Department of Decision Sciences, Bocconi Institute for Data Science and Analytics, Bocconi University, Milan Abstract Accurate tuning of hyperparameters is crucial to ensure that models can generalise effectively across different settings. We construct a variational approximation to a hierarchical Bayes procedure, and derive upper bounds for the contraction rate of the variational posterior in an abstract setting. The theory is applied to various Gaussian process priors and variational classes, resulting in minimax optimal rates. Our theoretical results are accompanied with numerical analysis both on synthetic and real world data sets. Keywords: variational inference, Bayesian model selection, Gaussian processes, nonparametric regression, adaptation, posterior contraction rates 1 Introduction A core challenge in Bayesian statistics is scalability, i.e. the computation of the posterior for large sample sizes. Variational Bayes approximation is a standard approach to speed up inference. Variational posteriors are random probability measures that minimise the Kullback-Leibler divergence between a suitable class of distributions and the otherwise hard to compute posterior. Typically, the variational class of distributions over which the optimisation takes place does not contain the original posterior, hence the variational procedure can be viewed as a projection onto this class. The projected variational distribution then approximates the posterior. During the approximation procedure one inevitably loses information and hence it is important to characterize the accuracy of the approach. Despite the wide use of variational approximations, their theoretical underpinning started to emerge only recently, see for instance Alquier and Ridgway (2020); Yang et al. (2020); Zhang and Gao (2020a); Ray and Szab o (2022). In a Bayesian procedure, the choice of prior reflects the presumed properties of the unknown parameter. In comparison to regular parametric models, where in view of the Bernstein-von Mises theorem the posterior is asymptotically normal, the prior plays a crucial role in the asymptotic behaviour of the posterior. In fact, the large-sample behaviour of the posterior typically depends intricately on the choice of prior hyperparam-eters, so it is vital that these are tuned correctly. The two classical approaches are hierarchical and empirical Bayes methods.

Complex systems with intricate causal dependencies challenge accurate prediction. Effective modeling requires precise physical process representation, integration of interdependent factors, and incorporation of multi-resolution observational data. These systems manifest in both static scenarios with instantaneous causal chains and temporal scenarios with evolving dynamics, complicating modeling efforts. Current methods struggle to simultaneously handle varying resolutions, capture physical relationships, model causal dependencies, and incorporate temporal dynamics, especially with inconsistently sampled data from diverse sources. We introduce Temporal-SVGDM: Score-based Variational Graphical Diffusion Model for Multi-resolution observations. Our framework constructs individual SDEs for each variable at its native resolution, then couples these SDEs through a causal score mechanism where parent nodes inform child nodes' evolution. This enables unified modeling of both immediate causal effects in static scenarios and evolving dependencies in temporal scenarios. In temporal models, state representations are processed through a sequence prediction model to predict future states based on historical patterns and causal relationships. Experiments on real-world datasets demonstrate improved prediction accuracy and causal understanding compared to existing methods, with robust performance under varying levels of background knowledge. Our model exhibits graceful degradation across different disaster types, successfully handling both static earthquake scenarios and temporal hurricane and wildfire scenarios, while maintaining superior performance even with limited data.

-- Certifying safety in dynamical systems is crucial, but barrier certificates -- widely used to verify that system trajectories remain within a safe region -- typically require explicit system models. When dynamics are unknown, data-driven methods can be used instead, yet obtaining a valid certificate requires rigorous uncertainty quantification. For this purpose, existing methods usually rely on full-state measurements, limiting their applicability. This paper proposes a novel approach for synthesizing barrier certificates for unknown systems with latent states and polynomial dynamics. A Bayesian framework is employed, where a prior in state-space representation is updated using input-output data via a targeted marginal Metropolis-Hastings sampler . The resulting samples are used to construct a candidate barrier certificate through a sum-of-squares program. It is shown that if the candidate satisfies the required conditions on a test set of additional samples, it is also valid for the true, unknown system with high probability. The approach and its probabilistic guarantees are illustrated through a numerical simulation. Ensuring the safety of dynamical systems is a critical concern in applications such as human-robot interaction, autonomous driving, and medical devices, where failures can lead to severe consequences. In such scenarios, safety constraints typically mandate that the system state remains within a predefined allowable region. Barrier certificates [1] provide a systematic framework for verifying safety by establishing mathematical conditions that guarantee that system trajectories remain within these regions.

--In this work, we investigate causal learning of independent causal mechanisms from a Bayesian perspective. Confirming previous claims from the literature, we show in a didactically accessible manner that unlabeled data (i.e., cause realizations) do not improve the estimation of the parameters defining the mechanism. Furthermore, we observe the importance of choosing an appropriate prior for the cause and mechanism parameters, respectively. Specifically, we show that a factorized prior results in a factorized posterior, which resonates with Janz-ing and Sch olkopf's definition of independent causal mechanisms via the Kolmogorov complexity of the involved distributions and with the concept of parameter independence of Heckerman et al. Impact Statement --Learning the effect from a given cause is an important problem in many engineering disciplines, specifically in the field of surrogate modeling, which aims to reduce the computational cost of numerical simulations. Causal learning, however, cannot make use of unlabeled data - i.e., cause realizations - if the mechanism that produces the effect is independent from the cause. In this work, we recover this well-known fact from a Bayesian perspective.

This paper presents the Greedy Min-Cut Bayesian Consensus (GMCBC) algorithm for the structural fusion of Bayesian Networks (BNs). The method is designed to preserve essential dependencies while controlling network complexity. It addresses the limitations of traditional fusion approaches, which often lead to excessively complex models that are impractical for inference, reasoning, or real-world applications. As the number and size of input networks increase, this issue becomes even more pronounced. GMCBC integrates principles from flow network theory into BN fusion, adapting the Backward Equivalence Search (BES) phase of the Greedy Equivalence Search (GES) algorithm and applying the Ford-Fulkerson algorithm for minimum cut analysis. This approach removes non-essential edges, ensuring that the fused network retains key dependencies while minimizing unnecessary complexity. Experimental results on synthetic Bayesian Networks demonstrate that GMCBC achieves near-optimal network structures. In federated learning simulations, GMCBC produces a consensus network that improves structural accuracy and dependency preservation compared to the average of the input networks, resulting in a structure that better captures the real underlying (in)dependence relationships. This consensus network also maintains a similar size to the original networks, unlike unrestricted fusion methods, where network size grows exponentially.

Heart disease remains a leading cause of mortality and morbidity worldwide, necessitating the development of accurate and reliable predictive models to facilitate early detection and intervention. While state of the art work has focused on various machine learning approaches for predicting heart disease, but they could not able to achieve remarkable accuracy. In response to this need, we applied nine machine learning algorithms XGBoost, logistic regression, decision tree, random forest, k-nearest neighbors (KNN), support vector machine (SVM), gaussian na\"ive bayes (NB gaussian), adaptive boosting, and linear regression to predict heart disease based on a range of physiological indicators. Our approach involved feature selection techniques to identify the most relevant predictors, aimed at refining the models to enhance both performance and interpretability. The models were trained, incorporating processes such as grid search hyperparameter tuning, and cross-validation to minimize overfitting. Additionally, we have developed a novel voting system with feature selection techniques to advance heart disease classification. Furthermore, we have evaluated the models using key performance metrics including accuracy, precision, recall, F1-score, and the area under the receiver operating characteristic curve (ROC AUC). Among the models, XGBoost demonstrated exceptional performance, achieving 99% accuracy, precision, F1-Score, 98% recall, and 100% ROC AUC. This study offers a promising approach to early heart disease diagnosis and preventive healthcare.

Predictive coding (PC) is an influential theory of information processing in the brain, providing a biologically plausible alternative to backpropagation. It is motivated in terms of Bayesian inference, as hidden states and parameters are optimised via gradient descent on variational free energy. However, implementations of PC rely on maximum \textit{a posteriori} (MAP) estimates of hidden states and maximum likelihood (ML) estimates of parameters, limiting their ability to quantify epistemic uncertainty. In this work, we investigate a Bayesian extension to PC that estimates a posterior distribution over network parameters. This approach, termed Bayesian Predictive coding (BPC), preserves the locality of PC and results in closed-form Hebbian weight updates. Compared to PC, our BPC algorithm converges in fewer epochs in the full-batch setting and remains competitive in the mini-batch setting. Additionally, we demonstrate that BPC offers uncertainty quantification comparable to existing methods in Bayesian deep learning, while also improving convergence properties. Together, these results suggest that BPC provides a biologically plausible method for Bayesian learning in the brain, as well as an attractive approach to uncertainty quantification in deep learning.

Real-world event sequences are often generated by different temporal point processes (TPPs) and thus have clustering structures. Nonetheless, in the modeling and prediction of event sequences, most existing TPPs ignore the inherent clustering structures of the event sequences, leading to the models with unsatisfactory interpretability. In this study, we learn structure-enhanced TPPs with the help of Gromov-Wasserstein (GW) regularization, which imposes clustering structures on the sequence-level embeddings of the TPPs in the maximum likelihood estimation framework.In the training phase, the proposed method leverages a nonparametric TPP kernel to regularize the similarity matrix derived based on the sequence embeddings. In large-scale applications, we sample the kernel matrix and implement the regularization as a Gromov-Wasserstein (GW) discrepancy term, which achieves a trade-off between regularity and computational efficiency.The TPPs learned through this method result in clustered sequence embeddings and demonstrate competitive predictive and clustering performance, significantly improving the model interpretability without compromising prediction accuracy.

Likelihood-free approaches are appealing for performing inference on complex dependence models, either because it is not possible to formulate a likelihood function, or its evaluation is very computationally costly. This is the case for several models available in the multivariate extremes literature, particularly for the most flexible tail models, including those that interpolate between the two key dependence classes of `asymptotic dependence' and `asymptotic independence'. We focus on approaches that leverage neural networks to approximate Bayes estimators. In particular, we explore the properties of neural Bayes estimators for parameter inference for several flexible but computationally expensive models to fit, with a view to aiding their routine implementation. Owing to the absence of likelihood evaluation in the inference procedure, classical information criteria such as the Bayesian information criterion cannot be used to select the most appropriate model. Instead, we propose using neural networks as neural Bayes classifiers for model selection. Our goal is to provide a toolbox for simple, fast fitting and comparison of complex extreme-value dependence models, where the best model is selected for a given data set and its parameters subsequently estimated using neural Bayes estimation. We apply our classifiers and estimators to analyse the pairwise extremal behaviour of changes in horizontal geomagnetic field fluctuations at three different locations.

Land cover classification involves the production of land cover maps, which determine the type of land through remote sensing imagery. Over recent years, such classification is being performed by machine learning classification models, which can give highly accurate predictions on land cover per pixel using large quantities of input training data. However, such models do not currently take account of input measurement uncertainty, which is vital for traceability in metrology. In this work we propose a Bayesian classification framework using generative modelling to take account of input measurement uncertainty. We take the specific case of Bayesian quadratic discriminant analysis, and apply it to land cover datasets from Copernicus Sentinel-2 in 2020 and 2021. We benchmark the performance of the model against more popular classification models used in land cover maps such as random forests and neural networks. We find that such Bayesian models are more trustworthy, in the sense that they are more interpretable, explicitly model the input measurement uncertainty, and maintain predictive performance of class probability outputs across datasets of different years and sizes, whilst also being computationally efficient.