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June 1996 Nonparametric hierarchical Bayes via sequential imputations
Jun S. Liu
Ann. Statist. 24(3): 911-930 (June 1996). DOI: 10.1214/aos/1032526949


We consider the empirical Bayes estimation of a distribution using binary data via the Dirichlet process. Let $\mathscr{D}(\alpha)$ denote a Dirichlet process with $\alpha$ being a finite measure on Instead of having direct samples from an unknown random distribution F from $\mathscr{D}(\alpha)$, we assume that only indirect binomial data are observable. This paper presents a new interpretation of Lo's formula, and thereby relates the predictive density of the observations based on a Dirichlet process model to likelihoods of much simpler models. As a consequence, the log-likelihood surface, as well as the maximum likelihood estimate of $c = \alpha([0, 1])$, is found when the shape of $\alpha$ a is assumed known, together with a formula for the Fisher information evaluated at the estimate. The sequential imputation method of Kong, Liu and Wong is recommended for overcoming computational difficulties commonly encountered in this area. The related approximation formulas are provided. An analysis of the tack data of Beckett and Diaconis, which motivated this study, is supplemented to illustrate our methods.


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Jun S. Liu. "Nonparametric hierarchical Bayes via sequential imputations." Ann. Statist. 24 (3) 911 - 930, June 1996.


Published: June 1996
First available in Project Euclid: 20 September 2002

zbMATH: 0880.62038
MathSciNet: MR1401830
Digital Object Identifier: 10.1214/aos/1032526949

Primary: 62G05
Secondary: 62E25 , 65U05

Keywords: Dirichlet process , Empirical Bayes , Gibbs sampler , importance sampling , Pólya urn , sensitivity analysis

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 3 • June 1996
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