Graphical Belief ModelingThis innovative volume explores graphical models using belief functions as a representation of uncertainty, offering an alternative approach to problems where probability proves inadequate. Graphical Belief Modeling makes it easy to compare the two approaches while evaluating their relative strengths and limitations. The author examines both theory and computation, incorporating practical notes from the author's own experience with the BELIEF software package. As one of the first volumes to apply the Dempster-Shafer belief functions to a practical model, a substantial portion of the book is devoted to a single example--calculating the reliability of a complex system. This special feature enables readers to gain a thorough understanding of the application of this methodology. The first section provides a description of graphical belief models and probablistic graphical models that form an important subset: the second section discusses the algorithm used in the manipulation of graphical models: the final segment of the book offers a complete description of the risk assessment example, as well as the methodology used to describe it. Graphical Belief Modeling offers researchers and graduate students in artificial intelligence and statistics more than just a new approach to an old reliability task: it provides them with an invaluable illustration of the process of graphical belief modeling. |
Contents
Overview of Graphical Belief Models | 1 |
Probability | 26 |
6 | 32 |
8a Parallel components | 38 |
Probability | 57 |
Basic Belief Functions | 71 |
Basic Belief Functions | 72 |
21 | 92 |
AN EXAMPLE | 217 |
Belief Function Models for Components | 239 |
Models for Simple Series and Parallel Systems | 261 |
Information Common Parameter Dependence | 285 |
Three Examples | 317 |
Belief Risk Assessment and Public Policy | 347 |
MODELLERS | 355 |
B The BELIEF Package and Other Software | 387 |
Graphical Models | 103 |
Graphical Models | 111 |
The Fusion and Propagation Algorithm | 145 |
Model Exploration | 183 |
References | 396 |
Index | 407 |
| 425 | |
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Common terms and phrases
Almond assumptions basic events Bayes belief and plausibility belief function models BELIEF package Bernoulli process binomial calculate Chapter combination conditional belief functions conditional independence conditional probability corresponding credibility interval decision problem defined describes directed graph discrete elimination order Equation estimates example factor fail failure beliefs failure probability failure rate Figure focal elements fusion and propagation graphical belief models hyperedge hypergraph imprecise independent influence diagrams information dependence junction tree large number LOCA logical belief function marginal distribution mass function Maximus method Möbius transforms Monte Carlo node notation number of events operations outcome space pivotal variables Poisson process probabilistic graphical models probability distributions probability models problem produces propagation algorithm provides random interval random variable represent sample Section setq Shafer Shenoy statistics structure subevent subset subsystem system failure Theorem tion tree model tree of cliques type parameters unlinked model vacuous belief function vagueness values