9780387955377

Summary

Bayesian nonparametrics has grown tremendously in the last three decades, especially in the last few years. This book is the first systematic treatment of Bayesian nonparametric methods and the theory behind them. While the book is of special interest to Bayesians, it will also appeal to statisticians in general because Bayesian nonparametrics offers a whole continuous spectrum of robust alternatives to purely parametric and purely nonparametric methods of classical statistics. The book is primarily aimed at graduate students and can be used as the text for a graduate course in Bayesian nonparametrics. Though the emphasis of the book is on nonparametrics, there is a substantial chapter onasymptotics of classical Bayesian parametric models.Jayanta Ghosh has been Director and Jawaharlal Nehru Professor at the Indian Statistical Institute and President of the International Statistical Institute. He is currently professor of statistics at Purdue University. He has been editor of Sankhya and served on the editorial boards of several journals including the Annals of Statistics. Apart from Bayesian analysis, his interests include asymptotics, stochastic modeling, high dimensional model selection, reliability and survival analysis and bioinformatics.R.V. Ramamoorthi is professor at the Department of Statistics and Probability at Michigan State University. He has published papers in the areas of sufficiency invariance, comparison of experiments, nonparametric survival analysis and Bayesian analysis. In addition to Bayesian nonparametrics, he is currently interested in Bayesian networks and graphical models. He is on the editorial board of Sankhya.

Author Biography

R. V. Ramamoorthi is a professor at the Department of Statistics and Probability at Michigan State University.

Introduction: Why Bayesian Nonparametrics-An Overview and Sum-mary	p. 1
Preliminaries and the Finite Dimensional Case	p. 9
Introduction	p. 9
Metric Spaces	p. 10
preliminaries	p. 10
Weak Convergence	p. 12
Posterior Distribution and Consistency	p. 15
Preliminaries	p. 15
Posterior Consistency and Posterior Robustness	p. 18
Doob's Theorem	p. 22
Wald-Type Conditions	p. 24
Asymptotic Normality of MLE and Bernstein-von Mises Theorem	p. 33
Ibragimov and Hasminski&ibreve; Conditions	p. 41
Nonsubjective Priors	p. 46
Fully Specified	p. 46
Discussion	p. 52
Conjugate and Hierarchical Priors	p. 52
Exchangeability, De Finetti's Theorem, Exponential Families	p. 54
<$>M({\cal X})<$> and Priors on <$>M({\cal X})<$>	p. 57
Introduction	p. 57
The Space M(X)	p. 58
(Prior) Probability Measures on <$>M({\cal X})<$>	p. 62
<$>{\cal X}<$> Finite	p. 62
<$>{\cal X} = {\op R}<$>	p. 64
Tail Free Priors	p. 70
Tail Free Priors and 0-1 Laws	p. 75
Space of Probability Measures on <$>M({\op R})<$>	p. 78
De Finetti's Theorem	p. 83
Dirichlet and Polya tree process	p. 87
Dirichlet and Polya tree process	p. 87
Finite Dimensional Dirichlet Distribution	p. 87
Dirichlet Distribution via Polya Urn Scheme	p. 94
Dirichlet Process on <$>M({\op R})<$>	p. 96
Construction and Properties	p. 96
The Sethuraman Construction	p. 103
Support of D_¿	p. 104
Convergence Properties of D_¿	p. 105
Elicitation and Some Applications	p. 107
Mutual Singularity of Dirichlet Priors	p. 110
Mixtures of Dirichlet Process	p. 113
Polya Tree Process	p. 114
The Finite Case	p. 114
<$>{\cal X} = {\op R}<$>	p. 116
Consistency Theorems	p. 121
Introduction	p. 121
Preliminaries	p. 122
Finite and Tail free case	p. 124
Posterior Consistency on Densities	p. 126
Schwartz Theorem	p. 126
L₁-Consistency	p. 132
Consistency via LeCam's inequality	p. 137
Density Estimation	p. 141
Introduction	p. 141
Polya Tree Priors	p. 142
Mixtures of Kernels	p. 143
Hierarchical Mixtures	p. 147
Random Histograms	p. 148
Weak Consistency	p. 150
L₁-Consistency	p. 156
Mixtures of Normal Kernel	p. 161
Dirichlet Mixtures: Weak Consistency	p. 161
Dirichlet Mixtures: L₁-Consistency	p. 169
Extensions	p. 172
Gaussian Process Priors	p. 174
Inference for Location Parameter	p. 181
Introduction	p. 181
The Diaconis-Freedman Example	p. 182
Consistency of the Posterior	p. 185
Polya Tree Priors	p. 189
Regression Problems	p. 197
Introduction	p. 197
Schwartz Theorem	p. 198
Exponentially Consistent Tests	p. 201
Prior Positivity of Neighborhoods	p. 206
Polya Tree Priors	p. 208
Dirichlet Mixture of Normals	p. 209
Binary Response Regression with Unknown Link	p. 212
Stochastic Regressor	p. 215
Simulations	p. 215
Uniform Distribution on Infinite-Dimensional Spaces	p. 221
Introduction	p. 221
Towards a Uniform Distribution	p. 222
The Jeffreys Prior	p. 222
Uniform Distribution via Sieves and Packing Numbers	p. 223
Technical Preliminaries	p. 224
The Jeffreys Prior Revisited	p. 225
Posterior Consistency for Noninformative Priors for Infinite-Dimensional Problems	p. 229
Convergence of Posterior at Optimal Rate	p. 231
Survival Analysis-Dirichlet Priors	p. 237
Introduction	p. 237
Dirichlet Prior	p. 238
Cumulative Hazard Function, Identifiability	p. 242
Priors via Distributions of (Z, ¿)	p. 247
Interval Censored Data	p. 249
Neutral to the Right Priors	p. 253
Introduction	p. 253
Neutral to the Right Priors	p. 254
Independent Increment Processes	p. 258
Basic Properties	p. 262
Beta Processes	p. 265
Definition and Construction	p. 265
Properties	p. 268
Posterior Consistency	p. 271
Exercises	p. 281
References	p. 285
Index	p. 300
Table of Contents provided by Publisher. All Rights Reserved.

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Bayesian Nonparametrics

0387955372

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Summary

Author Biography

Table of Contents

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