Posterior Distribution of a Dirichlet Process from Quantal Response Data

Abstract

The posterior distribution of a Dirichlet process from $N$ quantal responses at $r$ dosage levels has been recognized by Antoniak as a mixture of Dirichlet processes. The purpose of this paper is to develop a systematic procedure for computing finite-dimensional distributions of such mixtures which can be equivalently expressed as multivariate beta distributions with random parameter vectors. It is shown that the sequence of random parameter vectors of the updated beta posteriors from observations at increasing dosage levels evolves in a manner which is described by $r$ separate Markov chains. This description is then used to derive the asymptotic posterior distribution. The weak limits of the relevant Markov chains are shown to be solutions of certain stochastic differential equations and the random parameter vector of the posterior beta distribution is shown to be asymptotically normal, the mean vector and covariance matrix of which are given by recursion formulas.