Bayesian Deep Discrete Latent Structures

Abstract

Modern scientific datasets are increasingly high-dimensional, structured, and heterogeneous, posing significant challenges for statistical modeling and inference. Discrete latent variable models provide a natural framework for uncovering hidden structure in such data, but classical approaches often fall short in terms of scalability, flexibility, and theoretical robustness. This dissertation develops a new class of models, Bayesian deep discrete latent structures, that extend traditional latent variable frameworks through hierarchical modeling and deep probabilistic architectures.We explore this broad modeling paradigm in two complementary domains. First, we address the problem of analyzing multiplex network data, where multiple network observations are available over a common set of nodes. To uncover both shared and individual-specific structure, we introduce a Bayesian hierarchical model that captures population-level node hierarchies while performing multi-resolution clustering of individual networks. The model is supported by rigorous theoretical results, including identifiability and posterior consistency, and is equipped with efficient inference algorithms. Performance is demonstrated through simulations and real-world application to brain connectome data.Second, we consider the analysis of high-dimensional categorical data with associated covariates, a setting where standard latent class regression models often fail due to the curse of dimensionality. We propose a deep latent class modeling framework that improves flexibility and robustness by incorporating multi-layer probabilistic mechanisms. The model retains key theoretical properties and enjoys a Bayes oracle clustering guarantee, making it particularly suited for high-dimensional applications. We illustrate its effectiveness through simulated studies and an application to ecological species distribution modeling.Collectively, the methods developed in this dissertation advance the theoretical and computational foundations of Bayesian discrete latent structure modeling, offering general-purpose tools for extracting meaningful patterns from complex data.