The Latent Dirichlet Allocation (LDA) model is a popular method for creating mixed-membership clusters. Despite having been originally developed for text analysis, LDA has been used for a wide range of other applications. We propose a new formulation for the LDA model which incorporates covariates. In this model, a negative binomial regression is embedded within LDA, enabling straight-forward interpretation of the regression coefficients and the analysis of the quantity of cluster-specific elements in each sampling units (instead of the analysis being focused on modeling the proportion of each cluster, as in Structural Topic Models). We use slice sampling within a Gibbs sampling algorithm to estimate model parameters. We rely on simulations to show how our algorithm is able to successfully retrieve the true parameter values and the ability to make predictions for the abundance matrix using the information given by the covariates. The model is illustrated using real data sets from three different areas: text-mining of Coronavirus articles, analysis of grocery shopping baskets, and ecology of tree species on Barro Colorado Island (Panama). This model allows the identification of mixed-membership clusters in discrete data and provides inference on the relationship between covariates and the abundance of these clusters.

We revisit convergence rates for maximum likelihood estimation (MLE) under finite mixture models. The Wasserstein distance has become a standard loss function for the analysis of parameter estimation in these models, due in part to its ability to circumvent label switching and to accurately characterize the behaviour of fitted mixture components with vanishing weights. However, the Wasserstein metric is only able to capture the worst-case convergence rate among the remaining fitted mixture components. We demonstrate that when the log-likelihood function is penalized to discourage vanishing mixing weights, stronger loss functions can be derived to resolve this shortcoming of the Wasserstein distance. These new loss functions accurately capture the heterogeneity in convergence rates of fitted mixture components, and we use them to sharpen existing pointwise and uniform convergence rates in various classes of mixture models. In particular, these results imply that a subset of the components of the penalized MLE typically converge significantly faster than could have been anticipated from past work. We further show that some of these conclusions extend to the traditional MLE. Our theoretical findings are supported by a simulation study to illustrate these improved convergence rates.

We consider the analysis of probability distributions through their associated covariance operators from reproducing kernel Hilbert spaces. We show that the von Neumann entropy and relative entropy of these operators are intimately related to the usual notions of Shannon entropy and relative entropy, and share many of their properties. They come together with efficient estimation algorithms from various oracles on the probability distributions. We also consider product spaces and show that for tensor product kernels, we can define notions of mutual information and joint entropies, which can then characterize independence perfectly, but only partially conditional independence. We finally show how these new notions of relative entropy lead to new upper-bounds on log partition functions, that can be used together with convex optimization within variational inference methods, providing a new family of probabilistic inference methods.

We analyze the dynamics of a random sequential message passing algorithm for approximate inference with large Gaussian latent variable models in a student-teacher scenario. To model nontrivial dependencies between the latent variables, we assume random covariance matrices drawn from rotation invariant ensembles. Moreover, we consider a model mismatching setting, where the teacher model and the one used by the student may be different. By means of dynamical functional approach, we obtain exact dynamical mean-field equations characterizing the dynamics of the inference algorithm. We also derive a range of model parameters for which the sequential algorithm does not converge. The boundary of this parameter range coincides with the de Almeida Thouless (AT) stability condition of the replica symmetric ansatz for the static probabilistic model.

We introduce a general theory of epistemic random fuzzy sets for reasoning with fuzzy or crisp evidence. This framework generalizes both the Dempster-Shafer theory of belief functions, and possibility theory. Independent epistemic random fuzzy sets are combined by the generalized product-intersection rule, which extends both Dempster's rule for combining belief functions, and the product conjunctive combination of possibility distributions. We introduce Gaussian random fuzzy numbers and their multi-dimensional extensions, Gaussian random fuzzy vectors, as practical models for quantifying uncertainty about scalar or vector quantities. Closed-form expressions for the combination, projection and vacuous extension of Gaussian random fuzzy numbers and vectors are derived.

Bayesian inference is based on the fact that we often don't know the underlying distribution of data, so we need to build a model and then iteratively adjust it as we get more data. In parametric Bayesian inference you start with picking a general form of the probability distribution f(x;θ) defined by parameters θ. A good example of the distribution could be a Normal distribution with two parameters μ and σ 2. The probability of the data under a hypothetical distribution, assuming independent data examples, is: This function is called likelihood function. The parameter θ is itself a random variable, and its probability distribution can be found using Bayes' theorem: Here p(θ) is called posterior distribution, π(θ) is prior distribution and expresses our beliefs about parameter θ before we see any data. The term in the denominator is called evidence and represents probability of data.

Air pollution is one of the leading causes of mortality globally, resulting in millions of deaths each year. Efficient monitoring is important to measure exposure and enforce legal limits. New low-cost sensors can be deployed in greater numbers and in more varied locations, motivating the problem of efficient automated placement. Previous work suggests Bayesian optimisation is an appropriate method, but only considered a satellite data set, with data aggregated over all altitudes. It is ground-level pollution, that humans breathe, which matters most. We improve on those results using hierarchical models and evaluate our models on urban pollution data in London to show that Bayesian optimisation can be successfully applied to the problem.

Anomaly detection is a widely studied task for a broad variety of data types; among them, multiple time series appear frequently in applications, including for example, power grids and traffic networks. Detecting anomalies for multiple time series, however, is a challenging subject, owing to the intricate interdependencies among the constituent series. We hypothesize that anomalies occur in low density regions of a distribution and explore the use of normalizing flows for unsupervised anomaly detection, because of their superior quality in density estimation. Moreover, we propose a novel flow model by imposing a Bayesian network among constituent series. A Bayesian network is a directed acyclic graph (DAG) that models causal relationships; it factorizes the joint probability of the series into the product of easy-to-evaluate conditional probabilities. We call such a graph-augmented normalizing flow approach GANF and propose joint estimation of the DAG with flow parameters. We conduct extensive experiments on real-world datasets and demonstrate the effectiveness of GANF for density estimation, anomaly detection, and identification of time series distribution drift.

Generative models create latent representations, which distil information from big data in order to generate realistic and novel data points. In the long term, these models could be vital in developing accurate world models, as well as learning categorical and continuous features of a dataset in an unsupervised way. Currently, generative models are demonstrating their value in a variety of downstream tasks such as inpainting, super-resolution, and generating continuous exploration spaces for reinforcement learning. Generative Adversarial Networks (GANs) have represented the state of the art (SotA) for some time, however recently OpenAI has published results that make a strong case for a new era of Denoising Diffusion Probabilistic models dominating generative SotA applications. In this article, I shall introduce the theory behind this method and describe the contributions which have enabled this relatively unstudied technique to topple GANs.

Gene expression datasets consist of thousand of genes with relatively small samplesizes (i.e. are large-$p$-small-$n$). Moreover, dependencies of various orders co-exist in the datasets. In the Undirected probabilistic Graphical Model (UGM) framework the Glasso algorithm has been proposed to deal with high dimensional micro-array datasets forcing sparsity. Also, modifications of the default Glasso algorithm are developed to overcome the problem of complex interaction structure. In this work we advocate the use of a simple score-based Hill Climbing algorithm (HC) that learns Gaussian Bayesian Networks (BNs) leaning on Directed Acyclic Graphs (DAGs). We compare HC with Glasso and its modifications in the UGM framework on their capability to reconstruct GRNs from micro-array data belonging to the Escherichia Coli genome. We benefit from the analytical properties of the Joint Probability Density (JPD) function on which both directed and undirected PGMs build to convert DAGs to UGMs. We conclude that dependencies in complex data are learned best by the HC algorithm, presenting them most accurately and efficiently, simultaneously modelling strong local and weaker but significant global connections coexisting in the gene expression dataset. The HC algorithm adapts intrinsically to the complex dependency structure of the dataset, without forcing a specific structure in advance. On the contrary, Glasso and modifications model unnecessary dependencies at the expense of the probabilistic information in the network and of a structural bias in the JPD function that can only be relieved including many parameters.