Machine generated contents note: 1.Introduction -- 1.1.Graphical Models -- 1.2.Outline of the Book -- 1.2.1.Discrete Graphical Models and Their Parameterization -- 1.2.2.Binary vs Non-binary Variables -- 2.Conditional Independence and Cross-product Ratios -- 2.1.Notation and Terminology -- 2.1.1.Cross-classified Tables -- 2.2.Conditional Independence -- 2.3.Establishing Independence Relationships -- 3.Mobius Inversion -- 3.1.Preliminaries -- 3.1.1.Notation and Terminology -- 3.1.2.The Zeta and the Mobius Matrices -- 3.2.The Mobius Inversion Formula -- 3.2.1.Two Basic Lemmas -- 3.3.Mobius Inversion and Partially Ordered Sets -- 4.Undirected Graph Models -- 4.1.Graphs -- 4.2.Markov Properties for Undirected Graphs -- 4.3.The Log-linear Parameterization -- 4.4.Hierarchical Log-linear Models -- 4.5.Log-linear Graphical Models -- 4.6.Data, Estimation and Testing -- 4.7.Graph Decomposition and Decomposable Graphs -- 4.8.Local Computation Properties -- 4.9.Models for Decomposable Graphs -- 4.10.Log-linear Models and the Exponential Family -- 4.10.1.Basic Facts on the Theory of the Exponential Family -- 4.10.2.The Cross-classified Bernoulli Distribution -- 4.10.3.Exponential Family Representations of the Saturated Model -- 4.10.4.Exponential Family Representation of Hierarchical Log-linear Models -- 4.11.Modular Structure of the Asymptotic Variance of ML Estimates -- 4.11.1.The Variance Function and the Asymptotic Variance of ML Estimates -- 4.11.2.Variances in the Saturated Model -- 4.11.3.Variances in Hierarchical Log-linear Models -- 4.11.4.Decompositions and Decomposable Models -- 5.Bidirected Graph Models -- 5.1.Bidirected Graphs -- 5.2.Markov Properties for Bidirected Graphs -- 5.3.The Log-mean Linear Parameterization -- 5.4.Log-mean Linear Graphical Models -- 5.5.Example: Symptoms in Psychiatric Patients -- 5.6.Parsimonious Graphical Modeling -- 6.Directed Acyclic and Regression Graph Models -- 6.1.Directed Acyclic Graphs -- 6.2.Markov Properties for Directed Acyclic Graphs -- 6.3.Regression Graphs -- 6.4.Markov Properties for Regression Graphs -- 6.5.On the Interpretation of Models defined by Regression Graphs -- 6.6.The Log-hybrid Linear Parameterization -- 6.7.Log-hybrid Linear Graphical Models -- 6.8.Inference in Regression Graph Models.
Summary:
For advanced students of network data science, this compact account covers both well-established methodology and the theory of models recently introduced in the graphical model literature. It focuses on the discrete case where all variables involved are categorical and, in this context, it achieves a unified presentation of classical and recent results. --
