Bayesian

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.

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.

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.

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.

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.

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.

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.

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.

Overparameterized models often have continuous parameter symmetries, so different parameters define the same predictor. We show that PAC--Bayesian analysis should be performed on the quotient predictor space: pushing a prior and posterior to the quotient preserves the empirical and population Gibbs risks while removing the nonnegative KL contribution caused solely by how the two distributions differ among parameterizations of the same predictor. Quotienting alone does not determine which prior to use. We construct a canonical choice of one parameterization for each predictor and account for the geometric volume of its equivalent parameterizations. This transforms a neutral reference prior into a data-independent prior that reflects the model's implicit bias. It approximates the ideal but inadmissible posterior-matched prior, which would minimize the KL term by depending on the training data. The resulting certificate is tighter exactly when this geometry-induced prior has smaller KL divergence from the learned quotient posterior than the neutral prior. We test this prediction in Fourier regression with a Hadamard parameterization and in Query-Key attention, using ordinary SGD without an explicit regularizer. The implicit-bias prior reduces the mean quotient-space KL by $40.69\%$ and the mean PAC--Bayes certificate by $21.40\%$ in the Fourier-Hadamard experiment. The smaller, prior-scale-dependent improvement in Query-Key attention confirms the predicted conditional nature of the effect.

The soft-label Bayes-error estimator beta(z) = E[min(z, 1-z)] of Ishida et al. estimates the irreducible error of a binary task directly from probability-valued labels. Recent work by Ushio et al. showed that this estimator is fragile when the probabilities are not the true posterior: even perfectly calibrated soft labels can yield a substantially inaccurate estimate, and they propose isotonic calibration as a consistent remedy. We complement that line of work by characterizing exactly how the most widely used post-hoc calibration map -- temperature scaling -- distorts the proxy. We prove an exact, model-free identity reducing the temperature-scaled proxy to the classifier's margin distribution, from which we obtain (i) strict monotonicity in the temperature and (ii) a continuous bijection from the temperature axis onto the open interval (0, 1/2), so that a fixed classifier -- with fixed decisions and fixed 0-1 error -- can be made to report any proxy value whatsoever. Under a Gaussian model of the logits we further derive a two-parameter closed form for the entire proxy-versus-temperature curve. Across CIFAR-10, Fashion-MNIST, and SVHN (eight binary tasks), the proxy varies by 56x to 980x at constant test error, the closed form reproduces the empirical curve to within 0.018, and the calibration temperature that minimizes the expected calibration error does not coincide with any stable proxy value. Our results give a precise, predictive account of the distortion whose existence motivates calibration-based remedies, and they reinforce the practical recommendation that a proxy value is meaningful only together with the mechanism that produced its probabilities.

Bayesian inference provides a principled foundation for reasoning under uncertainty, but its computational cost hinders deployment on resource-constrained edge devices. In this paper, we present a hardware-oriented methodology for accelerating discrete Bayesian inference on commercial off-the-shelf embedded GPUs. We identify that the latency of a broad class of variational message-passing algorithms is dominated by tensor contractions. Our approach restructures the memory layout of these operations using two complementary merging strategies that produce compact, regularly-shaped primitives better suited for efficient GPU execution. We then introduce optional sparse array representations and a tensor-clustering scheme to reduce the memory footprint. We instantiate the methodology and produce optimized variants of three message-passing algorithms for Hidden Markov Models (HMMs), namely variational filtering, variational message passing, and marginal message passing. Furthermore, we complement this with a machine-learning-based autotuner that automatically selects the best-performing algorithmic variant for a given generative model specification. Benchmarked on an NVIDIA Jetson Orin AGX across 770 randomly sampled realistic Partially Observable Markov Decision Process (POMDP) configurations, our implementations achieve speedups of up to 5x, with typical gains of 2-2.5x, while producing numerically identical outputs to the baseline implementations.

Neural simulation-based inference enables parameter estimation for complex models, but typically requires the user to specify a simulator encoding a fixed model structure. We present a framework for joint model selection and parameter estimation that combines large language models for program synthesis with neural simulation-based inference. Given a natural language description of the system and data under investigation, an LLM proposes candidate simulator programs which are iteratively refined via feedback-driven mutation and evaluated using neural density estimation. The approach enables simulation-based inference over a pool of models, not just parameters within a fixed model. On benchmarks spanning deterministic dynamics, stochastic epidemic models, and dark matter substructure inference from gravitational-lensing images, the method identifies plausible model families from open-ended prompts, with accuracy that reflects the information content of the data and identifiability of candidate models.

We introduce Deep Adaptive Bayesian Screening (DABS), a method for performing adaptive factorial screening in high-dimensional discrete design spaces. DABS learns a policy network offline to sequentially select informative experiments, amortizing Bayesian Optimal Experimental Design. It handles binary designs, incorporates sparsity and interactions via a spike-and-slab prior with strong heredity. The model is trained using a contrastive lower bound on information about factor activity with nuisance effect sizes and noise variance analytically integrated out. Unlike prior amortized Bayesian design approaches, DABS also integrates Gibbs posterior inference at deployment, yielding posterior probabilities of factor activity and credible intervals on effect sizes. We demonstrate DABS on screening problems calibrated to real-world benchmarks and show it achieves superior accuracy and scalability over classical and Bayesian baselines under tight experimental budgets.

Attention collapse in autoregressive language models -- manifested as repetitive token loops where the model becomes trapped in self-reinforcing attractors -- is a persistent pathology that existing decoding-time heuristics fail to address at its root cause. We present a principled framework that penalises or compensates anomalous confidence arising from collapsed generation patterns, by comparing a token's observed frequency against its corpus prior through an adjacent-conditional probability construction. The resulting self-normalising penalty ratio $R=f(m,n,p)/f(np,n,p)$ requires no ad hoc standardisation and admits a closed-form logit offset with zero approximation error. The correction is isolated from the loss gradient and accumulated into a frozen output-layer bias via exponential moving average, enabling deployment as a repair mechanism for models that have already collapsed without requiring intrusive modifications to standard training pipelines. Experimental validation on a 1.5B-parameter model demonstrates that the frozen-bias mechanism can rescue a model already trapped in a collapsed attractor, reducing 2-gram repetition from 0.073 to near 0 while preserving generation quality.

Contact-rich manipulation requires pose estimates that are often more accurate than what depth-only sensing provides. Existing methods, relying on vision and contact, employ costly offline training procedures that need to be retrained for new environments and geometries. We propose BayesContact, a Simulation-Based Inference framework for visuo-tactile pose estimation in peg-in-hole insertion. BayesContact maintains a particle belief over object pose and fuses depth observations with force/torque-derived contact evidence. We employ simulation based forward models to approximate these observation likelihoods. For each pose hypothesis, a renderer predicts depth measurements and a physics simulator predicts contact outcomes under guarded probing actions; both are scored against real observations to update the belief. The resulting multimodal belief also enables information-gain-based probing for active disambiguation. Across simulated geometries and real-robot experiments, BayesContact improves pose observability and insertion success over vision-only inference by 30%

Prompt optimization adapts large language models (LLMs) without updating model parameters, but many automatic prompt optimizers remain heuristic search procedures over candidate instructions. This paper studies prompt optimization as Bayesian posterior sampling over discrete prompt tokens. We define a posterior distribution by combining a task likelihood term, which rewards prompts that explain input-output examples, with a language-model prior, which favors fluent instructions. This converts prompt optimization into an energy-based posterior sampling problem, for which gradients can be used to guide discrete Markov chain Monte Carlo (MCMC) proposals over vocabulary tokens. We refer to our framework as BayesPO, short for Bayesian Prompt Optimization. In this paper, BayesPO is instantiated with Markov chain Monte Carlo: it uses a Metropolis-Hastings corrected Gibbs-with-Langevin (GwL) proposal and integrates parallel tempering for global exploration of rugged LLM-induced energy landscapes. The concrete sampler further adapts the GwL sampler to the practical constraints of non-weight-tied LLM embeddings. Experiments with Qwen2.5 models show that the sampler discovers semantically meaningful prompts on diagnostic tasks, that parallel tempering helps escape a local optimum in a poetry completion task, and that post-optimizing APE prompts on 24 instruction-induction subtasks improves average accuracy from 60.04% to 63.23%. The study also reveals two main limitations: energy minimization may overfit small optimization sets, and the current sampler remains computationally expensive. These findings position Bayesian prompt sampling as a principled post-optimization tool and point to a promising direction for probabilistic prompt optimization.

Calibration of grey-box simulation models is a constrained optimization problem in which model evaluations are expensive, the parameter space can be high-dimensional, and the search must respect plausibility constraints. Although the simulation code is fully available to the analyst, the joint effect of multiple parameters remains difficult to predict analytically. Classical optimizers such as Nelder--Mead (NM) are simple to deploy but sample-inefficient, particularly under constraints. Modern Bayesian Optimization methods achieve competitive solutions with far fewer evaluations but require non-trivial modeling machinery for constraint handling. We introduce an agentic calibration method in which a large language model acts as the optimizer, with constraints incorporated as a plain-language section of the system prompt. We evaluate the agentic method, NM, and Bayesian Optimization (BO) on an anal cancer simulation model under both unconstrained and clinically constrained calibration. Under unconstrained calibration, the agentic method achieves substantially lower best error than BO and NM, while requiring fewer model evaluations. Under constrained calibration, the agentic method reaches comparable error levels and both outperform NM. These results are obtained at the cost of increased inference time per iteration. Agentic calibration achieves competitive performance with substantially fewer model evaluations, and constraint handling is essentially free at the modeller-facing interface through simple textual specifications rather than additional modelling machinery. The main trade-off lies in increased per-iteration inference cost, making the approach particularly suitable when simulation time dominates. Beyond performance, the per-iteration rationale makes the search auditable and explainable, so its decisions can be scrutinised and justified to third parties.

Logic and optimization can, in combination, make valuable contributions to rule-based AI. Logic is the obvious medium for encoding a rule base and drawing inferences from it, while optimization provides a powerful technology for computing inferences. Their combination has taken on new relevance amid a growing concern for transparency in AI. which is important for reproducibility, explainability, trustworthiness, and fairness. Rule-based AI provides a natural solution to transparency that is becoming increasingly practical due to today's highly advanced optimization methods. This article surveys several areas of logic-optimization partnership, including probabilistic logic, Bayesian logic, belief logics and Dempster-Shafer theory, nonmonotonic (default) logic, many-valued logics, and inference of logical formulas from noisy data based on Boolean regression. It shows how to compute projections, the fundamental problem of both logic and optimization, using decision diagrams and logic-based Benders decomposition. It describes the use of postoptimality analysis to explain how conclusions are reached, further enhancing transparency, as well as the role of optimization in answer set programming modulo theories. The paper concludes by suggesting possible future research directions.

Black-box models limit the adoption of artificial intelligence in medicine because their predictions are difficult to interpret and reproduce. We present a statistically grounded framework for interpretable, rule-based clinical classification using the Bernoulli Naïve Bayes (BNB) model. Supervised chi-square-guided binarization converts continuous variables into binary indicators by selecting thresholds that maximize association with the clinical outcome within the training folds, which allows BNB to operate on continuous medical data without sacrificing transparency. On three benchmark datasets, Pima Indians Diabetes, Wisconsin Breast Cancer, and Heart Failure Prediction, the framework reached areas under the receiver operating characteristic curve of 0.800, 0.984, and 0.919, respectively. Probabilistic reliability was assessed with a leakage-safe cross-validated calibration analysis reporting Brier score and calibration intercept and slope, and post-hoc beta calibration improved probability calibration across datasets. These results indicate that an interpretable, statistically motivated framework can perform comparably to more complex models while providing explicit decision rules expressed in clinical units and calibrated risk estimates. A complete worked example further shows that model inference can be reproduced from a printed reference table using only basic arithmetic, without software or proprietary tools, supporting trustworthy and auditable use of artificial intelligence in clinical settings.

Current evaluation of epistemic uncertainty relies on tasks such as out-ofdistribution detection and active learning. However, the Bayes-optimal decision strategies for these tasks do not coincide with the scores commonly used to quantify epistemic uncertainty. Building on the epistemic reject-option framework, we evaluate epistemic uncertainty using its ability to identify regret, the reducible error. Formulating selective prediction as a constrained optimization over coverage, expected risk, and regret, we prove the optimal selector is a thresholded convex combination of the ground-truth aleatoric and epistemic uncertainties. This theoretical unification exposes a weakness in recent uncertainty disentanglement literature: we demonstrate that standard correlation metrics between learned components do not necessarily predict their actual operational utility. We instead propose to evaluate the achievable risk, regret, coverage surface of the decomposition as a diagnostic for joint disentanglement and utility. Benchmarking standard methods on datasets with dense human annotations reveals that decision-theoretic rankings can disagree substantially with proxy-task rankings, including pairwise rank inversions between methods that are top-ranked on one criterion and bottom-ranked on other.

Many signal processing systems ultimately exist to {act}. Whenever the state variable that determines the action to be taken by a decision maker, or agent, is uncertain, the way that uncertainty is represented decides how well the agent performs and how much its performance can be trusted. This lecture note develops, from first principles and within a single decision-theoretic setting, the link between the {objective} and the knowledge of an agent and the form of uncertainty representation that is sufficient to act optimally. To start, assuming a known environment distribution, we show that a risk-neutral agent needs the posterior distribution over the state, whereas a risk-averse agent can rely without loss of optimality on a {prediction set} and a worst-case decision rule. We then turn to the case in which the environment is unknown, and identify three complementary approaches to address the resulting epistemic uncertainty: calibration of a fixed predictor, credal (ambiguity) sets with distributionally robust optimization, and Bayesian inference over model parameters. The common thread is that reliable decisions require an uncertainty representation matched to the decision objective and to the knowledge profile of the agent, together with a guarantee that certifies the utility the agent will actually obtain.

Longitudinal tumor measurements, dropout information, and genetic covariates provide complementary information about treatment response, but integrating these data sources within a single population modeling framework remains challenging. We extend the empirical Bayes variational autoencoder (EB-VAE) framework to joint longitudinal and time-to-event modeling and evaluate it on tumor growth data. The framework represents inter-individual variability using latent individual effects regularized by a covariate-conditioned empirical Bayes prior, while a decoder maps these latent effects to tumor-volume trajectories. To account for informative dropout, the decoder was augmented with a hazard model, yielding joint predictions of tumor growth and time to dropout. We further compared fully neural and hybrid semi-mechanistic decoder formulations and incorporated genomic covariates through a genetics-conditioned prior adaptation. The hybrid decoder recovered treatment-effect parameters broadly consistent with previously reported nonlinear mixed-effects estimates, while achieving prior predictive performance comparable to the neural decoder. The joint model reproduced both tumor-volume distributions and dropout patterns in held-out individuals, and genetic conditioning improved individual-level prior predictions in both cutaneous melanoma and breast cancer experiments. Stability selection identified several biologically plausible genetic indicators, including alterations in BRAF, NRAS, NF1, and MDM2. These results demonstrate that EB-VAE provides a flexible probabilistic framework for combining neural dynamics, mechanistic structure, time-to-event modeling, and high-dimensional covariates in pharmacometric applications.

Maneuvering target tracking in three-dimensional space remains a challenging problem due to complex motion dynamics and model mismatch. To address this, this paper proposes a hybrid model/data-driven algorithm named IMMNet, which integrates the interpretable structure of the interacting multiple model (IMM) algorithm with learnable neural components. Unlike end-to-end black-box methods, the proposed IMMNet algorithm not only can preserve the Bayesian inference mechanism that is essential for real-time radar applications, but also can adaptively learn motion patterns and noise characteristics from data. Extensive experiments demonstrate that the proposed IMMNet algorithm consistently outperforms the existing algorithms across various scenarios, validating it as a robust, interpretable, and practical solution for maneuvering target tracking.

Tabular foundation models (TFMs) generate predictions on structured data via in-context learning, without task-specific estimation. We ask whether TFMs can be effectively applied to discrete choice, a central demand estimation framework in marketing and operations, and find that directly applying TFMs yields limited performance. The gap is structural: TFMs assume row-independent observations, whereas discrete choice is inherently set-valued and subject to persistent consumer preference heterogeneity. We propose a reformulation that encodes both choice-set dependence and individual heterogeneity within a row-based learning framework. Evaluated on a yogurt scanner panel, individual-level heterogeneity encoding is the dominant driver of predictive accuracy. The best reformulation outperforms hierarchical Bayesian estimation by 8% in holdout log-likelihood and 3.6% in hit rate, running 16 times faster, a practical advantage for large-scale demand estimation. The advantage is largest in the medium-data regime (10--40 purchase occasions per consumer), where parametric Bayesian shrinkage most distorts estimates for atypical consumers. Fine-tuning on population choice data provides additional gains for consumers with shallow purchase histories, where in-context learning has limited individual-specific signal to condition on. These results establish a principled approach for applying foundation models to consumer choice problems more broadly.

Sensor-rich data-driven applications increasingly use Bayesian approaches to infer latent states of dynamic systems from noisy sensor measurements and physical models. Yet the computation of the likelihood remains an essential bottleneck for accurate posteriors and performant inference. This paper presents a Bayesian filtering technique that uses processor-native uncertainty tracking for both uncertainty propagation and inference. The technique implements deterministic hierarchical importance restructuring through a native operation, giving deterministic latency and bounded memory use for arbitrary models written as program code. Benchmarks across three nonlinear state-space systems compare the approach against particle filters and Monte-Carlo-based likelihood estimators. The technique enables deterministic approximate filtering with as high as 805$\times$ average speedup against direct Monte Carlo work at matched result quality for model evaluation, and Pareto-dominant accuracy-latency trade-offs for posterior inference while remaining competitive in RMSE with baseline particle filters.

3D Gaussian Splatting represents scenes as finite mixtures of anisotropic Gaussians whose number of components $K$ is set by heuristic density control or user caps. Variational Bayes Gaussian Splatting (VBGS) recast splat fitting as conjugate variational inference, but $K$ remains fixed. We replace the finite symmetric Dirichlet over mixture weights with a truncated stick-breaking Dirichlet-process prior -- and, as a theory-backed alternative, a sparse overfitted finite Dirichlet -- so that the number of occupied components adapts to the data while every update remains a closed-form coordinate-ascent step; a natural-gradient stochastic variant makes the per-step cost independent of the number of points. We give an exact monotonicity guarantee, a rigorous truncation-error bound correcting an anti-conservative large-$α$ approximation in common use, and an honest account of what the fitted number of components estimates. Empirically: (i) the effective complexity $\hat{K}$ adapts to scene complexity and recovers the true $K$ within $\pm 1$ on well-separated synthetic data with regime-appropriate concentration; (ii) a deconfounded comparison shows the DP prior's contribution is complexity selection, not per-component efficiency -- converged DP fits exceed single-pass fixed-$K$ VBGS by +2.7 dB at matched budgets yet tie an equally converged fixed-$K$ baseline, and on 3D scenes DP-Splat matches or exceeds VBGS's held-out color prediction with 5.9-7.6x fewer components; (iii) the posterior-predictive color variance is well calibrated on model-matched synthetic data; and (iv) the ordering suggested by exact-posterior asymptotics reverses under mean-field coordinate ascent: the DP prior resists over-splitting while the sparse finite mixture saturates its truncation, a gap between variational practice and posterior asymptotics documented across three orders of magnitude in $N$.

Numerical integration is a cornerstone of various scientific computing applications, such as engineering simulations and model evidence computations in probabilistic machine learning. Bayesian Quadrature uses Gaussian process surrogates that explicitly encode structural assumptions about the integrand to obtain integral estimates with quantified uncertainty. These surrogates are predominantly based on stationary covariance functions, which results in model misspecification for integrands exhibiting nonstationary behavior. We tackle this issue through an adaptively growing, tree-based partition of the integration domain into local stationary models. Our method recombines the local integral estimates through a hierarchy of GP conditioning that reintroduces cross-subdomain correlations, while model selection criteria control the tree growth to avoid unnecessary partitioning. The resulting algorithm is simple, requires no MCMC, and adapts its evaluation budget to local integrand complexity. On benchmark integration problems and a model evidence computation for an epidemiological model, Hierarchical Bayesian Quadrature achieves substantial gains over standard Bayesian Quadrature on nonstationary integrands while matching its performance on stationary ones.

Bayesian optimization (BO) is widely used for expensive black-box problems, yet practical performance depends not only on high-level algorithmic choices but also on how surrogate model training, input and output warping transformations, acquisition functions, and candidate search are implemented. We present tidyHEBO, a BoTorch-native single-objective optimizer designed for robust general-purpose optimization. tidyHEBO jointly fits Yeo-Johnson output warping with the Gaussian-process surrogate, evaluates acquisition functions on the original objective scale using deterministic quadrature or MC-samples, and performs constrained cumulative Pareto search over multiple acquisition criteria. Without any Olympus-specific hyperparameter tuning - using only default optimizer configurations - tidyHEBO ranked first among the evaluated methods on the Olympus benchmark. It achieved the best average ranks for typical performance (average rank 1.53), worst-tail performance (1.21), and run-to-run variability (2.00), measured by median nAUC, CVaR_nAUC, and IQR_nAUC, respectively. Using the same default configuration, tidyHEBO also performed strongly on synthetic and Needle-in-a-Haystack problems and closely matched HEBO on Bayesmark (92.64 versus 93.34) while exceeding GP with logarithmic expected improvement and random search. Adaptive batching reduced feedback rounds while revealing a controllable trade-off between parallelization and optimization quality as the batch cap increased. These results characterize tidyHEBO as a robust, reproducible general-purpose optimizer for a broad range of practical optimization problems, including scientific applications and hyperparameter tuning.

Bayesian optimization routinely warm-starts a target experiment with data from related source tasks, and the multi-task Gaussian process is the textbook surrogate for the job. We revisit this default in a controlled setting and find that it misestimates the cross-task correlation even in the simplest non-trivial case, affinely related source and target tasks, where a working transfer learning method should obviously succeed. We trace the failure to two independent structural mechanisms. Per-task standardization, the textbook fix for the affine slice ambiguity, propagates a finite-sample alignment error into the recovered correlation. The marginal likelihood itself identifies the correlation only at a per-sample rate that a Gaussian process at non-overlapping designs further dilutes. We propose three conservative remedies that follow from the analysis: promoting per-task means and scales to model parameters, restricting the task covariance to non-negative correlations, and co-locating part of the source and target designs. Across synthetic multi-task problems and surrogate-based hyperparameter tuning transfer, these remedies recover the target-only baseline on the simple instances, while the broader failure persists on harder instances and across most rank-based and latent-context variants.

Scientific experiments are often designed to maximize information gain, yet in many applications the primary objective is to support reliable downstream decision-making. Existing decision-aware experimental design and active learning methods typically assume well-specified outcome models and implicitly rely on the stability of the optimal decision under real-world perturbations. In practice, however, experimental outcomes are frequently influenced by hidden or weakly modeled effects, which can substantially alter decision optimality and lead to misleading conclusions. We study sequential adversarially robust decision-aware experimental design, where data acquisition has to take into account information gain against plausible worst-case unexpected effects, modeled here as variation in adversarial variables. Building on Bayesian decision theory, we formalize an adversarially robust optimal decision under this setting and derive a principled Bayesian experimental design criterion. The criterion explicitly targets decision stability rather than nominal optimality. Experiments on synthetic and real-world scientific datasets show that conventional decision-aware design can converge rapidly to high confidence yet fragile decisions, while our robustness-aware approach yields decisions that are significantly more stable and reliable under adversarial variation.

Policy-based approaches to Bayesian experimental design (BED) allow the learning of deep policy networks that adaptively make intelligent design decisions based on previously collected data. However, the training of such policies is often held back by a fundamental challenge: the double intractability of the expected information gain (EIG). This necessitates expensive or complex approximations that restrict the effort one can invest in optimising the policy itself. To address this, we show that the double intractability of the EIG can be isolated from the policy learning by first solving a score matching problem that is independent of the policy used, then using the learned score approximation to train the policy in a singly intractable manner. This turns the key multiplicative cost into an additive one and reduces the computational burden on the policy training itself, making it far cheaper to train the policy multiple times when needed, e.g. for architecture search, hyperparameter tuning, or avoiding local optima. In our experiments we train multiple competitive policies without inducing a multiplicative cost in likelihood evaluations, which can increase performance by allowing us to select the best policy even without performing hyperparameter or architecture searches.

Parameterized quantum circuits (PQCs) are increasingly used as policies and value functions in quantum reinforcement learning, yet it remains unclear when and why quantum policies generalize. We give a PAC-Bayesian account in which generalization is governed not by the raw number of circuit parameters, but by the effective dimension of the Fisher geometry induced by the circuit. This quantity is inflated by entanglement, making entangling connectivity an independent axis of complexity.In controlled experiments that fix the number of trainable rotations and vary only entanglement, we find that circuits with larger Fisher effective dimension exhibit larger train-test gaps, while parameter count is a weak predictor. The resulting bound acts primarily as a ranking certificate: it correctly orders circuits with identical parameter count, which parameter-counting bounds cannot do. We validate this mechanism across supervised classification, quantum contextual bandits, and value-function generalization, where entangled circuits consistently generalize worse than non-entangled circuits of equal parameter count, with gaps shrinking as sample size increases.Our strongest evidence comes from low-variance decision models, including single-observable classifiers, value heads, and one-step policies. In end-to-end multi-step policy learning, entanglement effects remain statistically significant but high return variance leaves the full ordering only partially resolved. Partial-correlation analysis shows that Fisher effective dimension screens off entangling pattern, and controls for training accuracy, readout, and optimizer rule out major optimization confounders. The effect also persists on an IBM Heron quantum processor under real noise. Overall, our results reframe quantum policy design around an entanglement--generalization trade-off rather than expressivity alone.

The uncanny valley is a long-standing empirical rule in humanoid robot design: making robots more human-like can reduce, rather than increase, affinity. Yet existing guidelines, such as adopting robot-like appearances, avoiding excessive realism, and reducing cross-modal mismatches, remain difficult to use for algorithmic design because they are not expressed as manipulable variables. Here, we propose a hierarchical Bayesian generative model that operationalizes these guidelines as mathematical design variables. The model represents affinity toward humanoid robots as posterior-weighted negative category-conditional surprise and explains category ambiguity and perceptual mismatch as increases in surprise. It maps uncanny-valley mechanisms onto four variables: deviation from the predicted robot-category mean, inconsistency in human likeness across modalities, prediction uncertainty, and observational uncertainty. Simulations showed that category ambiguity and appearance--motion mismatch can produce affinity reductions, and that uncertainty reshapes the valley. In a human-subject experiment with robot--human morphing images, we manipulated prediction uncertainty using blurred prior robot stimuli and observational uncertainty using blurred evaluation stimuli. Increased observational uncertainty attenuated the decrease in familiarity ratings at intermediate human likeness, whereas low prediction uncertainty increased ratings for robot-like appearances. This framework turns empirical uncanny-valley heuristics into a computational basis for algorithmically evaluating and optimizing humanoid robot appearance and behavior.

In many decision-making settings, new interventions are acceptable only if they do not reduce outcomes below some established threshold. For example, in clinical medicine, new treatments are often acceptable only if they do not worsen outcomes relative to an established standard of care. Safe Bayesian optimization maximizes an objective subject to safety constraints. In the setting that we consider here, safety is defined relative to a known baseline policy whose outcomes are counterfactual and therefore unobserved. Thus, the counterfactual outcomes of the baseline policy must be estimated and those (uncertain) estimates must be used to safely optimize the objective. We address this estimation problem by using conformal prediction to construct valid uncertainty intervals for counterfactual baseline outcomes, and we show how these intervals can be integrated into safe Bayesian optimization to ensure that constraint violations occur at or below a user-specified rate. We also show how to adapt these conformal estimates to different kinds of covariate shift. We provide a safety proof, experimental evidence, and a sensitivity analysis.

Simulation-Based Inference (SBI) is critical for scientific discovery, with generative models offering a promising path toward efficient inference. However, existing methods struggle with effective multimodal modeling. They often rely on brute-force fusion strategies that ignore the structural disparities between parameters and observations, thus limiting estimation fidelity. In this work, we introduce FUSE (Feynman-Kac steered mUlti-modal flow matching for efficient Simulation-based posterior Estimation). Unlike prior work, FUSE employs a dual-track architecture that preserves the distinct features of multimodal inputs while facilitating dynamic interaction. Additionally, we propose an FK-steered sampling strategy that leverages intermediate observation likelihoods to guide the generative trajectories, effectively improving the sample quality during inference. Our approach outperforms state-of-the-art baselines on standard SBI benchmarks, producing posteriors that closely match ground-truth MCMC. Furthermore, in a real-world exoplanet orbital estimation task, FUSE successfully resolves complex parameter degeneracies that challenge existing methods, highlighting its potential to accelerate complex scientific discoveries in astrophysics and beyond.

Minimum Bayes Risk (MBR) decoding yields more robust and higher-quality text generation than maximum a posteriori (MAP) decoding by selecting hypotheses that maximize expected utility over sampled pseudo-references. However, there exists a discrepancy in the design: hypothesis selection calculates expected utility scores conditioned on given pseudo-references, while commonly used evaluation metrics, e.g., BLEU and COMET, are asymmetric. Therefore, it is important to consider both hypothesis-to-reference and reference-to-hypothesis directional effects. In this study, we introduce a noisy channel decomposition of MBR decoding that naturally incorporates bidirectional effects to account for these asymmetries. We decompose MBR decoding into four interacting components: hypothesis-to-reference likelihood, reference-to-hypothesis likelihood, hypothesis prior, and reference prior. This decomposition provides a unified interpretation of existing MBR variants and enables metric- and task-specific interpretability by isolating the contribution of each channel. Our comprehensive analysis reveals that channel-wise contributions exhibit distinct characteristics across metrics while remaining consistent across tasks, and suggests that appropriate channel weighting may lead to improvements over original MBR decoding.

Scientists often seek to draw causal inferences from structured data that is not independently and identically distributed, such as spatial data, network data, or molecular data. We develop geometric causal models (GCMs), a framework for causal inference from dependent data that exploits underlying symmetries of the data generating process. For example, in spatial data, we consider processes that are symmetric under translations, or in graph data, symmetric under permutations of the nodes. We show how symmetries, formalized with group theory, can enable causal identification and estimation. We deploy ergodic theory for amenable groups to establish identification, and combine geometric deep learning with scalable Bayesian inference for estimation. We recover i.i.d. causal models and do-calculus when the data is a sequence and the symmetry is permutation equivariance, and find novel types of causal models when we use alternate structures and symmetries. As an example, we construct a causal model that satisfies the symmetries of DNA. This GCM enables new estimators for the effects of genetic variation, combining deep functional genomics models to describe outcomes and DNA language models to describe propensities. We illustrate on semisynthetic data.