The Annals of Statistics

Posterior Distribution of a Dirichlet Process from Quantal Response Data

P. K. Bhattacharya

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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.

Article information

Ann. Statist., Volume 9, Number 4 (1981), 803-811.

First available in Project Euclid: 12 April 2007

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Primary: 62E20: Asymptotic distribution theory
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 62G99: None of the above, but in this section 62P10: Applications to biology and medical sciences

Dirichlet process mixtures quantal response data asymptotic posterior distribution Markov chain weak convergence stochastic differential equation


Bhattacharya, P. K. Posterior Distribution of a Dirichlet Process from Quantal Response Data. Ann. Statist. 9 (1981), no. 4, 803--811. doi:10.1214/aos/1176345520.

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