AN APPLICATION OF THE LAPLACE METHOD TO FINITE MIXTURE DISTRIBUTIONS

An exact Bayesian analysis of finite mixture distributions is often computationally infeasible, because the number of terms in the posterior density grows exponentially with the sample size. A modification of the Laplace method is presented and applied to estimation of posterior functions in a Bayesian analysis of finite mixture distributions. The procedure, which involves computations similar to those required in maximum likelihood estimation, is shown to have high asymptotic accuracy for finite mixtures of certain exponential-family densities. For these mixture densities, the posterior density is also shown to be asymptotically normal. An approximation of the posterior density of the number of components is presented. The method is applied to Duncan's barley data and to a distribution of lake chemistry data for north-central Wisconsin.