Bayesian Theory

by Bernardo, José; M.; Smith, Adrian F. M.

• ISBN13: 9780471494645
• ISBN10: 047149464X
• Edition: 1st
• Format: Paperback
• Copyright: 2000-05-18
• Publisher: WILEY

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