Bayesian Theory

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Edition: 1st
Format: Paperback
Pub. Date: 2000-05-18
Publisher(s): WILEY
List Price: $124.74

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Summary

This highly acclaimed text, now available in paperback, provides a thorough account of key concepts and theoretical results, with particular emphasis on viewing statistical inference as a special case of decision theory. Information-theoretic concepts play a central role in the development of the theory, which provides, in particular, a detailed discussion of the problem of specification of so-called prior ignorance. The work is written from the authorss committed Bayesian perspective, but an overview of non-Bayesian theories is also provided, and each chapter contains a wide-ranging critical re-examination of controversial issues. The level of mathematics used is such that most material is accessible to readers with knowledge of advanced calculus. In particular, no knowledge of abstract measure theory is assumed, and the emphasis throughout is on statistical concepts rather than rigorous mathematics. The book will be an ideal source for all students and researchers in statistics, mathematics, decision analysis, economic and business studies, and all branches of science and engineering, who wish to further their understanding of Bayesian statistics

Author Biography

About the Authors Jose M. Bernardo received his PhD from University College London and has subsequently been at the University of Valencia, Spain, where he is currently Professor of Statistics and special scientific advisor to the Governor of the State of Valencia. Adrian F. M. Smith received his PhD from University College London and is currently at Imperial College London, where he is Professor of Statistics and Head of the Department of Mathematics

Table of Contents

Introduction
1(12)
Thomas Bayes
1(1)
The subjectivist view of probability
2(1)
Bayesian Statistics in perspective
3(2)
An overview of Bayesian Theory
5(4)
Scope
5(1)
Foundations
5(1)
Generalisations
6(1)
Modelling
7(1)
Inference
7(1)
Remodelling
8(1)
Basic formulae
8(1)
Non-Bayesian theories
9(1)
A Bayesian reading list
9(4)
Foundations
13(92)
Beliefs and actions
13(3)
Decision problems
16(7)
Basic elements
16(2)
Formal representation
18(5)
Coherence and quantification
23(10)
Events, options and preferences
23(1)
Coherent preferences
23(5)
Quantification
28(5)
Beliefs and probabilities
33(16)
Representation of beliefs
33(5)
Revision of beliefs and Bayes' theorem
38(7)
Conditional independence
45(2)
Sequential revision of beliefs
47(2)
Actions and utilities
49(7)
Bounded sets of consequences
49(1)
Bounded decision problems
50(4)
General decision problems
54(2)
Sequential decision problems
56(11)
Complex decision problems
56(3)
Backward induction
59(4)
Design of experiments
63(4)
Inference and information
67(14)
Reporting beliefs as a decision problem
67(2)
The utility of a probability distribution
69(6)
Approximation and discrepancy
75(2)
Information
77(4)
Discussion and further references
81(24)
Operational definitions
81(2)
Quantitative coherence theories
83(2)
Related theories
85(7)
Critical issues
92(13)
Generalisations
105(60)
Generalised representation of beliefs
105(4)
Motivation
105(1)
Countable additivity
106(3)
Review of probability theory
109(32)
Random quantities and distributions
109(5)
Some particular univariate distributions
114(11)
Convergence and limit theorems
125(2)
Random vectors, Bayes' theorem
127(6)
Some particular multivariate distributions
133(8)
Generalised options and utilities
141(9)
Motivation and preliminaries
141(4)
Generalised preferences
145(2)
The value of information
147(3)
Generalised information measures
150(10)
The general problem of reporting beliefs
150(1)
The utility of a general probability distribution
151(3)
Generalised approximation and discrepancy
154(3)
Generalised information
157(3)
Discussion and further references
160(5)
The role of mathematics
160(1)
Critical issues
161(4)
Modelling
165(76)
Statistical models
165(2)
Beliefs and models
165(2)
Exchangeability and related concepts
167(5)
Dependence and independence
167(1)
Exchangeability and partial exchangeability
168(4)
Models via exchangeability
172(9)
The Bernoulli and binomial models
172(4)
The multinomial model
176(1)
The general model
177(4)
Models via invariance
181(9)
The normal model
181(4)
The multivariate normal model
185(2)
The exponential model
187(2)
The geometric model
189(1)
Models via sufficient statistics
190(19)
Summary statistics
190(1)
Predictive sufficiency and parametric sufficiency
191(6)
Sufficiency and the exponential family
197(10)
Information measures and the exponential family
207(2)
Models via partial exchangeability
209(17)
Models for extended data structures
209(2)
Several samples
211(6)
Structured layouts
217(2)
Covariates
219(3)
Hierarchical models
222(4)
Pragmatic aspects
226(9)
Finite and infinite exchangeability
226(2)
Parametric and nonparametric models
228(1)
Model elaboration
229(4)
Model simplification
233(1)
Prior distributions
234(1)
Discussion and further references
235(6)
Representation theorems
235(1)
Subjectivity and objectivity
236(1)
Critical issues
237(4)
Inference
241(136)
The Bayesian paradigm
241(24)
Observables, beliefs and models
241(1)
The role of Bayes' theorem
242(1)
Predictive and parametric inference
243(4)
Sufficiency, ancillarity and stopping rules
247(8)
Decisions and inference summaries
255(8)
Implementation issues
263(2)
Conjugate analysis
265(20)
Conjugate families
265(4)
Canonical conjugate analysis
269(10)
Approximations with conjugate families
279(6)
Asymptotic analysis
285(13)
Discrete asymptotics
286(1)
Continuous asymptotics
287(8)
Asymptotics under transformations
295(3)
Reference analysis
298(41)
Reference decisions
299(3)
One-dimensional reference distributions
302(14)
Restricted reference distributions
316(4)
Nuisance parameters
320(13)
Multiparameter problems
333(6)
Numerical approximations
339(17)
Laplace approximation
340(6)
Iterative quadrature
346(2)
Importance sampling
348(2)
Sampling-importance-resampling
350(3)
Markov chain Monte Carlo
353(3)
Discussion and further references
356(21)
An historical footnote
356(1)
Prior ignorance
357(10)
Robustness
367(4)
Hierarchical and empirical Bayes
371(2)
Further methodological developments
373(1)
Critical issues
374(3)
Remodelling
377(50)
Model comparison
377(32)
Ranges of models
377(6)
Perspectives on model comparison
383(3)
Model comparison as a decision problem
386(3)
Zero-one utilities and Bayes factors
389(6)
General utilities
395(8)
Approximation by cross-validation
403(4)
Covariate selection
407(2)
Model rejection
409(8)
Model rejection through model comparison
409(3)
Discrepancy measures for model rejection
412(1)
Zero-one discrepancies
413(2)
General discrepancies
415(2)
Discussion and further references
417(10)
Overview
417(1)
Modelling and remodelling
418(1)
Critical issues
418(9)
A. SUMMARY OF BASIC FORMULAE 427(16)
Probability distributions
427(9)
Inferential processes
436(7)
B. NON-BAYESIAN THEORIES 443(46)
Overview
443(2)
Alternative approaches
445(15)
Classical decision theory
445(4)
Frequentist procedures
449(5)
Likelihood inference
454(2)
Fiducial and related theories
456(4)
Stylised inference problems
460(18)
Point estimation
460(5)
Interval estimation
465(4)
Hypothesis testing
469(6)
Significance testing
475(3)
Comparative issues
478(11)
Conditional and unconditional inference
478(1)
Nuisance parameters and marginalisation
479(3)
Approaches to prediction
482(3)
Aspects of asymptotics
485(1)
Model choice criteria
486(3)
References 489(66)
Subject Index 555(18)
Author Index 573

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