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Nonparametric hierarchical Bayes via sequential imputations. (English) Zbl 0880.62038

Summary: We consider the empirical Bayes estimation of a distribution using binary data via the Dirichlet process. Let $𝒟\left(\alpha \right)$ denote a Dirichlet process with $\alpha$ being a finite measure on $\left[0,1\right]$. Instead of having direct samples from an unknown random distribution $F$ from $𝒟\left(\alpha \right)$, we assume that only indirect binomial data are observable.

This paper presents a new interpretation of Lo’s formula [A. Y. Lo, ibid. 12, 351-357 (1984; Zbl 0557.62036)] 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 $c=\alpha \left(\left[0,1\right]\right)$, is found when the shape of $\alpha$ is assumed known, together with a formula for the Fisher information evaluated at the estimate.

The sequential imputation method of A. Kong, J. S. Liu and W. H. Wong [J. Am. Stat. Assoc. 89, No. 425, 278-288 (1994)] is recommended for overcoming computational difficulties commonly encountered in this area. The related approximation formulas are provided. An analysis of the tack data of L. Beckett and P. Diaconis [Adv. Math. 103, No. 1, 107-128 (1994; Zbl 0805.62085)] which motivated this study, is supplemented to illustrate our methods.

##### MSC:
 62G05 Nonparametric estimation 62C12 Empirical decision procedures; empirical Bayes procedures 65C05 Monte Carlo methods 65C99 Probabilistic methods, simulation and stochastic differential equations (numerical analysis)