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We commonly assume that data are a homogeneous set of observations when learning the structure of Bayesian networks. However, they often comprise different data sets that are related but not homogeneous because they have been collected in different ways or from different populations. In our previous work (Azzimonti, Corani and Scutari, 2021), we proposed a closed-form Bayesian Hierarchical Dirichlet score for discrete data that pools information across related data sets to learn a single encompassing network structure, while taking into account the differences in their probabilistic structures. In this paper, we provide an analogous solution for learning a Bayesian network from continuous data using mixed-effects models to pool information across the related data sets. We study its structural, parametric, predictive and classification accuracy and we show that it outperforms both conditional Gaussian Bayesian networks (that do not perform any pooling) and classical Gaussian Bayesian networks (that disregard the heterogeneous nature of the data). The improvement is marked for low sample sizes and for unbalanced data sets.

Bayesian additive regression trees (BART) is a non-parametric method to approximate functions. It is a black-box method based on the sum of many trees where priors are used to regularize inference, mainly by restricting trees' learning capacity so that no individual tree is able to explain the data, but rather the sum of trees. We discuss BART in the context of probabilistic programming languages (PPLs), specifically we introduce a BART implementation extending PyMC, a Python library for probabilistic programming. We present a few examples of models that can be built using this probabilistic programming-oriented version of BART, discuss recommendations for sample diagnostics and selection of model hyperparameters, and finally we close with limitations of the current approach and future extensions.

The probabilistic satisfiability of a logical expression is a fundamental concept known as the partition function in statistical physics and field theory, an evaluation of a related graph's Tutte polynomial in mathematics, and the Moore-Shannon network reliability of that graph in engineering. It is the crucial element for decision-making under uncertainty. Not surprisingly, it is provably hard to compute exactly or even to approximate. Many of these applications are concerned only with a subset of problems for which the solutions are monotonic functions. Here we extend the weak- and strong-coupling methods of statistical physics to heterogeneous satisfiability problems and introduce a novel approach to constructing lower and upper bounds on the approximation error for monotonic problems. These bounds combine information from both perturbative analyses to produce bounds that are tight in the sense that they are saturated by some problem instance that is compatible with all the information contained in either approximation.

Employing a forward Markov diffusion chain to gradually map the data to a noise distribution, diffusion probabilistic models learn how to generate the data by inferring a reverse Markov diffusion chain to invert the forward diffusion process. To achieve competitive data generation performance, they demand a long diffusion chain that makes them computationally intensive in not only training but also generation. To significantly improve the computation efficiency, we propose to truncate the forward diffusion chain by abolishing the requirement of diffusing the data to random noise. Consequently, we start the inverse diffusion chain from an implicit generative distribution, rather than random noise, and learn its parameters by matching it to the distribution of the data corrupted by the truncated forward diffusion chain. Experimental results show our truncated diffusion probabilistic models provide consistent improvements over the non-truncated ones in terms of the generation performance and the number of required inverse diffusion steps.

Graph convolutional networks (GCNs) can successfully learn the graph signal representation by graph convolution. The graph convolution depends on the graph filter, which contains the topological dependency of data and propagates data features. However, the estimation errors in the propagation matrix (e.g., the adjacency matrix) can have a significant impact on graph filters and GCNs. In this paper, we study the effect of a probabilistic graph error model on the performance of the GCNs. We prove that the adjacency matrix under the error model is bounded by a function of graph size and error probability. We further analytically specify the upper bound of a normalized adjacency matrix with self-loop added. Finally, we illustrate the error bounds by running experiments on a synthetic dataset and study the sensitivity of a simple GCN under this probabilistic error model on accuracy.

Bayesian neural networks (BNNs) promise improved generalization under covariate shift by providing principled probabilistic representations of epistemic uncertainty. However, weight-based BNNs often struggle with high computational complexity of large-scale architectures and datasets. Node-based BNNs have recently been introduced as scalable alternatives, which induce epistemic uncertainty by multiplying each hidden node with latent random variables, while learning a point-estimate of the weights. In this paper, we interpret these latent noise variables as implicit representations of simple and domain-agnostic data perturbations during training, producing BNNs that perform well under covariate shift due to input corruptions. We observe that the diversity of the implicit corruptions depends on the entropy of the latent variables, and propose a straightforward approach to increase the entropy of these variables during training. We evaluate the method on out-of-distribution image classification benchmarks, and show improved uncertainty estimation of node-based BNNs under covariate shift due to input perturbations. As a side effect, the method also provides robustness against noisy training labels.

Spatial connectivity is an important consideration when modelling infectious disease data across a geographical region. Connectivity can arise for many reasons, including shared characteristics between regions, and human or vector movement. Bayesian hierarchical models include structured random effects to account for spatial connectivity. However, conventional approaches require the spatial structure to be fully defined prior to model fitting. By applying penalised smoothing splines to coordinates, we create 2-dimensional smooth surfaces describing the spatial structure of the data whilst making minimal assumptions about the structure. The result is a non-stationary surface which is setting specific. These surfaces can be incorporated into a hierarchical modelling framework and interpreted similarly to traditional random effects. Through simulation studies we show that the splines can be applied to any continuous connectivity measure, including measures of human movement, and that the models can be extended to explore multiple sources of spatial structure in the data. Using Bayesian inference and simulation, the relative contribution of each spatial structure can be computed and used to generate hypotheses about the drivers of disease. These models were found to perform at least as well as existing modelling frameworks, whilst allowing for future extensions and multiple sources of spatial connectivity.

Covariance matrix estimation is a fundamental statistical task in many applications, but the sample covariance matrix is sub-optimal when the sample size is comparable to or less than the number of features. Such high-dimensional settings are common in modern genomics, where covariance matrix estimation is frequently employed as a method for inferring gene networks. To achieve estimation accuracy in these settings, existing methods typically either assume that the population covariance matrix has some particular structure, for example sparsity, or apply shrinkage to better estimate the population eigenvalues. In this paper, we study a new approach to estimating high-dimensional covariance matrices. We first frame covariance matrix estimation as a compound decision problem. This motivates defining a class of decision rules and using a nonparametric empirical Bayes g-modeling approach to estimate the optimal rule in the class. Simulation results and gene network inference in an RNA-seq experiment in mouse show that our approach is comparable to or can outperform a number of state-of-the-art proposals.

Message passing Graph Neural Networks (GNNs) provide a powerful modeling framework for relational data. However, the expressive power of existing GNNs is upper-bounded by the 1-Weisfeiler-Lehman (1-WL) graph isomorphism test, which means GNNs that are not able to predict node clustering coefficients and shortest path distances, and cannot differentiate between different d-regular graphs. Here we develop a class of message passing GNNs, named Identity-aware Graph Neural Networks (ID-GNNs), with greater expressive power than the 1-WL test. ID-GNN offers a minimal but powerful solution to limitations of existing GNNs. ID-GNN extends existing GNN architectures by inductively considering nodes' identities during message passing. To embed a given node, ID-GNN first extracts the ego network centered at the node, then conducts rounds of heterogeneous message passing, where different sets of parameters are applied to the center node than to other surrounding nodes in the ego network. We further propose a simplified but faster version of ID-GNN that injects node identity information as augmented node features. Altogether, both versions of ID-GNN represent general extensions of message passing GNNs, where experiments show that transforming existing GNNs to ID-GNNs yields on average 40% accuracy improvement on challenging node, edge, and graph property prediction tasks; 3% accuracy improvement on node and graph classification benchmarks; and 15% ROC AUC improvement on real-world link prediction tasks. Additionally, ID-GNNs demonstrate improved or comparable performance over other task-specific graph networks.

Modeling multivariate time series has long been a subject that has attracted researchers from a diverse range of fields including economics, finance, and traffic. A basic assumption behind multivariate time series forecasting is that its variables depend on one another but, upon looking closely, it is fair to say that existing methods fail to fully exploit latent spatial dependencies between pairs of variables. In recent years, meanwhile, graph neural networks (GNNs) have shown high capability in handling relational dependencies. GNNs require well-defined graph structures for information propagation which means they cannot be applied directly for multivariate time series where the dependencies are not known in advance. In this paper, we propose a general graph neural network framework designed specifically for multivariate time series data. Our approach automatically extracts the uni-directed relations among variables through a graph learning module, into which external knowledge like variable attributes can be easily integrated. A novel mix-hop propagation layer and a dilated inception layer are further proposed to capture the spatial and temporal dependencies within the time series. The graph learning, graph convolution, and temporal convolution modules are jointly learned in an end-to-end framework. Experimental results show that our proposed model outperforms the state-of-the-art baseline methods on 3 of 4 benchmark datasets and achieves on-par performance with other approaches on two traffic datasets which provide extra structural information.