Bayesian Statistics : An Introduction

by Peter M. Lee

• ISBN13: 9780340814055
• ISBN10: 0340814055
• Edition: 3rd
• Format: Paperback
• Copyright: 2004-05-13
• Publisher: Hodder Education Publishers

Summary

Bayesian Statistics is the school of thought that uses all information surrounding the likelihood of an event rather than just that collected experimentally. Among statisticians the Bayesian approach continues to gain adherents and this new edition of Peter Lee's well-established introduction maintains the clarity of exposition and use of examples for which this text is known and praised. In addition, there is extended coverage of the Metropolis-Hastings algorithm as well as an introduction to the use of BUGS (Bayesian Inference Using Gibbs Sampling) as this is now the standard computational tool for such numerical work. Other alterations include new material on generalized linear modelling and Bernardo's theory of reference points.

Author Biography

Peter M. Lee is Provost of Wentworth College, University of York, UK.

Table of Contents

Preface to the third edition	xi
Preface to the first edition	xiii
1 Preliminaries	1	(30)
1.1 Probability and Bayes' theorem	1	(7)
1.2 Examples on Bayes' theorem	8	(2)
1.3 Random variables	10	(5)
1.4 Several random variables	15	(5)
1.5 Means and variances	20	(7)
1.6 Exercises	27	(4)
2 Bayesian inference for the normal distribution	31	(42)
2.1 Nature of Bayesian inference	31	(3)
2.2 Normal prior and likelihood	34	(4)
2.3 Several normal observations with a normal prior	38	(3)
2.4 Dominant likelihoods	41	(2)
2.5 Locally uniform priors	43	(3)
2.6 Highest density regions	46	(1)
2.7 Normal variance	47	(3)
2.8 HDRs for the normal variance	50	(2)
2.9 The role of sufficiency	52	(4)
2.10 Conjugate prior distributions	56	(4)
2.11 The exponential family	60	(2)
2.12 Normal mean and variance both unknown	62	(4)
2.13 Conjugate joint prior for the normal distribution	66	(3)
2.14 Exercises	69	(4)
3 Some other common distributions	73	(44)
3.1 The binomial distribution	73	(6)
3.2 Reference prior for the binomial likelihood	79	(3)
3.3 Jeffreys' rule	82	(5)
3.4 The Poisson distribution	87	(3)
3.5 The uniform distribution	90	(3)
3.6 Reference prior for the uniform distribution	93	(2)
3.7 The tramcar problem	95	(1)
3.8 The first digit problem: invariant priors	96	(3)
3.9 The circular normal distribution	99	(4)
3.10 Approximations based on the likelihood	103	(6)
3.11 Reference posterior distributions	109	(5)
3.12 Exercises	114	(3)
4 Hypothesis testing	117	(22)
4.1 Hypothesis testing	117	(4)
4.2 One-sided hypothesis tests	121	(2)
4.3 Lindley's method	123	(1)
4.4 Point (or sharp) null hypotheses with prior information	124	(3)
4.5 Point null hypotheses for the normal distribution	127	(6)
4.6 The Doogian philosophy	133	(1)
4.7 Exercises	134	(5)
5 Two-sample problems	139	(20)
5.1 Two-sample problems - both variances unknown	139	(3)
5.2 Variances unknown but equal	142	(3)
5.3 Variances unknown and equal (Behrens-Fisher problem)	145	(2)
5.4 The Behrens-Fisher controversy	147	(2)
5.5 Inferences concerning a variance ratio	149	(3)
5.6 Comparison of two proportions: the 2 x 2 table	152	(2)
5.7 Exercises	154	(5)
6 Correlation, regression and the analysis of variance	159	(34)
6.1 Theory of the correlation coefficient	159	(6)
6.2 Examples on the use of the correlation coefficient	165	(1)
6.3 Regression and the bivariate normal model	166	(5)
6.4 Conjugate prior for the bivariate regression model	171	(3)
6.5 Comparison of several means - the one-way model	174	(8)
6.6 The two-way layout	182	(3)
6.7 The general linear model	185	(3)
6.8 Exercises	188	(5)
7 Other topics	193	(28)
7.1 The likelihood principle	193	(4)
7.2 The stopping rule principle	197	(3)
7.3 Informative stopping rules	200	(2)
7.4 The likelihood principle and reference priors	202	(2)
7.5 Bayesian decision theory	204	(5)
7.6 Bayes linear methods	209	(2)
7.7 Decision theory and hypothesis testing	211	(2)
7.8 Empirical Bayes methods	213	(2)
7.9 Exercises	215	(6)
8 Hierarchical models	221	(24)
8.1 The idea of a hierarchical model	221	(4)
8.2 The hierarchical normal model	225	(4)
8.3 The baseball example	229	(1)
8.4 The Stein estimator	230	(4)
8.5 Bayesian analysis for an unknown overall mean	234	(3)
8.6 The general linear model revisited	237	(5)
8.7 Exercises	242	(3)
9 The Gibbs sampler and other numerical methods	245	(40)
9.1 Introduction to numerical methods	245	(1)
9.2 The EM algorithm	246	(7)
9.3 Data augmentation by Monte Carlo	253	(2)
9.4 The Gibbs sampler	255	(12)
9.5 Rejection sampling	267	(1)
9.6 The Metropolis-Hastings algorithm	267	(6)
9.7 Introduction to WinBUGS	273	(5)
9.8 Generalized linear models	278	(3)
9.9 Exercises	281	(4)
Appendix A: Common statistical distributions	285	(22)
A.1 Normal distribution	286	(1)
A.2 Chi-squared distribution	286	(1)
A.3 Normal approximation to chi-squared	287	(1)
A.4 Gamma distribution	287	(1)
A.5 Inverse chi-squared distribution	288	(1)
A.6 Inverse chi distribution	289	(1)
A.7 Log chi-squared distribution	290	(1)
A.8 Student's t distribution	291	(1)
A.9 Normal/chi-squared distribution	291	(1)
A.10 Beta distribution	292	(1)
A.11 Binomial distribution	293	(1)
A.12 Poisson distribution	294	(1)
A.13 Negative binomial distribution	295	(1)
A.14 Hypergeometric distribution	296	(1)
A.15 Uniform distribution	296	(1)
A.16 Pareto distribution	297	(2)
A.17 Circular normal distribution	299	(1)
A.18 Behrens' distribution	300	(1)
A.19 Snedecor's F distribution	301	(1)
A.20 Fisher's z distribution	302	(1)
A.21 Cauchy distribution	303	(1)
A.22 The probability that one beta variable is greater than another	303	(1)
A.23 Bivariate normal distribution	304	(1)
A.24 Multivariate normal distribution	304	(1)
A.25 Distribution of the correlation coefficient	305	(2)
Appendix B: Tables	307	(22)
B.1 Percentage points of the Behrens-Fisher distribution	308	(2)
B.2 HDRs for the chi-squared distribution	310	(2)
B.3 HDRs for the inverse chi-squared distribution	312	(2)
B.4 Chi-squared corresponding to HDRs for log chi-squared	314	(2)
B.5 Values of F corresponding to HDRs for log F	316	(13)
Appendix C: R programs	329	(6)
Appendix D: Further reading	335	(2)
References	337	(10)
Index	347