Bayesian optimal experimental design is a principled framework for conducting experiments that leverages Bayesian inference to quantify how much information one can expect to gain from selecting a certain design. However, accurate Bayesian inference relies on the assumption that one's statistical model of the data-generating process is correctly specified. If this assumption is violated, Bayesian methods can lead to poor inference and estimates of information gain. Generalised Bayesian (or Gibbs) inference is a more robust probabilistic inference framework that replaces the likelihood in the Bayesian update by a suitable loss function. In this work, we present Generalised Bayesian Optimal Experimental Design (GBOED), an extension of Gibbs inference to the experimental design setting which achieves robustness in both design and inference. Using an extended information-theoretic framework, we derive a new acquisition function, the Gibbs expected information gain (Gibbs EIG). Our empirical results demonstrate that GBOED enhances robustness to outliers and incorrect assumptions about the outcome noise distribution.

Hierarchical Bayesian models based on Gaussian processes a re considered useful for describing complex nonlinear statistical dependen cies among variables in real-world data. However, effective Monte Carlo algorithm s for inference with these models have not yet been established, except for sever al simple cases. In this study, we show that, compared with the slow inference ac hieved with existing program libraries, the performance of Riemannian-m anifold Hamiltonian Monte Carlo (RMHMC) can be drastically improved by optimisi ng the computation order according to the model structure and dynamical ly programming the eigendecomposition. This improvement cannot be achieved w hen using an existing library based on a naive automatic differentiator. W e nu merically demonstrate that RMHMC effectively samples from the posterior, allowin g the calculation of model evidence, in a Bayesian logistic regression on simula ted data and in the estimation of propensity functions for the American nation al medical expenditure data using several Bayesian multiple-kernel models. These results lay a foundation for implementing effective Monte Carlo algorithms for analysing real-world data with Gaussian processes, and highlight the need to deve lop a customisable library set that allows users to incorporate dynamically pr ogrammed objects and finely optimises the mode of automatic differentiation depe nding on the model structure.

Uncertainty quantification (UQ) for adaptively collected data, such as that coming from adaptive experiments, bandits, or reinforcement learning, is necessary for critical elements of data collection such as ensuring safety and conducting after-study inference. The data's adaptivity creates significant challenges for frequentist UQ, yet Bayesian UQ remains the same as if the data were independent and identically distributed (i.i.d.), making it an appealing and commonly used approach. Bayesian UQ requires the (correct) specification of a prior distribution while frequentist UQ does not, but for i.i.d. data the celebrated Bernstein-von Mises theorem shows that as the sample size grows, the prior 'washes out' and Bayesian UQ becomes frequentist-valid, implying that the choice of prior need not be a major impediment to Bayesian UQ as it makes no difference asymptotically. This paper for the first time extends the Bernstein-von Mises theorem to adaptively collected data, proving asymptotic equivalence between Bayesian UQ and Wald-type frequentist UQ in this challenging setting. Our result showing this asymptotic agreement does not require the standard stability condition required by works studying validity of Wald-type frequentist UQ; in cases where stability is satisfied, our results combined with these prior studies of frequentist UQ imply frequentist validity of Bayesian UQ. Counterintuitively however, they also provide a negative result that Bayesian UQ is not asymptotically frequentist valid when stability fails, despite the fact that the prior washes out and Bayesian UQ asymptotically matches standard Wald-type frequentist UQ. We empirically validate our theory (positive and negative) via a range of simulations.

Supply chain disruptions and volatile demand pose significant challenges to the UK automotive industry, which relies heavily on Just-In-Time (JIT) manufacturing. While qualitative studies highlight the potential of integrating Artificial Intelligence (AI) with traditional optimization, a formal, quantitative demonstration of this synergy is lacking. This paper introduces a novel stochastic learning-optimization framework that integrates Bayesian inference with inventory optimization for supply chain management (SCM). We model a two-echelon inventory system subject to stochastic demand and supply disruptions, comparing a traditional static optimization policy against an adaptive policy where Bayesian learning continuously updates parameter estimates to inform stochastic optimization. Our simulations over 365 periods across three operational scenarios demonstrate that the integrated approach achieves 7.4\% cost reduction in stable environments and 5.7\% improvement during supply disruptions, while revealing important limitations during sudden demand shocks due to the inherent conservatism of Bayesian updating. This work provides mathematical validation for practitioner observations and establishes a formal framework for understanding AI-driven supply chain resilience, while identifying critical boundary conditions for successful implementation.

This paper presents a comprehensive analysis of hyperparameter estimation within the empirical Bayes framework (EBF) for sparse learning. By studying the influence of hyperpriors on the solution of EBF, we establish a theoretical connection between the choice of the hyperprior and the sparsity as well as the local optimality of the resulting solutions. We show that some strictly increasing hyperpriors, such as half-Laplace and half-generalized Gaussian with the power in $(0,1)$, effectively promote sparsity and improve solution stability with respect to measurement noise. Based on this analysis, we adopt a proximal alternating linearized minimization (PALM) algorithm with convergence guaranties for both convex and concave hyperpriors. Extensive numerical tests on two-dimensional image deblurring problems demonstrate that introducing appropriate hyperpriors significantly promotes the sparsity of the solution and enhances restoration accuracy. Furthermore, we illustrate the influence of the noise level and the ill-posedness of inverse problems to EBF solutions.

Learning robot policies that capture multimodality in the training data has been a long-standing open challenge for behavior cloning. Recent approaches tackle the problem by modeling the conditional action distribution with generative models. One of these approaches is Diffusion Policy, which relies on a diffusion model to denoise random points into robot action trajectories. While achieving state-of-the-art performance, it has two main drawbacks that may lead the robot out of the data distribution during policy execution. First, the stochasticity of the denoising process can highly impact on the quality of generated trajectory of actions. Second, being a supervised learning approach, it can learn data outliers from the dataset used for training. Recent work focuses on mitigating these limitations by combining Diffusion Policy either with large-scale training or with classical behavior cloning algorithms. Instead, we propose KDPE, a Kernel Density Estimation-based strategy that filters out potentially harmful trajectories output of Diffusion Policy while keeping a low test-time computational overhead. For Kernel Density Estimation, we propose a manifold-aware kernel to model a probability density function for actions composed of end-effector Cartesian position, orientation, and gripper state. KDPE overall achieves better performance than Diffusion Policy on simulated single-arm tasks and real robot experiments. Additional material and code are available on our project page at https://hsp-iit.github.io/KDPE/.

In the era of smart manufacturing, predictive maintenance (PdM) plays a pivotal role in improving equipment reliability and reducing operating costs. In this paper, we propose a novel Markov Decision Process (MDP) framework that integrates advanced soft computing techniques - Fourier Neural Operator (FNO), Denoising Autoencoder (DAE), Graph Neural Network (GNN), and Proximal Policy Optimisation (PPO) - to address the multidimensional challenges of predictive maintenance in complex manufacturing systems. Specifically, the proposed framework innovatively combines the powerful frequency-domain representation capability of FNOs to capture high-dimensional temporal patterns; DAEs to achieve robust, noise-resistant latent state embedding from complex non-Gaussian sensor data; and GNNs to accurately represent inter-device dependencies for coordinated system-wide maintenance decisions. Furthermore, by exploiting PPO, the framework ensures stable and efficient optimisation of long-term maintenance strategies to effectively handle uncertainty and non-stationary dynamics. Experimental validation demonstrates that the approach significantly outperforms multiple deep learning baseline models with up to 13% cost reduction, as well as strong convergence and inter-module synergy. The framework has considerable industrial potential to effectively reduce downtime and operating expenses through data-driven strategies.

The complexity of human cognition has meant that psychology makes more use of theory and conceptual models than perhaps any other biomedical field. To enable precise quantitative study of the full breadth of phenomena in psychological and psychiatric medicine as well as cognitive aspects of AI safety, there is a need for a mathematical formulation which is both mathematically precise and equally accessible to experts from numerous fields. In this paper we formalize human psychodynamics via the diagrammatic framework of process theory, describe its key properties, and explain the links between a diagrammatic representation and central concepts in analysis of cognitive processes in contexts such as psychotherapy, neurotechnology, AI alignment, AI agent representation of individuals in autonomous negotiations, developing human-like AI systems, and other aspects of AI safety.

Electricity price forecasting has become a critical tool for decision-making in energy markets, particularly as the increasing penetration of renewable energy introduces greater volatility and uncertainty. Historically, research in this field has been dominated by point forecasting methods, which provide single-value predictions but fail to quantify uncertainty. However, as power markets evolve due to renewable integration, smart grids, and regulatory changes, the need for probabilistic forecasting has become more pronounced, offering a more comprehensive approach to risk assessment and market participation. This paper presents a review of probabilistic forecasting methods, tracing their evolution from Bayesian and distribution based approaches, through quantile regression techniques, to recent developments in conformal prediction. Particular emphasis is placed on advancements in probabilistic forecasting, including validity-focused methods which address key limitations in uncertainty estimation. Additionally, this review extends beyond the Day-Ahead Market to include the Intra-Day and Balancing Markets, where forecasting challenges are intensified by higher temporal granularity and real-time operational constraints. We examine state of the art methodologies, key evaluation metrics, and ongoing challenges, such as forecast validity, model selection, and the absence of standardised benchmarks, providing researchers and practitioners with a comprehensive and timely resource for navigating the complexities of modern electricity markets.

ABSTRACT Data-driven astrophysics currently relies on the detection and characterisation of correlations between objects' properties, which are then used to test physical theories that make predictions for them. This process fails to utilise information in the data that forms a crucial part of the theories' predictions, namely which variables are directly correlated (as opposed to accidentally correlated through others), the directions of these determinations, and the presence or absence of confounders that correlate variables in the dataset but are themselves absent from it. We propose to recover this information through causal discovery, a well-developed methodology for inferring the causal structure of datasets that is however almost entirely unknown to astrophysics. INTRODUCTION Understanding the physical processes that shape galaxies is a central goal of astrophysics. Empirical progress has traditionally relied on identifying correlations between observed properties, which can then be interpreted in light of theoretical models for galaxy formation and used to constrain them. The advent of large surveys and powerful machine learning techniques has greatly expanded our ability to find such statistical associations, uncovering intricate patterns across high-dimensional parameter spaces. However, correlation alone cannot determine causal influences among variables: which properties are actually responsible for determining others, in what direction this influence goes, and whether there exist confounding variables that are not included in the dataset but influence those that are.