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E-raamat: Probabilistic Graphical Models

(Stanford University), (Hebrew University)
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A general framework for constructing and using probabilistic models of complex systems that would enable a computer to use available information for making decisions.

Most tasks require a person or an automated system to reason—to reach conclusions based on available information. The framework of probabilistic graphical models, presented in this book, provides a general approach for this task. The approach is model-based, allowing interpretable models to be constructed and then manipulated by reasoning algorithms. These models can also be learned automatically from data, allowing the approach to be used in cases where manually constructing a model is difficult or even impossible. Because uncertainty is an inescapable aspect of most real-world applications, the book focuses on probabilistic models, which make the uncertainty explicit and provide models that are more faithful to reality.

Probabilistic Graphical Models discusses a variety of models, spanning Bayesian networks, undirected Markov networks, discrete and continuous models, and extensions to deal with dynamical systems and relational data. For each class of models, the text describes the three fundamental cornerstones: representation, inference, and learning, presenting both basic concepts and advanced techniques. Finally, the book considers the use of the proposed framework for causal reasoning and decision making under uncertainty.

The main text in each chapter provides the detailed technical development of the key ideas. Most chapters also include boxes with additional material: skill boxes, which describe techniques; case study boxes, which discuss empirical cases related to the approach described in the text, including applications in computer vision, robotics, natural language understanding, and computational biology; and concept boxes, which present significant concepts drawn from the material in the chapter. Instructors (and readers) can group chapters in various combinations, from core topics to more technically advanced material, to suit their particular needs.

Adaptive Computation and Machine Learning series
Acknowledgments xxiii
List of Figures
xxv
List of Algorithms
xxxi
List of Boxes
xxxiii
Introduction
1(14)
Motivation
1(1)
Structured Probabilistic Models
2(4)
Probabilistic Graphical Models
3(2)
Representation, Inference, Learning
5(1)
Overview and Roadmap
6(6)
Overview of
Chapters
6(3)
Reader's Guide
9(2)
Connection to Other Disciplines
11(1)
Historical Notes
12(3)
Foundations
15(28)
Probability Theory
15(19)
Probability Distributions
15(3)
Basic Concepts in Probability
18(1)
Random Variables and Joint Distributions
19(4)
Independence and Conditional Independence
23(2)
Querying a Distribution
25(2)
Continuous Spaces
27(4)
Expectation and Variance
31(3)
Graphs
34(5)
Nodes and Edges
34(1)
Subgraphs
35(1)
Paths and Trails
36(1)
Cycles and Loops
36(3)
Relevant Literature
39(1)
Exercises
39(4)
I Representation
43(242)
The Bayesian Network Representation
45(58)
Exploiting Independence Properties
45(6)
Independent Random Variables
45(1)
The Conditional Parameterization
46(2)
The Naive Bayes Model
48(3)
Bayesian Networks
51(17)
The Student Example Revisited
52(4)
Basic Independencies in Bayesian Networks
56(4)
Graphs and Distributions
60(8)
Independencies in Graphs
68(10)
D-separation
69(3)
Soundness and Completeness
72(2)
An Algorithm for d-Separation
74(2)
I-Equivalence
76(2)
From Distributions to Graphs
78(14)
Minimal I-Maps
78(3)
Perfect Maps
81(2)
Finding Perfect Maps
83(9)
Summary
92(1)
Relevant Literature
93(3)
Exercises
96(7)
Undirected Graphical Models
103(54)
The Misconception Example
103(3)
Parameterization
106(8)
Factors
106(2)
Gibbs Distributions and Markov Networks
108(2)
Reduced Markov Networks
110(4)
Markov Network Independencies
114(8)
Basic Independencies
114(3)
Independencies Revisited
117(3)
From Distributions to Graphs
120(2)
Parameterization Revisited
122(12)
Finer-Grained Parameterization
123(5)
Overparameterization
128(6)
Bayesian Networks and Markov Networks
134(8)
From Bayesian Networks to Markov Networks
134(4)
From Markov Networks to Bayesian Networks
138(1)
Chordal Graphs
139(3)
Partially Directed Models
142(9)
Conditional Random Fields
142(6)
Chain Graph Models
148(3)
Summary and Discussion
151(1)
Relevant Literature
152(1)
Exercises
153(4)
Local Probabilistic Models
157(42)
Tabular CPDs
157(1)
Deterministic CPDs
158(4)
Representation
158(1)
Independencies
159(3)
Context-Specific CPDs
162(13)
Representation
162(9)
Independencies
171(4)
Independence of Causal Influence
175(10)
The Noisy-Or Model
175(3)
Generalized Linear Models
178(4)
The General Formulation
182(2)
Independencies
184(1)
Continuous Variables
185(6)
Hybrid Models
189(2)
Conditional Bayesian Networks
191(2)
Summary
193(1)
Relevant Literature
194(1)
Exercises
195(4)
Template-Based Representations
199(48)
Introduction
199(1)
Temporal Models
200(12)
Basic Assumptions
201(1)
Dynamic Bayesian Networks
202(5)
State-Observation Models
207(5)
Template Variables and Template Factors
212(4)
Directed Probabilistic Models for Object-Relational Domains
216(12)
Plate Models
216(6)
Probabilistic Relational Models
222(6)
Undirected Representation
228(4)
Structural Uncertainty
232(8)
Relational Uncertainty
233(2)
Object Uncertainty
235(5)
Summary
240(2)
Relevant Literature
242(1)
Exercises
243(4)
Gaussian Network Models
247(14)
Multivariate Gaussians
247(4)
Basic Parameterization
247(2)
Operations on Gaussians
249(1)
Independencies in Gaussians
250(1)
Gaussian Bayesian Networks
251(3)
Gaussian Markov Random Fields
254(3)
Summary
257(1)
Relevant Literature
258(1)
Exercises
258(3)
The Exponential Family
261(24)
Introduction
261(1)
Exponential Families
261(5)
Linear Exponential Families
263(3)
Factored Exponential Families
266(3)
Product Distributions
266(1)
Bayesian Networks
267(2)
Entropy and Relative Entropy
269(4)
Entropy
269(3)
Relative Entropy
272(1)
Projections
273(9)
Comparison
274(3)
M-Projections
277(5)
I-Projections
282(1)
Summary
282(1)
Relevant Literature
283(1)
Exercises
283(2)
II Inference
285(410)
Exact Inference: Variable Elimination
287(58)
Analysis of Complexity
288(4)
Analysis of Exact Inference
288(2)
Analysis of Approximate Inference
290(2)
Variable Elimination: The Basic Ideas
292(4)
Variable Elimination
296(10)
Basic Elimination
297(6)
Dealing with Evidence
303(3)
Complexity and Graph Structure: Variable Elimination
306(9)
Simple Analysis
306(1)
Graph-Theoretic Analysis
306(4)
Finding Elimination Orderings
310(5)
Conditioning
315(10)
The Conditioning Algorithm
315(3)
Conditioning and Variable Elimination
318(4)
Graph-Theoretic Analysis
322(1)
Improved Conditioning
323(2)
Inference with Structured CPDs
325(11)
Independence of Causal Influence
325(4)
Context-Specific Independence
329(6)
Discussion
335(1)
Summary and Discussion
336(1)
Relevant Literature
337(1)
Exercises
338(7)
Exact Inference: Clique Trees
345(36)
Variable Elimination and Clique Trees
345(3)
Cluster Graphs
346(1)
Clique Trees
346(2)
Message Passing: Sum Product
348(16)
Variable Elimination in a Clique Tree
349(6)
Clique Tree Calibration
355(6)
A Calibrated Clique Tree as a Distribution
361(3)
Message Passing: Belief Update
364(8)
Message Passing with Division
364(4)
Equivalence of Sum-Product and Belief Update Messages
368(1)
Answering Queries
369(3)
Constructing a Clique Tree
372(4)
Clique Trees from Variable Elimination
372(2)
Clique Trees from Chordal Graphs
374(2)
Summary
376(1)
Relevant Literature
377(1)
Exercises
378(3)
Inference as Optimization
381(106)
Introduction
381(5)
Exact Inference Revisited
382(2)
The Energy Functional
384(2)
Optimizing the Energy Functional
386(1)
Exact Inference as Optimization
386(5)
Fixed-Point Characterization
388(2)
Inference as Optimization
390(1)
Propagation-Based Approximation
391(39)
A Simple Example
391(5)
Cluster-Graph Belief Propagation
396(3)
Properties of Cluster-Graph Belief Propagation
399(2)
Analyzing Convergence
401(3)
Constructing Cluster Graphs
404(7)
Variational Analysis
411(3)
Other Entropy Approximations
414(14)
Discussion
428(2)
Propagation with Approximate Messages
430(18)
Factorized Messages
431(2)
Approximate Message Computation
433(3)
Inference with Approximate Messages
436(6)
Expectation Propagation
442(3)
Variational Analysis
445(3)
Discussion
448(1)
Structured Variational Approximations
448(25)
The Mean Field Approximation
449(7)
Structured Approximations
456(13)
Local Variational Methods
469(4)
Summary and Discussion
473(2)
Relevant Literature
475(2)
Exercises
477(10)
Particle-Based Approximate Inference
487(64)
Forward Sampling
488(4)
Sampling from a Bayesian Network
488(2)
Analysis of Error
490(1)
Conditional Probability Queries
491(1)
Likelihood Weighting and Importance Sampling
492(13)
Likelihood Weighting: Intuition
492(2)
Importance Sampling
494(4)
Importance Sampling for Bayesian Networks
498(6)
Importance Sampling Revisited
504(1)
Markov Chain Monte Carlo Methods
505(21)
Gibbs Sampling Algorithm
505(2)
Markov Chains
507(5)
Gibbs Sampling Revisited
512(3)
A Broader Class of Markov Chains
515(3)
Using a Markov Chain
518(8)
Collapsed Particles
526(10)
Collapsed Likelihood Weighting
527(4)
Collapsed MCMC
531(5)
Deterministic Search Methods
536(4)
Summary
540(1)
Relevant Literature
541(3)
Exercises
544(7)
MAP Inference
551(54)
Overview
551(3)
Computational Complexity
551(1)
Overview of Solution Methods
552(2)
Variable Elimination for (Marginal) MAP
554(8)
Max-Product Variable Elimination
554(2)
Finding the Most Probable Assignment
556(3)
Variable Elimination for Marginal MAP
559(3)
Max-Product in Clique Trees
562(5)
Computing Max-Marginals
562(2)
Message Passing as Reparameterization
564(1)
Decoding Max-Marginals
565(2)
Max-Product Belief Propagation in Loopy Cluster Graphs
567(10)
Standard Max-Product Message Passing
567(5)
Max-Product BP with Counting Numbers
572(3)
Discussion
575(2)
MAP as a Linear Optimization Problem
577(11)
The Integer Program Formulation
577(2)
Linear Programming Relaxation
579(2)
Low-Temperature Limits
581(7)
Using Graph Cuts for MAP
588(7)
Inference Using Graph Cuts
588(4)
Nonbinary Variables
592(3)
Local Search Algorithms
595(2)
Summary
597(1)
Relevant Literature
598(3)
Exercises
601(4)
Inference in Hybrid Networks
605(46)
Introduction
605(3)
Challenges
605(1)
Discretization
606(1)
Overview
607(1)
Variable Elimination in Gaussian Networks
608(7)
Canonical Forms
609(2)
Sum-Product Algorithms
611(1)
Gaussian Belief Propagation
612(3)
Hybrid Networks
615(15)
The Difficulties
615(3)
Factor Operations for Hybrid Gaussian Networks
618(3)
EP for CLG Networks
621(5)
An ``Exact'' CLG Algorithm
626(4)
Nonlinear Dependencies
630(12)
Linearization
631(6)
Expectation Propagation with Gaussian Approximation
637(5)
Particle-Based Approximation Methods
642(4)
Sampling in Continuous Spaces
642(1)
Forward Sampling in Bayesian Networks
643(1)
MCMC Methods
644(1)
Collapsed Particles
645(1)
Nonparametric Message Passing
646(1)
Summary and Discussion
646(1)
Relevant Literature
647(2)
Exercises
649(2)
Inference in Temporal Models
651(44)
Inference Tasks
652(1)
Exact Inference
653(7)
Filtering in State-Observation Models
653(1)
Filtering as Clique Tree Propagation
654(1)
Clique Tree Inference in DBNs
655(1)
Entanglement
656(4)
Approximate Inference
660(15)
Key Ideas
661(1)
Factored Belief State Methods
662(3)
Particle Filtering
665(10)
Deterministic Search Techniques
675(1)
Hybrid DBNs
675(13)
Continuous Models
676(8)
Hybrid Models
684(4)
Summary
688(2)
Relevant Literature
690(2)
Exercises
692(3)
III Learning
695(312)
Learning Graphical Models: Overview
697(20)
Motivation
697(1)
Goals of Learning
698(4)
Density Estimation
698(2)
Specific Prediction Tasks
700(1)
Knowledge Discovery
701(1)
Learning as Optimization
702(9)
Empirical Risk and Overfitting
703(6)
Discriminative versus Generative Training
709(2)
Learning Tasks
711(4)
Model Constraints
712(1)
Data Observability
712(2)
Taxonomy of Learning Tasks
714(1)
Relevant Literature
715(2)
Parameter Estimation
717(66)
Maximum Likelihood Estimation
717(5)
The Thumbtack Example
717(3)
The Maximum Likelihood Principle
720(2)
MLE for Bayesian Networks
722(11)
A Simple Example
723(1)
Global Likelihood Decomposition
724(1)
Table-CPDs
725(3)
Gaussian Bayesian Networks
728(3)
Maximum Likelihood Estimation as M-Projection
731(2)
Bayesian Parameter Estimation
733(8)
The Thumbtack Example Revisited
733(4)
Priors and Posteriors
737(4)
Bayesian Parameter Estimation in Bayesian Networks
741(13)
Parameter Independence and Global Decomposition
742(4)
Local Decomposition
746(2)
Priors for Bayesian Network Learning
748(3)
MAP Estimation
751(3)
Learning Models with Shared Parameters
754(15)
Global Parameter Sharing
755(5)
Local Parameter Sharing
760(2)
Bayesian Inference with Shared Parameters
762(1)
Hierarchical Priors
763(6)
Generalization Analysis
769(7)
Asymptotic Analysis
769(1)
PAC-Bounds
770(6)
Summary
776(1)
Relevant Literature
777(1)
Exercises
778(5)
Structure Learning in Bayesian Networks
783(66)
Introduction
783(3)
Problem Definition
783(2)
Overview of Methods
785(1)
Constraint-Based Approaches
786(4)
General Framework
786(1)
Independence Tests
787(3)
Structure Scores
790(17)
Likelihood Scores
791(3)
Bayesian Score
794(3)
Marginal Likelihood for a Single Variable
797(2)
Bayesian Score for Bayesian Networks
799(2)
Understanding the Bayesian Score
801(3)
Priors
804(3)
Score Equivalence
807(1)
Structure Search
807(17)
Learning Tree-Structured Networks
808(1)
Known Order
809(2)
General Graphs
811(10)
Learning with Equivalence Classes
821(3)
Bayesian Model Averaging
824(8)
Basic Theory
824(2)
Model Averaging Given an Order
826(2)
The General Case
828(4)
Learning Models with Additional Structure
832(6)
Learning with Local Structure
833(4)
Learning Template Models
837(1)
Summary and Discussion
838(2)
Relevant Literature
840(3)
Exercises
843(6)
Partially Observed Data
849(94)
Foundations
849(13)
Likelihood of Data and Observation Models
849(4)
Decoupling of Observation Mechanism
853(3)
The Likelihood Function
856(4)
Identifiability
860(2)
Parameter Estimation
862(35)
Gradient Ascent
863(5)
Expectation Maximization (EM)
868(19)
Comparison: Gradient Ascent versus EM
887(6)
Approximate Inference
893(4)
Bayesian Learning with Incomplete Data
897(11)
Overview
897(2)
MCMC Sampling
899(5)
Variational Bayesian Learning
904(4)
Structure Learning
908(17)
Scoring Structures
909(8)
Structure Search
917(3)
Structural EM
920(5)
Learning Models with Hidden Variables
925(8)
Information Content of Hidden Variables
926(2)
Determining the Cardinality
928(2)
Introducing Hidden Variables
930(3)
Summary
933(1)
Relevant Literature
934(1)
Exercises
935(8)
Learning Undirected Models
943(64)
Overview
943(1)
The Likelihood Function
944(5)
An Example
944(2)
Form of the Likelihood Function
946(1)
Properties of the Likelihood Function
947(2)
Maximum (Conditional) Likelihood Parameter Estimation
949(9)
Maximum Likelihood Estimation
949(1)
Conditionally Trained Models
950(4)
Learning with Missing Data
954(2)
Maximum Entropy and Maximum Likelihood
956(2)
Parameter Priors and Regularization
958(3)
Local Priors
958(3)
Global Priors
961(1)
Learning with Approximate Inference
961(8)
Belief Propagation
962(5)
MAP-Based Learning
967(2)
Alternative Objectives
969(9)
Pseudolikelihood and Its Generalizations
970(4)
Contrastive Optimization Criteria
974(4)
Structure Learning
978(18)
Structure Learning Using Independence Tests
979(2)
Score-Based Learning: Hypothesis Spaces
981(1)
Objective Functions
982(3)
Optimization Task
985(7)
Evaluating Changes to the Model
992(4)
Summary
996(2)
Relevant Literature
998(3)
Exercises
1001(6)
IV Actions and Decisions
1007(128)
Causality
1009(48)
Motivation and Overview
1009(5)
Conditioning and Intervention
1009(3)
Correlation and Causation
1012(2)
Causal Models
1014(3)
Structural Causal Identifiability
1017(9)
Query Simplification Rules
1017(3)
Iterated Query Simplification
1020(6)
Mechanisms and Response Variables
1026(5)
Partial Identifiability in Functional Causal Models
1031(3)
Counterfactual Queries
1034(5)
Twinned Networks
1034(3)
Bounds on Counterfactual Queries
1037(2)
Learning Causal Models
1039(13)
Learning Causal Models without Confounding Factors
1040(3)
Learning from Interventional Data
1043(4)
Dealing with Latent Variables
1047(3)
Learning Functional Causal Models
1050(2)
Summary
1052(1)
Relevant Literature
1053(1)
Exercises
1054(3)
Utilities and Decisions
1057(26)
Foundations: Maximizing Expected Utility
1057(5)
Decision Making Under Uncertainty
1057(3)
Theoretical Justification
1060(2)
Utility Curves
1062(4)
Utility of Money
1063(1)
Attitudes Toward Risk
1064(1)
Rationality
1065(1)
Utility Elicitation
1066(3)
Utility Elicitation Procedures
1066(1)
Utility of Human Life
1067(2)
Utilities of Complex Outcomes
1069(10)
Preference and Utility Independence
1069(3)
Additive Independence Properties
1072(7)
Summary
1079(1)
Relevant Literature
1080(2)
Exercises
1082(1)
Structured Decision Problems
1083(48)
Decision Trees
1083(3)
Representation
1083(2)
Backward Induction Algorithm
1085(1)
Influence Diagrams
1086(7)
Basic Representation
1087(1)
Decision Rules
1088(2)
Time and Recall
1090(1)
Semantics and Optimality Criterion
1091(2)
Backward Induction in Influence Diagrams
1093(5)
Decision Trees for Influence Diagrams
1094(2)
Sum-Max-Sum Rule
1096(2)
Computing Expected Utilities
1098(7)
Simple Variable Elimination
1098(2)
Multiple Utility Variables: Simple Approaches
1100(1)
Generalized Variable Elimination
1101(4)
Optimization in Influence Diagrams
1105(12)
Optimizing a Single Decision Rule
1105(1)
Iterated Optimization Algorithm
1106(2)
Strategic Relevance and Global Optimality
1108(9)
Ignoring Irrelevant Information
1117(2)
Value of Information
1119(5)
Single Observations
1120(2)
Multiple Observations
1122(2)
Summary
1124(1)
Relevant Literature
1125(3)
Exercises
1128(3)
Epilogue
1131(4)
A Background Material
1135(36)
Information Theory
1135(6)
Compression and Entropy
1135(2)
Conditional Entropy and Information
1137(1)
Relative Entropy and Distances Between Distributions
1138(3)
Convergence Bounds
1141(3)
Central Limit Theorem
1142(1)
Convergence Bounds
1143(1)
Algorithms and Algorithmic Complexity
1144(8)
Basic Graph Algorithms
1144(1)
Analysis of Algorithmic Complexity
1145(2)
Dynamic Programming
1147(1)
Complexity Theory
1148(4)
Combinatorial Optimization and Search
1152(7)
Optimization Problems
1152(1)
Local Search
1152(6)
Branch and Bound Search
1158(1)
Continuous Optimization
1159(12)
Characterizing Optima of a Continuous Function
1159(2)
Gradient Ascent Methods
1161(4)
Constrained Optimization
1165(4)
Convex Duality
1169(2)
Bibliography 1171(38)
Notation Index 1209(4)
Subject Index 1213