Contents. Acknowledgments. List of Figures. List of Algorithms

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1 Contents Acknowledgments xxiii List of Figures xxv List of Algorithms xxxi List of Boxes xxxiii 1 Introduction Motivation Structured Probabilistic Models Probabilistic Graphical Models Representation, Inference, Learning Overview and Roadmap Overview of Chapters Reader s Guide Connection to Other Disciplines Historical Notes 12 2 Foundations Probability Theory Probability Distributions Basic Concepts in Probability Random Variables and Joint Distributions Independence and Conditional Independence Querying a Distribution Continuous Spaces Expectation and Variance Graphs Nodes and Edges Subgraphs Paths and Trails 36

2 x CONTENTS Cycles and Loops Relevant Literature Exercises 39 I Representation 43 3 The Bayesian Network Representation Exploiting Independence Properties Independent Random Variables The Conditional Parameterization The Naive Bayes Model Bayesian Networks The Student Example Revisited Basic Independencies in Bayesian Networks Graphs and Distributions Independencies in Graphs D-separation Soundness and Completeness An Algorithm for d-separation I-Equivalence From Distributions to Graphs Minimal I-Maps Perfect Maps Finding Perfect Maps Summary Relevant Literature Exercises 96 4 Undirected Graphical Models The Misconception Example Parameterization Factors Gibbs Distributions and Markov Networks Reduced Markov Networks Markov Network Independencies Basic Independencies Independencies Revisited From Distributions to Graphs Parameterization Revisited Finer-Grained Parameterization Overparameterization Bayesian Networks and Markov Networks From Bayesian Networks to Markov Networks From Markov Networks to Bayesian Networks 138

3 CONTENTS xi Chordal Graphs Partially Directed Models Conditional Random Fields Chain Graph Models Summary and Discussion Relevant Literature Exercises Local Probabilistic Models Tabular CPDs Deterministic CPDs Representation Independencies Context-Specific CPDs Representation Independencies Independence of Causal Influence The Noisy-Or Model Generalized Linear Models The General Formulation Independencies Continuous Variables Hybrid Models Conditional Bayesian Networks Summary Relevant Literature Exercises Template-Based Representations Introduction Temporal Models Basic Assumptions Dynamic Bayesian Networks State-Observation Models Template Variables and Template Factors Directed Probabilistic Models for Object-Relational Domains Plate Models Probabilistic Relational Models Undirected Representation Structural Uncertainty Relational Uncertainty Object Uncertainty Summary Relevant Literature Exercises 243

4 xii CONTENTS 7 Gaussian Network Models Multivariate Gaussians Basic Parameterization Operations on Gaussians Independencies in Gaussians Gaussian Bayesian Networks Gaussian Markov Random Fields Summary Relevant Literature Exercises The Exponential Family Introduction Exponential Families Linear Exponential Families Factored Exponential Families Product Distributions Bayesian Networks Entropy and Relative Entropy Entropy Relative Entropy Projections Comparison M-Projections I-Projections Summary Relevant Literature Exercises 283 II Inference Variable Elimination Analysis of Complexity Analysis of Exact Inference Analysis of Approximate Inference Variable Elimination: The Basic Ideas Variable Elimination Basic Elimination Dealing with Evidence Complexity and Graph Structure: Variable Elimination Simple Analysis Graph-Theoretic Analysis Finding Elimination Orderings Conditioning 315

5 CONTENTS xiii The Conditioning Algorithm Conditioning and Variable Elimination Graph-Theoretic Analysis Improved Conditioning Inference with Structured CPDs Independence of Causal Influence Context-Specific Independence Discussion Summary and Discussion Relevant Literature Exercises Clique Trees Variable Elimination and Clique Trees Cluster Graphs Clique Trees Message Passing: Sum Product Variable Elimination in a Clique Tree Clique Tree Calibration A Calibrated Clique Tree as a Distribution Message Passing: Belief Update Message Passing with Division Equivalence of Sum-Product and Belief Update Messages Answering Queries Constructing a Clique Tree Clique Trees from Variable Elimination Clique Trees from Chordal Graphs Summary Relevant Literature Exercises Inference as Optimization Introduction Exact Inference Revisited The Energy Functional Optimizing the Energy Functional Exact Inference as Optimization Fixed-Point Characterization Inference as Optimization Propagation-Based Approximation A Simple Example Cluster-Graph Belief Propagation Properties of Cluster-Graph Belief Propagation Analyzing Convergence Constructing Cluster Graphs 404

6 xiv CONTENTS Variational Analysis Other Entropy Approximations Discussion Propagation with Approximate Messages Factorized Messages Approximate Message Computation Inference with Approximate Messages Expectation Propagation Variational Analysis Discussion Structured Variational Approximations The Mean Field Approximation Structured Approximations Local Variational Methods Summary and Discussion Relevant Literature Exercises Particle-Based Approximate Inference Forward Sampling Sampling from a Bayesian Network Analysis of Error Conditional Probability Queries Likelihood Weighting and Importance Sampling Likelihood Weighting: Intuition Importance Sampling Importance Sampling for Bayesian Networks Importance Sampling Revisited Markov Chain Monte Carlo Methods Gibbs Sampling Algorithm Markov Chains Gibbs Sampling Revisited A Broader Class of Markov Chains Using a Markov Chain Collapsed Particles Collapsed Likelihood Weighting Collapsed MCMC Deterministic Search Methods Summary Relevant Literature Exercises MAP Inference Overview Computational Complexity 551

7 CONTENTS xv Overview of Solution Methods Variable Elimination for (Marginal) MAP Max-Product Variable Elimination Finding the Most Probable Assignment Variable Elimination for Marginal MAP Max-Product in Clique Trees Computing Max-Marginals Message Passing as Reparameterization Decoding Max-Marginals Max-Product Belief Propagation in Loopy Cluster Graphs Standard Max-Product Message Passing Max-Product BP with Counting Numbers Discussion MAP as a Linear Optimization Problem The Integer Program Formulation Linear Programming Relaxation Low-Temperature Limits Using Graph Cuts for MAP Inference Using Graph Cuts Nonbinary Variables Local Search Algorithms Summary Relevant Literature Exercises Inference in Hybrid Networks Introduction Challenges Discretization Overview Variable Elimination in Gaussian Networks Canonical Forms Sum-Product Algorithms Gaussian Belief Propagation Hybrid Networks The Difficulties Factor Operations for Hybrid Gaussian Networks EP for CLG Networks An Exact CLG Algorithm Nonlinear Dependencies Linearization Expectation Propagation with Gaussian Approximation Particle-Based Approximation Methods Sampling in Continuous Spaces Forward Sampling in Bayesian Networks 643

8 xvi CONTENTS MCMC Methods Collapsed Particles Nonparametric Message Passing Summary and Discussion Relevant Literature Exercises Inference in Temporal Models Inference Tasks Exact Inference Filtering in State-Observation Models Filtering as Clique Tree Propagation Clique Tree Inference in DBNs Entanglement Approximate Inference Key Ideas Factored Belief State Methods Particle Filtering Deterministic Search Techniques Hybrid DBNs Continuous Models Hybrid Models Summary Relevant Literature Exercises 692 III Learning Learning Graphical Models: Overview Motivation Goals of Learning Density Estimation Specific Prediction Tasks Knowledge Discovery Learning as Optimization Empirical Risk and Overfitting Discriminative versus Generative Training Learning Tasks Model Constraints Data Observability Taxonomy of Learning Tasks Relevant Literature Parameter Estimation Maximum Likelihood Estimation 717

9 CONTENTS xvii The Thumbtack Example The Maximum Likelihood Principle MLE for Bayesian Networks A Simple Example Global Likelihood Decomposition Table-CPDs Gaussian Bayesian Networks Maximum Likelihood Estimation as M-Projection Bayesian Parameter Estimation The Thumbtack Example Revisited Priors and Posteriors Bayesian Parameter Estimation in Bayesian Networks Parameter Independence and Global Decomposition Local Decomposition Priors for Bayesian Network Learning MAP Estimation Learning Models with Shared Parameters Global Parameter Sharing Local Parameter Sharing Bayesian Inference with Shared Parameters Hierarchical Priors Generalization Analysis Asymptotic Analysis PAC-Bounds Summary Relevant Literature Exercises Structure Learning in Bayesian Networks Introduction Problem Definition Overview of Methods Constraint-Based Approaches General Framework Independence Tests Structure Scores Likelihood Scores Bayesian Score Marginal Likelihood for a Single Variable Bayesian Score for Bayesian Networks Understanding the Bayesian Score Priors Score Equivalence Structure Search Learning Tree-Structured Networks 808

10 xviii CONTENTS Known Order General Graphs Learning with Equivalence Classes Bayesian Model Averaging Basic Theory Model Averaging Given an Order The General Case Learning Models with Additional Structure Learning with Local Structure Learning Template Models Summary and Discussion Relevant Literature Exercises Partially Observed Data Foundations Likelihood of Data and Observation Models Decoupling of Observation Mechanism The Likelihood Function Identifiability Parameter Estimation Gradient Ascent Expectation Maximization (EM) Comparison: Gradient Ascent versus EM Approximate Inference Bayesian Learning with Incomplete Data Overview MCMC Sampling Variational Bayesian Learning Structure Learning Scoring Structures Structure Search Structural EM Learning Models with Hidden Variables Information Content of Hidden Variables Determining the Cardinality Introducing Hidden Variables Summary Relevant Literature Exercises Learning Undirected Models Overview The Likelihood Function An Example 944

11 CONTENTS xix Form of the Likelihood Function Properties of the Likelihood Function Maximum (Conditional) Likelihood Parameter Estimation Maximum Likelihood Estimation Conditionally Trained Models Learning with Missing Data Maximum Entropy and Maximum Likelihood Parameter Priors and Regularization Local Priors Global Priors Learning with Approximate Inference Belief Propagation MAP-Based Learning Alternative Objectives Pseudolikelihood and Its Generalizations Contrastive Optimization Criteria Structure Learning Structure Learning Using Independence Tests Score-Based Learning: Hypothesis Spaces Objective Functions Optimization Task Evaluating Changes to the Model Summary Relevant Literature Exercises 1001 IV Actions and Decisions Causality Motivation and Overview Conditioning and Intervention Correlation and Causation Causal Models Structural Causal Identifiability Query Simplification Rules Iterated Query Simplification Mechanisms and Response Variables Partial Identifiability in Functional Causal Models Counterfactual Queries Twinned Networks Bounds on Counterfactual Queries Learning Causal Models Learning Causal Models without Confounding Factors Learning from Interventional Data 1043

12 xx CONTENTS Dealing with Latent Variables Learning Functional Causal Models Summary Relevant Literature Exercises Utilities and Decisions Foundations: Maximizing Expected Utility Decision Making Under Uncertainty Theoretical Justification Utility Curves Utility of Money Attitudes Toward Risk Rationality Utility Elicitation Utility Elicitation Procedures Utility of Human Life Utilities of Complex Outcomes Preference and Utility Independence Additive Independence Properties Summary Relevant Literature Exercises Structured Decision Problems Decision Trees Representation Backward Induction Algorithm Influence Diagrams Basic Representation Decision Rules Time and Recall Semantics and Optimality Criterion Backward Induction in Influence Diagrams Decision Trees for Influence Diagrams Sum-Max-Sum Rule Computing Expected Utilities Simple Variable Elimination Multiple Utility Variables: Simple Approaches Generalized Variable Elimination Optimization in Influence Diagrams Optimizing a Single Decision Rule Iterated Optimization Algorithm Strategic Relevance and Global Optimality Ignoring Irrelevant Information 1117

13 CONTENTS xxi 23.7 Value of Information Single Observations Multiple Observations Summary Relevant Literature Exercises Epilogue 1131 A Background Material 1135 A.1 Information Theory 1135 A.1.1 Compression and Entropy 1135 A.1.2 Conditional Entropy and Information 1137 A.1.3 Relative Entropy and Distances Between Distributions 1138 A.2 Convergence Bounds 1141 A.2.1 Central Limit Theorem 1142 A.2.2 Convergence Bounds 1143 A.3 Algorithms and Algorithmic Complexity 1144 A.3.1 Basic Graph Algorithms 1144 A.3.2 Analysis of Algorithmic Complexity 1145 A.3.3 Dynamic Programming 1147 A.3.4 Complexity Theory 1148 A.4 Combinatorial Optimization and Search 1152 A.4.1 Optimization Problems 1152 A.4.2 Local Search 1152 A.4.3 Branch and Bound Search 1158 A.5 Continuous Optimization 1159 A.5.1 Characterizing Optima of a Continuous Function 1159 A.5.2 Gradient Ascent Methods 1161 A.5.3 Constrained Optimization 1165 A.5.4 Convex Duality 1169 Bibliography 1171 Notation Index 1209 Subject Index 1213