Latent Structure And Causality

This book covers selected topics in statistical modeling and causal inference, focusing on two classes of models in particular: (i) latent structure models; (ii) causal graphical models. The book also provides a self-contained development of computational methods and a brief review of statistical theory for inference under these models.After an introduction to the Expectation-Maximization (EM) algorithm on incomplete data, the book provides detailed coverage of a few widely used hidden variable models, including mixture models, hidden Markov models, and stochastic block models. EM and variation EM algorithms are developed for parameter estimation under these models, with a comparison to their Bayesian inference counterparts. We make further extensions of these models to related problems, such as clustering, motif discovery, Kalman filtering, and exchangeable random graphs. Additionally, conditional independence structures are utilized to learn the latent structures of the above models, which can be represented graphically.This notion generalizes naturally to the second part of the book, focusing on graphical models that use graph separation to encode conditional independence in a joint distribution. We cover a variety of graphical models, including undirected graphs, directed acyclic graphs (DAGs), chain graphs, and acyclic directed mixed graphs (ADMGs), and various Markov properties for these models. Recent methods that learn the structure of a graphical model from data are also reviewed. DAGs and Bayesian networks are a particularly important class of mathematical models for causality. The book introduces causal inference with DAGs and structural equation models, and provides a detailed review of recent research work on causal discovery via structure learning of DAGs. Finally, causal bandit problems with sequential interventions are reviewed.