Bayesian Networks and Decision Graphs

by Jensen, Finn V.; Nielsen, Thomas D.

Edition: 2nd

ISBN13: 9780387682815

ISBN10: 0387682813

Format: Hardcover

Pub. Date: 2007-04-01

Publisher(s): Springer Nature

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Summary

Probabilistic graphical models and decision graphs are powerful modeling tools for reasoning and decision making under uncertainty. As modeling languages they allow a natural specification of problem domains with inherent uncertainty, and from a computational perspective they support efficient algorithms for automatic construction and query answering. This includes belief updating, finding the most probable explanation for the observed evidence, detecting conflicts in the evidence entered into the network, determining optimal strategies, analyzing for relevance, and performing sensitivity analysis. The book introduces probabilistic graphical models and decision graphs, including Bayesian networks and influence diagrams. The reader is introduced to the two types of frameworks through examples and exercises, which also instruct the reader on how to build these models. The book is a new edition of Bayesian Networks and Decision Graphs by Finn V. Jensen. The new edition is structured into two parts. The first part focuses on probabilistic graphical models. Compared with the previous book, the new edition also includes a thorough description of recent extensions to the Bayesian network modeling language, advances in exact and approximate belief updating algorithms, and methods for learning both the structure and the parameters of a Bayesian network. The second part deals with decision graphs, and in addition to the frameworks described in the previous edition, it also introduces Markov decision processes and partially ordered decision problems. The authors also provide a well-founded practical introduction to Bayesian networks, object-oriented Bayesian networks, decision trees, influence diagrams (and variants hereof), and Markov decision processes. give practical advice on the construction of Bayesian networks, decision trees, and influence diagrams from domain knowledge. give several examples and exercises exploiting computer systems for dealing with Bayesian networks and decision graphs. present a thorough introduction to state-of-the-art solution and analysis algorithms. The book is intended as a textbook, but it can also be used for self-study and as a reference book.

Author Biography

Finn V. Jensen is a professor at the department of computer science at Aalborg University, Denmark.

Preface	p. V
Prerequisites on Probability Theory	p. 1
Two Perspectives on Probability Theory	p. 1
Fundamentals of Probability Theory	p. 2
Conditional Probabilities	p. 4
Probability Calculus	p. 5
Conditional Independence	p. 6
Probability Calculus for Variables	p. 7
Calculations with Probability Tables: An Example	p. 11
An Algebra of Potentials	p. 13
Random Variables	p. 15
Continuous Distributions	p. 15
Exercises	p. 16
Probabilistic Graphical Models
Causal and Bayesian Networks	p. 23
Reasoning Under Uncertainty	p. 23
Car Start Problem	p. 23
A Causal Perspective on the Car Start Problem	p. 24
Causal Networks and d-Separation	p. 26
d-separation	p. 30
Bayesian Networks	p. 32
Definition of Bayesian Networks	p. 32
The Chain Rule for Bayesian Networks	p. 35
Inserting Evidence	p. 39
Calculating Probabilities in Practice	p. 41
Graphical Models - Formal Languages for Model Specification	p. 42
Summary	p. 44
Bibliographical Notes	p. 45
Exercises	p. 45
Building Models	p. 51
Catching the Structure	p. 51
Milk Test	p. 52
Cold or Angina?	p. 54
Insemination	p. 55
A Simplified Poker Game	p. 57
Naive Bayes Models	p. 58
Causality	p. 60
Determining the Conditional Probabilities	p. 60
Milk Test	p. 60
Stud Farm	p. 62
Poker Game	p. 66
Transmission of Symbol Strings	p. 68
Cold or Angina?	p. 71
Why Causal Networks?	p. 72
Modeling Methods	p. 73
Undirected Relations	p. 73
Noisy-Or	p. 75
Divorcing	p. 78
Noisy Functional Dependence	p. 80
Expert Disagreements	p. 81
Object-Oriented Bayesian Networks	p. 84
Dynamic Bayesian Networks	p. 91
How to Deal with Continuous Variables	p. 93
Interventions	p. 96
Special Features	p. 97
Joint Probability Tables	p. 98
Most-Probable Explanation	p. 98
Data Conflict	p. 98
Sensitivity Analysis	p. 99
Summary	p. 100
Bibliographical Notes	p. 101
Exercises	p. 102
Belief Updating in Bayesian Networks	p. 109
Introductory Examples	p. 109
A Single Marginal	p. 110
Different Evidence Scenarios	p. 111
All Marginals	p. 114
Graph-Theoretic Representation	p. 115
Task and Notation	p. 115
Domain Graphs	p. 116
Triangulated Graphs and Join Trees	p. 119
Join Trees	p. 122
Propagation in Junction Trees	p. 124
Lazy Propagation in Junction Trees	p. 127
Exploiting the Information Scenario	p. 130
Barren Nodes	p. 130
d-Separation	p. 131
Nontriangulated Domain Graphs	p. 132
Triangulation of Graphs	p. 134
Triangulation of Dynamic Bayesian Networks	p. 137
Exact Propagation with Bounded Space	p. 140
Recursive Conditioning	p. 140
Stochastic Simulation in Bayesian Networks	p. 145
Probabilistic Logic Sampling	p. 146
Likelihood Weighting	p. 148
Gibbs Sampling	p. 150
Loopy Belief Propagation	p. 152
Summary	p. 154
Bibliographical Notes	p. 156
Exercises	p. 157
Analysis Tools for Bayesian Networks	p. 167
IEJ Trees	p. 168
Joint Probabilities and A-Saturated Junction Trees	p. 169
A-Saturated Junction Trees	p. 169
Configuration of Maximal Probability	p. 171
Axioms for Propagation in Junction Trees	p. 173
Data Conflict	p. 174
Insemination	p. 175
The Conflict Measure conf	p. 175
Conflict or Rare Case	p. 176
Tracing of Conflicts	p. 177
Other Approaches to Conflict Detection	p. 179
SE Analysis	p. 179
Example and Definitions	p. 179
h-Saturated Junction Trees and SE Analysis	p. 182
Sensitivity to Parameters	p. 184
One-Way Sensitivity Analysis	p. 187
Two-Way Sensitivity Analysis	p. 188
Summary	p. 188
Bibliographical Notes	p. 190
Exercises	p. 191
Parameter Estimation	p. 196
Complete Data	p. 195
Maximum Likelihood Estimation	p. 196
Bayesian Estimation	p. 197
Incomplete Data	p. 200
Approximate Parameter Estimation: The EM Algorithm	p. 201
Why We Cannot Perform Exact Parameter Estimation	p. 207
Adaptation	p. 207
Fractional Updating	p. 210
Fading	p. 211
Specification of an Initial Sample Size	p. 212
Example: Strings of Symbols	p. 213
Adaptation to Structure	p. 214
Fractional Updating as an Approximation	p. 215
Tuning	p. 218
Example	p. 220
Determining grad dist(x, y) as a Function of t	p. 222
Summary	p. 223
Bibliographical Notes	p. 225
Exercises	p. 226
Learning the Structure of Bayesian Networks	p. 229
Constraint-Based Learning Methods	p. 230
From Skeleton to DAG	p. 231
From Independence Tests to Skeleton	p. 234
Example	p. 235
Constraint-Based Learning on Data Sets	p. 237
Ockham's Razor	p. 240
Score-Based Learning	p. 241
Score Functions	p. 242
Search Procedures	p. 245
Chow-Liu Trees	p. 250
Bayesian Score Functions	p. 253
Summary	p. 258
Bibliographical Notes	p. 260
Exercises	p. 261
Bayesian Networks as Classifiers	p. 265
Naive Bayes Classifiers	p. 266
Evaluation of Classifiers	p. 268
Extensions of Naive Bayes Classifiers	p. 270
Classification Trees	p. 272
Summary	p. 274
Bibliographical Notes	p. 275
Exercises	p. 276
Decision Graphs
Graphical Languages for Specification of Decision Problems	p. 279
One-Shot Decision Problems	p. 280
Fold or Call?	p. 281
Mildew	p. 282
One Decision in General	p. 283
Utilities	p. 284
Instrumental Rationality	p. 287
Decision Trees	p. 290
A Couple of Examples	p. 293
Coalesced Decision Trees	p. 295
Solving Decision Trees	p. 296
Influence Diagrams	p. 302
Extended Poker Model	p. 302
Definition of Influence Diagrams	p. 305
Repetitive Decision Problems	p. 308
Asymmetric Decision Problems	p. 310
Different Sources of Asymmetry	p. 314
Unconstrained Influence Diagrams	p. 316
Sequential Influence Diagrams	p. 322
Decision Problems with Unbounded Time Horizons	p. 324
Markov Decision Processes	p. 324
Partially Observable Markov Decision Processes	p. 330
Summary	p. 332
Bibliographical Notes	p. 337
Exercises	p. 337
Solution Methods for Decision Graphs	p. 343
Solutions to Influence Diagrams	p. 343
The Chain Rule for Influence Diagrams	p. 345
Strategies and Expected Utilities	p. 346
An Example	p. 352
Variable Elimination	p. 353
Strong Junction Trees	p. 355
Required Past	p. 358
Policy Networks	p. 360
Node Removal and Arc Reversal	p. 362
Node Removal	p. 362
Arc Reversal	p. 363
An Example	p. 365
Solutions to Unconstrained Influence Diagrams	p. 367
Minimizing the S-DAG	p. 367
Determining Policies and Step Functions	p. 371
Decision Problems Without a Temporal Ordering: Troubleshooting	p. 373
Action Sequences	p. 373
A Greedy Approach	p. 375
Call Service	p. 378
Questions	p. 378
Solutions to Decision Problems with Unbounded Time Horizon	p. 380
A Basic Solution	p. 380
Value Iteration	p. 381
Policy Iteration	p. 385
Solving Partially Observable Markov Decision Processes	p. 388
Limited Memory Influence Diagrams	p. 392
Summary	p. 395
Bibliographical Notes	p. 400
Exercises	p. 401
Methods for Analyzing Decision Problems	p. 407
Value of Information	p. 407
Test for Infected Milk?	p. 407
Myopic Hypothesis-Driven Data Request	p. 409
Non-Utility-Based Value Functions	p. 411
Finding the Relevant Past and Future of a Decision Problem	p. 413
Identifying the Required Past	p. 415
Sensitivity Analysis	p. 420
Example	p. 421
One-Way Sensitivity Analysis in General	p. 423
Summary	p. 426
Bibliographical Notes	p. 427
Exercises	p. 427
List of Notation	p. 429
References	p. 431
Index	p. 441
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