Latent Structure and Causality: Inference from Data (1版)

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【簡介】 Inferring latent structure and causality is crucial for understanding underlying patterns and relationships hidden in the data. This book covers selected models for latent structures and causal networks and inference methods for these models.After an introduction to the EM algorithm on incomplete data, the book provides a detailed coverage of a few widely used latent structure 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 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. Conditional independence structures are utilized to infer the latent structures in the above models, which can be represented graphically. This notion generalizes naturally to the second part on graphical models that use graph separation to encode conditional independence. 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 reviewed and discussed. In particular, DAGs and Bayesian networks are an important class of mathematical models for causality. After an introduction to causal inference with DAGs and structural equation models, we provide a detailed review of recent research on causal discovery via structure learning of graphs. Finally, we briefly introduce the causal bandit problem with sequential intervention.

Graph Theory and Additive Combinatorics: Exploring Structure and Randomness (1版)

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【簡介】 Using the dichotomy of structure and pseudorandomness as a central theme, this accessible text provides a modern introduction to extremal graph theory and additive combinatorics. Readers will explore central results in additive combinatorics-notably the cornerstone theorems of Roth, Szemerédi, Freiman, and Green-Tao-and will gain additional insights into these ideas through graph theoretic perspectives. Topics discussed include the Turán problem, Szemerédi's graph regularity method, pseudorandom graphs, graph limits, graph homomorphism inequalities, Fourier analysis in additive combinatorics, the structure of set addition, and the sum-product problem. Important combinatorial, graph theoretic, analytic, Fourier, algebraic, and geometric methods are highlighted. Students will appreciate the chapter summaries, many figures and exercises, and freely available lecture videos on MIT OpenCourseWare. Meant as an introduction for students and researchers studying combinatorics, theoretical computer science, analysis, probability, and number theory, the text assumes only basic familiarity with abstract algebra, analysis, and linear algebra. 【目錄】