BAYESIAN AND LIKELIHOOD INFERENCE FROM EQUALLY WEIGHTED MIXTURES

TOM LEONARD^{1}, JOHN S. J. HSU^{2}, KAM-WAH TSUI^{1} AND JAMES F. MURRAY^{3}

1Department of Statistics, University of Wisconsin-Madison, 1210 West Dayton Street, Madison, WI 53706-1693, U.S.A.

2Department of Statistics and Applied Probability, University of California - Santa Barbara,

Santa Barbara, CA 93106-3110, U.S.A.

3Graduate Program in Hospital and Health Administration, University of Iowa, Iowa City, IA 52242, U.S.A.

(Received September 28, 1992; revised July 27, 1993)

Abstract. Equally weighted mixture models are recommended for situations where it is required to draw precise finite sample inferences requiring population parameters, but where the population distribution is not constrained to belong to a simple parametric family. They lead to an alternative procedure to the Laird-DerSimonian maximum likelihood algorithm for unequally weighted mixture models. Their primary purpose lies in the facilitation of exact Bayesian computations via importance sampling. Under very general sampling and prior specifications, exact Bayesian computations can be based upon an application of importance sampling, referred to as Permutable Bayesian Marginalization (PBM). An importance function based upon a truncated multivariate t-distribution is proposed, which refers to a generalization of the maximum likelihood procedure. The estimation of discrete distributions, by binomial mixtures, and inference for survivor distributions, via mixtures of exponential or Weibull distributions, are considered. Equally weighted mixture models are also shown to lead to an alternative Gibbs sampling methodology to the Lavine-West approach.

Key words and phrases: Equally weighted mixtures, survivor distribution, maximum likelihood, EM algorithm, binomial mixtures, Bayesian marginalization, importance sampling, Gibbs sampler.