Summary: We consider the empirical Bayes estimation of a distribution using binary data via the Dirichlet process. Let denote a Dirichlet process with being a finite measure on . Instead of having direct samples from an unknown random distribution from , 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 , is found when the shape of 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.