruction. In addition to the formulation and theoretical analysis, we report two simulation studies on image slices from the AAPM Mayo Clinic Low-Dose CT dataset: a mechanism-validation pilot using transparent surrogate estimators, and a feasibility study in which a trained Normal-Inverse-Gamma evidential network drives the full closed loop. On held-out patients, the learned prior mean reduces reconstruction error by roughly 70 percent relative to filtered back-projection, the learned epistemic uncertainty is error-predictive and supports selective trust, and the physics-guided update restores measurement consistency while the adaptive-precision reconstruction matches or exceeds a validation-tuned fixed prior and remains markedly more robust to prior-strength misspecification. Together these results demonstrate the complete ELECTRIC closed-loop pipeline, while identifying formal uncertainty calibration and joint training as the principal directions for future work.

Large scale Bayesian nonparametrics (BNP) learner such as Stochastic Variational Inference (SVI) can handle datasets with large class number and large training size at fractional cost. Like its predecessor, SVI rely on the assumption of conjugate variational posterior to approximate the true posterior. A more challenging problem is to consider large scale learning on non-conjugate posterior. Recent works in this direction are mostly associated with using Monte Carlo methods for approximating the learner. However, these works are usually demonstrated on non-BNP related task and less complex models such as logistic regression, due to higher computational complexity. In order to overcome the issue faced by SVI, we develop a novel approach based on the recently proposed constant stepsize stochastic gradient ascent to allow large scale learning on non-conjugate posterior. Unlike SVI, our new learner does not require closed- form expression for the variational posterior expectatations. Our only requirement is that the variational posterior is differentiable. In order to ensure convergence in stochastic settings, SVI rely on decaying step-sizes to slow its learning. Inspired by SVI and Adam, we propose the novel use of adaptive stepsizes in our method to significantly improve its learning. We show that our proposed methods is compatible with ResNet features when applied to large class number datasets such as MIT67 and SUN397. Finally, we compare our proposed learner with several recent works such as deep clustering algorithms and showed we were able to produce on-par or outperform the state-of-the-art methods in terms of clustering measures.

Diffusion Policies (DPs) are able to perform complex manipulation tasks. However, DPs are typically trained by minimizing a denoising objective, which provides limited control over generalization in the finite-data regimes common in robotics. In this letter, we propose PAC-DP, an approach that increases the performance of DPs in robotic manipulation tasks. By modeling the DP as a Bayesian neural network, and defining a PAC-Bayes generalization bound, we derive a novel training objective that augments the standard denoising loss with a Kullback-Leibler divergence regularizer between the posterior and prior parameter distributions. From the theoretical perspective, our approach provides a principled approach to regularize the training of DPs without significantly increasing the training time. From the practical point of view, experimental results demonstrate improved denoising performance, lower variational negative log-likelihood, and higher success rates across multiple robotic manipulation benchmarks. Crucially, the largest improvements are observed in low-data training regimes and complex tasks, establishing PAC-DP as a theoretically grounded framework for robot policy learning.

Autonomous laboratories automate experimental execution, but a campaign must also decide which recovery pathway merits optimization. We formulate this as a sequential decision problem with a discrete pathway-identification stage and a continuous within-pathway optimization stage under heterogeneous experimental costs. Our implementation, Coactive learning, combines a cost-sensitive Bayesian hypothesis-discrimination policy motivated by EC2 (Golovin et al., 2010) with Gaussian-process Bayesian optimization (Srinivas et al., 2010). Under explicitly stated assumptions, the expected spend of one fixed-budget campaign attempt is bounded by the expected pathway-identification cost plus the capped within-pathway optimization budget. We evaluate the method on synthetic benchmarks constrained by selected results reported for PNNL's CICERO selective-precipitation study (Ritchhart et al., 2026). The method performs comparably to an oracle-pathway Bayesian-optimization reference and to a strong split-plate baseline that discriminates pathways with its first plate, without receiving an oracle label for the correct pathway. It is given a candidate hypothesis space and a diagnostic likelihood model. On an NdFeB-inspired instance, it avoids the simulated penalty of a commit-first baseline that initially selects a plausible but inferior hydroxide pathway. This hypothetical wrong-first-commitment scenario is motivated by the hydroxide-oxalate performance contrast reported by CICERO. We characterize the sensitivity of these conclusions to the assumed cost model. The code and benchmark are open source.

Self-driving laboratories increasingly rely on multi-fidelity Bayesian optimization (MFBO) to balance cheap, approximate evaluations against scarce, expensive ones, with a predictive surrogate at its core. Gaussian processes (GPs) are the default choice, but they scale poorly as data accumulate and assume a smooth landscape that molecular and materials search spaces routinely violate. Transfer learning offers an alternative suited to this regime: it learns a representation from abundant cheap data and adapts it to sparse expensive data. Despite its use in property prediction, transfer learning has not been tested as the engine of a closed-loop optimization. Here we benchmark eleven transfer-learning surrogates against four GP methods under an identical selection rule, fidelity budget, and model size, across nine tasks spanning synthetic functions to real chemistry and materials problems. GPs win on smooth, low-dimensional functions but perform worst on molecular and materials problems, where transfer-learning surrogates reach substantially better solutions using far less computation. Because acquisition policy is held fixed across surrogates, this advantage is attributable to the surrogate itself. Uncertainty-driven exploration is not reliably beneficial, and calibration does not predict optimization performance, so greedy exploitation of the transfer-learned mean is the more robust default. Transfer learning is therefore the surrogate of choice for molecular and materials MFBO.

Longitudinal medical imaging captures temporal evolution of lesions, yet extracting the underlying dynamical parameters governing this evolution remains challenging. We propose an inverse Bayesian framework for inferring lesion dynamics from longitudinal spectral CT. We decompose spectral feature ($x$) evolution into three components: \begin{equation*} \frac{dx_i}{dt} = A_i x_i + B \cdot n + C \cdot Δx_{\text{sat}} \end{equation*} where $A_i$ captures intrinsic dynamics (lesion-autonomous evolution), $B$ captures local environment tumour burden (organ tumour burden through satellite count coupling), and $C$ captures environment/satellite state change (i.e., whether surrounding lesions move similarly or not). We demonstrate the framework on photon-counting NSCLC CT data from metastases, recovering distinct dynamical regimes: lung lesions exhibit significant satellite count coupling ($B=-0.34$, $p<0.05$) suggesting competitive dynamics, while liver lesions show synergistic satellite behaviour coupling ($C\approx+1.0$, $p<0.05$). Synthetic validation confirms parameter recovery, and cross-coupling analysis validates that our method detects non-zero coupling when present. This work establishes inverse dynamical inference as a principled methodology for extracting interpretable parameters from longitudinal imaging, moving beyond static feature extraction toward mechanistic characterisation of lesion behaviour. The code and data are available at: https://github.com/lukasf98/inverse-bayesian-inference

Brain-computer interfaces (BCIs) have long sought calibration-free operation, but classifiers are typically benchmarked by discrimination alone, blind to whether predicted probabilities are well calibrated - a meaningful gap given nonstationary electroencephalogram (EEG) signals and the risk of overconfident point-estimate classifiers under distribution shift. We conducted a large-scale study contrasting Bayesian complete-pooling models against frequentist baselines for cross-subject, left-hand versus right-hand motor imagery EEG classification across 20 datasets. Six frequentist pipelines were each paired with an analogous Bayesian pipeline sharing identical feature engineering, fit via Markov chain Monte Carlo posterior sampling. Our primary metric was the Brier score, decomposed into reliability and resolution, alongside AUROC for discrimination and Shannon entropy for sharpness. Each metric was analyzed via random-effects meta-analysis (REML, Knapp-Hartung adjustment), verified by leave-one-out influence analysis. Bayesian complete-pooling produced statistically but not practically significant improvements in reliability and increases in predictive uncertainty (lower sharpness); Brier score, resolution, and discrimination showed no significant differences. Between-study heterogeneity was low across all metrics, though the reliability result was sensitive to leave-one-out removal. We additionally profiled computational cost, finding that Bayesian pipelines consumed roughly thirteen times more energy than their frequentist counterparts, a cost that remains modest relative to common household appliances. These results suggest that Bayesian complete-pooling alone offers limited practical benefit for cross-subject motor imagery classification, and that partial-pooling across subjects and sessions is a more promising direction for future work.

Verbalized Machine Learning (VML) parameterizes a model as a natural-language prompt that an LLM evaluates as f(x; theta). The framework is interpretable, but it commits to a single hypothesis with no measure of uncertainty, and that hypothesis varies substantially across optimization runs on the same data. We propose the Verbalized Particle Posterior (VPP), which treats verbalized learning as a Bayesian inference problem: maintain a population of natural-language hypotheses as particles, update them with Metropolis-Hastings (VPP-MH) or Sequential Monte Carlo (VPP-SMC), and predict by Bayesian model averaging. Both algorithms treat the LLM as a black box, requiring no access to logits or gradients. A distinctive consequence follows. In classical Bayesian learning, model selection sits outside the posterior; in VPP both model structure and parameters share a single language space, and the posterior ranges over both. We evaluate VPP on regression, classification, and rule-discovery benchmarks. It improves over a single VML run on every benchmark and matches or exceeds an oracle-best ensemble of independent VML runs on most, while eliminating the catastrophic single-run failures that VML occasionally produces. Because each particle is a human-readable hypothesis, the posterior is itself something a reader can inspect, seeing in plain text which explanations the data supported and which it ruled out.

Models trained on long-tailed data using standard softmax tend to exhibit higher training error and a larger generalisation gap for classes with fewer training samples. We characterise this class-wise disparity as the preference issue and quantify it using a new metric, the model imbalance level $I$. To understand this issue, we analyse how imbalanced training data adversely affects class-wise gradients under standard softmax training. This paper then develops a finite-data Generalised Balanced Softmax (GBS) framework for analysing and mitigating the preference issue. The framework uses the training-time logit adjustment $z_{nc}+β\log|N_c|$, which is algebraically identical to the training-time logit-adjusted loss of Menon et al. (2021) when $τ=β$. The case $β=1$ also coincides with Balanced Softmax and with the unit adjustment supported by the Fisher-consistency argument under the true data distribution, corresponding to an idealised infinite-data setting. Building on this existing loss family, this paper uses a heuristic power-law assumption to motivate the adjustable coefficient and studies how $β$ affects trained models. Across the evaluated long-tailed benchmarks, $β=1$ does not attain the highest average testing recall on most datasets, showing that a different coefficient can be preferable when training on finite data. The selected values of $β$ reduce $I$ and improve average testing recall relative to the $β=1$ reference, while retaining negligible computational overhead and compatibility with existing representation-learning frameworks.

Empirical Bayes (EB) performs simultaneous inference across many related latent variables. Classical EB assumes that the likelihood p(x | z) is tractable. In many scientific applications, however, the likelihood is available only through a simulator. This paper develops EB for such implicit likelihoods. We introduce simulation-based empirical Bayes (SBEB), which connects nonparametric EB to simulation-based inference (SBI). SBEB computes EB estimates without an explicit density by using the observed data, simulator samples, and an amortized inference network. SBEB iteratively refines the fitted EB prior toward the population prior. With several scientific simulators and real-world data, we demonstrate that SBEB improves accuracy over SBI with a fixed prior.

Near 1310 nm photonic-crystal surface-emitting lasers (PCSELs) are attractive narrow-beam sources for optical communication and sensing, but their final design refinement is costly. Small geometry changes simultaneously shift the band-edge resonance, cavity leakage, far-field divergence, and the numerical stability of a high-$Q$ decay fit, while every full-wave trial requires a time-domain simulation. We couple a commercial finite-difference time-domain solver to a reliability-aware Bayesian optimization (BO) loop over eight local design variables. Each completed simulation updates the surrogate used to choose the next geometry. Candidate ranking combines wavelength and beam-quality requirements with a reliability-adjusted metric $Q_{\mathrm{eff}}$ derived from the solver-reported relative fit-error estimate $dQ/Q$. Across three 80-evaluation runs from the same reference model, BO produced 5--15 candidates per run that passed the joint filter. Designs reconstructed from fresh model copies retained $Q_{\mathrm{eff}}=4.33\times10^6$--$7.76\times10^6$, a 60--108-fold increase over the baseline metric, at 1308.23--1310.90~nm with approximately $0.84^{\circ}$ divergence. Under equal budgets, BO gave the highest mean strict-filter yield (9.0 candidates), compared with differential evolution (7.0) and Latin-hypercube sampling (1.5), although the controls occasionally matched the peak $Q_{\mathrm{eff}}$. Field maps, resonance spectra, and local perturbations further identify an index-related wavelength handle and a hole-size-related leakage handle. The resulting FDTD budget produces a pool of wavelength-compatible, narrow-beam, and reproducible high-$Q$ PCSEL candidates without trusting a single optimistic decay fit.

Simulation-based inference (SBI) with machine learning is an increasingly important tool for solving inverse problems in science and engineering, including parameter inference and the inversion of detector effects. We provide an overview of the Bayesian and frequentist statistical frameworks, describe how machine-learning-based SBI methods, such as neural posterior estimation and neural likelihood estimation, can be used for parameter estimation within these frameworks, and show that the same methods can also be applied to Empirical Bayes or unfolding tasks. We also discuss how to validate inference results and the limitations of SBI with machine learning.

Entry-only automatic fare collection systems record boardings but not alightings, preventing direct construction of origin-destination (OD) matrices. This study develops a Hierarchical Bayesian Latent-Destination (HBLD) model that combines station-hour boarding and inferred alighting demand with passenger card histories. Trip-chain destinations are treated as noisy evidence with a reliability parameter, allowing destination uncertainty to propagate into OD flows. The model was applied to 838,305 bus tap-ins collected in Changzhou in May 2025 and linked to stop-network and hourly weather data. It estimates destination distributions over feasible downstream and reverse-direction through-terminal stops using network, time-of-day, weather, and smoothed historical demand effects. A Bayesian personalization layer uses prior card trips and reverts to the shared trip-level distribution when history is unavailable. Fitted by stochastic variational inference and evaluated on the final week, HBLD outperformed the strongest baseline. Observed boarding patterns consistently improved prediction, especially without card history, while inferred alighting patterns helped only when trip-chain evidence was strongly trusted. The model captured travel consistent with through-terminal riding and bus-assisted road crossing and estimated destinations for trips unresolved by deterministic chaining. Because true alightings were unavailable, scores measure agreement with trip-chain outputs rather than actual destination accuracy. HBLD provides uncertainty-aware destination predictions and OD matrices for service management, planning, scheduling, and resource allocation.

Diffusion policies generate multimodal robot action sequences from demonstrations, but steering them toward deployment-time constraints typically relies on differentiable guidance costs. This excludes many practical safety constraints, such as binary collision checks, joint limits, and black-box rollout costs that are nondifferentiable. We propose Gradient-free Robot Action generation via Combined diffusion-MPPI posterior mean Estimation (GRACE), which guides a pretrained diffusion policy with Model Predictive Path Integral (MPPI) control using only forward cost evaluations. Building on the common score-ascent structure of diffusion and MPPI, GRACE constructs a cost-conditioned guidance posterior at each reverse step and estimates its mean with a single MPPI update centered at the diffusion reverse mean. For differentiable costs, GRACE recovers conventional gradient guidance under a first-order, matched-covariance approximation. GRACE attains higher success rates than diffusion-based and sampling-based baselines in simulation. On a real 7-DoF manipulator, GRACE avoids a deployment-time obstacle that the unguided prior collides with in every trial. Code and experiment videos are available at https://anonymous.4open.science/w/grace-70BB/.

Bayesian online learning promises uncertainty-aware prediction on data streams, but its performance hinges on inferential choices, including learning rates, prior distributions and variational families, which are usually fixed before seeing the stream. We address this by treating Bayesian update rules as experts and aggregating the Bayesian experts according to sequential predictive losses. We prove that the resulting aggregate competes with the best expert in hindsight at an aggregation cost determined by how each expert's per-round performance is evaluated. We instantiate the framework in online conformal inference and Gaussian process regression. The conformal inference application yields a smoothed Bayesian counterpart of adaptive conformal inference with long-run randomized coverage, while the Gaussian process application gives an oracle inequality in cumulative predictive Kullback-Leibler risk and adaptation to unknown Hölder smoothness up to logarithmic factors. Experiments show that the aggregate tracks strong experts without oracle expert selection.

Sophisticated Inference is a variant of active inference often associated with recursive belief modeling and tree search. We argue that its central computational role is simpler: within a planning horizon, it makes active inference closed-loop by allowing future actions to depend on future states and observations. This closed-loop structure can be represented in the epistemic-prior variational free energy framework. Epistemic priors supply the active-inference objective, while a joint posterior over future states and actions supplies the state-contingent control structure. We evaluate this decomposition in the Reactivity Maze, a stochastic benchmark designed to separate epistemic incentive from inner-horizon closed-loop control. The comparison includes three variational objectives with the same state-action posterior family, an action-state factorized active inference objective, Sophisticated Inference, and standard Expected Free Energy planning. The results show that neither ingredient is sufficient on its own. Methods without an epistemic component do not seek information, while methods that prevent future actions from depending on future states cannot turn information into reliable goal-reaching. By contrast, both Sophisticated Inference and full-joint epistemic-prior active inference solve the environment by combining epistemic drive with closed-loop inference. These results show that the advantage associated with Sophisticated Inference need not be specific to tree search itself. It arises from the closed-loop form of active inference, and this form can be represented in epistemic-prior variational inference when the posterior keeps future actions dependent on future states.

Gaussian process (GP) modeling is widely used in computational science and engineering. However, fitting a GP to high-dimensional inputs remains challenging due to the curse of dimensionality. While various methods have been proposed to reduce input dimensionality, they typically follow a two-stage approach, performing dimension reduction and GP fitting separately. We introduce a Bayesian framework that seamlessly integrates dimensionality reduction with GP modeling and inference. Our approach, built on a hierarchical Bayesian model with priors on the Stiefel manifold, enforces orthonormality on the projection matrix and enables posterior inference via Hamiltonian Monte Carlo with geodesic flow. Additionally, we extend this framework by incorporating Deep Gaussian Processes (DGP) with built-in dimension reduction, providing a more flexible and powerful tool for complex datasets. Through extensive numerical studies, we demonstrate that while the proposed Bayesian method incurs higher computational costs, it improves predictive performance and uncertainty quantification, providing a principled and robust alternative to existing methods.

Modeling galaxy-galaxy strong gravitational lenses to infer the brightness of the source galaxy and the mass distribution of the foreground galaxy is computationally challenging, particularly for high-resolution, high signal-to-noise ratio observations. In this regime, high-dimensional representations of both the source and the foreground mass distribution are necessary to model the data down to the noise level. This inference problem has been challenging for both traditional and machine learning-based methods because of its high dimensionality and its non-linearity in the foreground mass distribution. We present a method to generate joint posterior samples of the source galaxy and foreground mass distribution as pixelated images conditioned on observations. The method combines diffusion-based generative modeling and recurrent inference machines. It can model realistic gravitational lensing simulations with background and foreground galaxies drawn from cosmological hydrodynamical simulations down to the noise level.

Diffusion-based methods have achieved remarkable empirical success in solving inverse problems. However, many existing posterior samplers either lack rigorous theoretical guarantees or incur substantial computational overhead. We propose a simple and efficient algorithm, called \pddim, for solving linear inverse problems with diffusion priors via a DDIM-type sampler. Our method requires only lightweight, coordinate-wise modifications to the standard DDIM update, while explicitly incorporating the measurement model. The key idea is to perform posterior sampling separately along each singular direction of the measurement operator: for each direction, the sampler follows the learned diffusion prior when the observation signal-to-noise ratio (SNR) is below the corresponding diffusion SNR, and switches to a calibrated measurement-based predictor otherwise. We prove that the proposed sampler converges to the Bayesian posterior conditioned on the measurements. Empirical results show that the proposed sampler performs favorably against existing diffusion-based posterior samplers across a range of image restoration tasks, achieving the best performance on the majority of evaluation metrics considered. Overall, our results convert posterior sampling for noisy linear inverse problems to simple coordinate-wise DDIM updates, yielding an efficient, easy-to-implement algorithm with provable posterior consistency.

With growing surgeon shortages, automating surgical sub-tasks such as tissue dissection offers a promising step toward reducing workload and expanding patient access. Prior work has relied on hand-crafted incision policies that cannot quantify uncertainty or has relied on simulation-based methods that require strong modeling assumptions. We instead view tissue attachment identification as an inherently probabilistic problem and propose a Bayesian approach that avoids explicit tissue modeling. Our method uses a Sequential Bayesian Hilbert Map (SBHM) to represent the likelihood that each tissue point is attached to the underlying resection surface. An ensemble of learned classifiers predicts attachment likelihoods from spatial data acquired during robotic tissue retraction, with each classifier serving as a noisy information source to update the SBHM. To plan the next retraction, we devise Bayesian Retraction Optimization (BRO) to select the most informative action under safety constraints. As the SBHM refines over time, regions with high attachment likelihood are selectively incised. We validate our method in simulation across diverse tissue geometries and acquisition strategies, and demonstrate zero-shot transfer to real robotic dissection experiments.

In bearing vibration datasets, most samples receive predicted fault probabilities close to 0 or 1, while samples with intermediate (gray-zone) probabilities are rare. Such borderline samples are important because they reflect conditions in which maintenance decisions may require additional inspection or a conservative response and are useful for studying decision boundaries. To address this scarcity, this paper proposes and compares two approaches that generate vibration signals whose predicted fault probability matches a target probability of 0.25, 0.50, or 0.75. We use the average output of a heterogeneous ensemble classifier with different architectures and random initializations as a fixed, gradient-accessible probability oracle. The first, training-based approach, Probability-Regularized Generative Adversarial Network (PR-GAN), extends Wasserstein Generative Adversarial Network with Gradient Penalty (WGAN-GP) and edits a real signal through a residual generator while pushing the classifier output toward the target probability. The second is a training-free, per-sample Wachter-style counterfactual (CF) procedure that directly optimizes each input signal to reach the target probability while remaining close to the source signal. We evaluate both methods on the Case Western Reserve University (CWRU) and Paderborn bearing datasets using mean absolute target-probability error, time-domain total variation, and frequency-domain log power spectral density (log-PSD) differences. Across all settings, CF reaches the target with a mean absolute probability error of 0.005-0.008 and a within-tolerance success rate of 1.000 on retained samples, whereas PR-GAN's mean error is 0.046-0.059 with success rates between 0.501 and 0.680. CF therefore steers the probability more reliably and requires smaller average L1 changes, whereas PR-GAN has a lower reported runtime in most settings.