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.

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.

Explicit latent variable models provide a flexible yet powerful framework for data synthesis, enabling controlled manipulation of generative factors. With latent variables drawn from a tractable probability density function that can be further constrained, these models enable continuous and semantically rich exploration of the output space by navigating their latent spaces. Structured latent representations are typically obtained through the joint minimization of regularization loss functions. In variational information bottleneck models, reconstruction loss and Kullback-Leibler Divergence (KLD) are often linearly combined with an auxiliary Attribute-Regularization (AR) loss. However, balancing KLD and AR turns out to be a very delicate matter. When KLD dominates over AR, generative models tend to lack controllability; when AR dominates over KLD, the stochastic encoder is encouraged to violate the standard normal prior. We explore this trade-off in the context of symbolic music generation with explicit control over continuous musical attributes. We show that existing approaches struggle to jointly minimize both regularization objectives, whereas suitable attribute transformations can help achieve both controllability and regularization of the target latent dimensions.

Bayesian inference, while foundational to probabilistic reasoning, is often hampered by the computational intractability of posterior distributions, particularly through the challenging evidence integral. Conventional approaches like Markov Chain Monte Carlo (MCMC) and Variational Inference (VI) face significant scalability and efficiency limitations. This paper introduces a novel, unifying framework for fast Bayesian updates by leveraging harmonic analysis. We demonstrate that representing the prior and likelihood in a suitable orthogonal basis transforms the Bayesian update rule into a spectral convolution. Specifically, the Fourier coefficients of the posterior are shown to be the normalized convolution of the prior and likelihood coefficients. To achieve computational feasibility, we introduce a spectral truncation scheme, which, for smooth functions, yields an exceptionally accurate finite-dimensional approximation and reduces the update to a circular convolution. This formulation allows us to exploit the Fast Fourier Transform (FFT), resulting in a deterministic algorithm with O(N log N) complexity -- a substantial improvement over the O(N^2) cost of naive methods. We establish rigorous mathematical criteria for the applicability of our method, linking its efficiency to the smoothness and spectral decay of the involved distributions. The presented work offers a paradigm shift, connecting Bayesian computation to signal processing and opening avenues for real-time, sequential inference in a wide class of problems.

Large Language Models (LLMs) have demonstrated remarkable capabilities across numerous tasks, yet principled explanations for their underlying mechanisms and several phenomena, such as scaling laws, hallucinations, and related behaviors, remain elusive. In this work, we revisit the classical relationship between compression and prediction, grounded in Kolmogorov complexity and Shannon information theory, to provide deeper insights into LLM behaviors. By leveraging the Kolmogorov Structure Function and interpreting LLM compression as a two-part coding process, we offer a detailed view of how LLMs acquire and store information across increasing model and data scales -- from pervasive syntactic patterns to progressively rarer knowledge elements. Motivated by this theoretical perspective and natural assumptions inspired by Heap's and Zipf's laws, we introduce a simplified yet representative hierarchical data-generation framework called the Syntax-Knowledge model. Under the Bayesian setting, we show that prediction and compression within this model naturally lead to diverse learning and scaling behaviors observed in LLMs. In particular, our theoretical analysis offers intuitive and principled explanations for both data and model scaling laws, the dynamics of knowledge acquisition during training and fine-tuning, factual knowledge hallucinations in LLMs. The experimental results validate our theoretical predictions.

Ordinal categorical data are routinely encountered in a wide range of practical applications. When the primary goal is to construct a regression model for ordinal outcomes, cumulative link models represent one of the most popular choices to link the cumulative probabilities of the response with a set of covariates through a parsimonious linear predictor, shared across response categories. When the number of observations grows, standard sampling algorithms for Bayesian inference scale poorly, making posterior computation increasingly challenging in large datasets. In this article, we propose three scalable algorithms for approximating the posterior distribution of the regression coefficients in cumulative probit models relying on Variational Bayes and Expectation Propagation. We compare the proposed approaches with inference based on Markov Chain Monte Carlo, demonstrating superior computational performance and remarkable accuracy; finally, we illustrate the utility of the proposed algorithms on a challenging case study to investigate the structure of a criminal network.