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BAYESIAN AND LIKELIHOOD INFERENCE

FROM EQUALLY WEIGHTED MIXTURES

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

^{1} *Department of Statistics, University of Wisconsin-Madison,*

1210 West Dayton Street, Madison, WI 53706-1693, U.S.A.

^{2} *Department of Statistics and Applied Probability, University of California - Santa Barbara,*

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

^{3} *Graduate 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.

**Source**
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