Bayesian hierarchical models are well-suited to analyzing the often noisy data from electroencephalography experiments in cognitive neuroscience: these models provide an intuitive framework to account for structures and correlations in the data, and they allow a straightforward handling of uncertainty. In a typical neurolinguistic experiment, event-related potentials show only very small effect sizes and frequentist approaches to data analysis fail to establish the significance of some of these effects. Here, we present a Bayesian approach to analyzing event-related potentials using as an example data from an experiment which relates word surprisal and neural response. Our model is able to estimate the effect of word surprisal on most components of the event-related potential and provides a richer description of the data. The Bayesian framework also allows easier comparison between estimates based on surprisal values calculated using different language models.

In recent decades, science and engineering have been revolutionized by a momentous growth in the amount of available data. However, despite the unprecedented ease with which data are now collected and stored, labeling data by supplementing each feature with an informative tag remains to be challenging. Illustrative tasks where the labeling process requires expert knowledge or is tedious and time-consuming include labeling X-rays with a diagnosis, protein sequences with a protein type, texts by their topic, tweets by their sentiment, or videos by their genre. In these and numerous other examples, only a few features may be manually labeled due to cost and time constraints. How can we best propagate label information from a small number of expensive labeled features to a vast number of unlabeled ones? This is the question addressed by semi-supervised learning (SSL). This article overviews recent foundational developments on graph-based Bayesian SSL, a probabilistic framework for label propagation using similarities between features. SSL is an active research area and a thorough review of the extant literature is beyond the scope of this article. Our focus will be on topics drawn from our own research that illustrate the wide range of mathematical tools and ideas that underlie the rigorous study of the statistical accuracy and computational efficiency of graph-based Bayesian SSL.

Recent years have witnessed an upsurge of interest in the problem of anomaly detection on attributed networks due to its importance in both research and practice. Although various approaches have been proposed to solve this problem, two major limitations exist: (1) unsupervised approaches usually work much less efficiently due to the lack of supervisory signal, and (2) existing anomaly detection methods only use local contextual information to detect anomalous nodes, e.g., one- or two-hop information, but ignore the global contextual information. Since anomalous nodes differ from normal nodes in structures and attributes, it is intuitive that the distance between anomalous nodes and their neighbors should be larger than that between normal nodes and their neighbors if we remove the edges connecting anomalous and normal nodes. Thus, hop counts based on both global and local contextual information can be served as the indicators of anomaly. Motivated by this intuition, we propose a hop-count based model (HCM) to detect anomalies by modeling both local and global contextual information. To make better use of hop counts for anomaly identification, we propose to use hop counts prediction as a self-supervised task. We design two anomaly scores based on the hop counts prediction via HCM model to identify anomalies. Besides, we employ Bayesian learning to train HCM model for capturing uncertainty in learned parameters and avoiding overfitting. Extensive experiments on real-world attributed networks demonstrate that our proposed model is effective in anomaly detection.

Bayesian structure learning allows one to capture uncertainty over the causal directed acyclic graph (DAG) responsible for generating given data. In this work, we present Tractable Uncertainty for STructure learning (TRUST), a framework for approximate posterior inference that relies on probabilistic circuits as the representation of our posterior belief. In contrast to sample-based posterior approximations, our representation can capture a much richer space of DAGs, while also being able to tractably reason about the uncertainty through a range of useful inference queries. We empirically show how probabilistic circuits can be used as an augmented representation for structure learning methods, leading to improvement in both the quality of inferred structures and posterior uncertainty. Experimental results on conditional query answering further demonstrate the practical utility of the representational capacity of TRUST.

Unsupervised and semi-supervised ML methods such as variational autoencoders (VAE) have become widely adopted across multiple areas of physics, chemistry, and materials sciences due to their capability in disentangling representations and ability to find latent manifolds for classification and regression of complex experimental data. Like other ML problems, VAEs require hyperparameter tuning, e.g., balancing the Kullback Leibler (KL) and reconstruction terms. However, the training process and resulting manifold topology and connectivity depend not only on hyperparameters, but also their evolution during training. Because of the inefficiency of exhaustive search in a high-dimensional hyperparameter space for the expensive to train models, here we explored a latent Bayesian optimization (zBO) approach for the hyperparameter trajectory optimization for the unsupervised and semi-supervised ML and demonstrate for joint-VAE with rotational invariances. We demonstrate an application of this method for finding joint discrete and continuous rotationally invariant representations for MNIST and experimental data of a plasmonic nanoparticles material system. The performance of the proposed approach has been discussed extensively, where it allows for any high dimensional hyperparameter tuning or trajectory optimization of other ML models.

This work discusses how to derive upper bounds for the expected generalisation error of supervised learning algorithms by means of the chaining technique. By developing a general theoretical framework, we establish a duality between generalisation bounds based on the regularity of the loss function, and their chained counterparts, which can be obtained by lifting the regularity assumption from the loss onto its gradient. This allows us to re-derive the chaining mutual information bound from the literature, and to obtain novel chained information-theoretic generalisation bounds, based on the Wasserstein distance and other probability metrics. We show on some toy examples that the chained generalisation bound can be significantly tighter than its standard counterpart, particularly when the distribution of the hypotheses selected by the algorithm is very concentrated.

We propose a message passing algorithm, based on variational Bayesian inference, for low-rank tensor completion with automatic rank determination in the canonical polyadic format when additional side information (SI) is given. The SI comes in the form of low-dimensional subspaces the contain the fiber spans of the tensor (columns, rows, tubes, etc.). We validate the regularization properties induced by SI with extensive numerical experiments on synthetic and real-world data and present the results about tensor recovery and rank determination. The results show that the number of samples required for successful completion is significantly reduced in the presence of SI. We also discuss the origin of a bump in the phase transition curves that exists when the dimensionality of SI is comparable with that of the tensor.

In plant breeding the presence of a genotype by environment (GxE) interaction has a strong impact on cultivation decision making and the introduction of new crop cultivars. The combination of linear and bilinear terms has been shown to be very useful in modelling this type of data. A widely-used approach to identify GxE is the Additive Main Effects and Multiplicative Interaction Effects (AMMI) model. However, as data frequently can be high-dimensional, Markov chain Monte Carlo (MCMC) approaches can be computationally infeasible. In this article, we consider a variational inference approach for such a model. We derive variational approximations for estimating the parameters and we compare the approximations to MCMC using both simulated and real data. The new inferential framework we propose is on average two times faster whilst maintaining the same predictive performance as MCMC.

In Bayesian structure learning, we are interested in inferring a distribution over the directed acyclic graph (DAG) structure of Bayesian networks, from data. Defining such a distribution is very challenging, due to the combinatorially large sample space, and approximations based on MCMC are often required. Recently, a novel class of probabilistic models, called Generative Flow Networks (GFlowNets), have been introduced as a general framework for generative modeling of discrete and composite objects, such as graphs. In this work, we propose to use a GFlowNet as an alternative to MCMC for approximating the posterior distribution over the structure of Bayesian networks, given a dataset of observations. Generating a sample DAG from this approximate distribution is viewed as a sequential decision problem, where the graph is constructed one edge at a time, based on learned transition probabilities. Through evaluation on both simulated and real data, we show that our approach, called DAG-GFlowNet, provides an accurate approximation of the posterior over DAGs, and it compares favorably against other methods based on MCMC or variational inference.

Real-world applications of reinforcement learning for recommendation and experimentation faces a practical challenge: the relative reward of different bandit arms can evolve over the lifetime of the learning agent. To deal with these non-stationary cases, the agent must forget some historical knowledge, as it may no longer be relevant to minimise regret. We present a solution to handling non-stationarity that is suitable for deployment at scale, to provide business operators with automated adaptive optimisation. Our solution aims to provide interpretable learning that can be trusted by humans, whilst responding to non-stationarity to minimise regret. To this end, we develop an adaptive Bayesian learning agent that employs a novel form of dynamic memory. It enables interpretability through statistical hypothesis testing, by targeting a set point of statistical power when comparing rewards and adjusting its memory dynamically to achieve this power. By design, the agent is agnostic to different kinds of non-stationarity. Using numerical simulations, we compare its performance against an existing proposal and show that, under multiple non-stationary scenarios, our agent correctly adapts to real changes in the true rewards. In all bandit solutions, there is an explicit trade-off between learning and achieving maximal performance. Our solution sits on a different point on this trade-off when compared to another similarly robust approach: we prioritise interpretability, which relies on more learning, at the cost of some regret. We describe the architecture of a large-scale deployment of automatic optimisation-as-a-service where our agent achieves interpretability whilst adapting to changing circumstances.