A Technique for Estimating Marginal Posterior Densities in Hierarchical Models Using Mixtures of Conditional Densities (Revision of DP# 89-07)

Authors: 
Valen E. Johnson
Duke University

Nov 30 1990

A technique called quantile integration is proposed for the estimation of marginal posterior densities arising in Bayesian models having hierarchical representations. The method is based on approximating marginal densities as mixtures of conditional densities, where the conditioning variables are selected deterministically from the mixing distributions. The form of the approximation makes it easy to implement, and the resulting approximations are computationally efficient to obtain. The technique leads to particularly simple approximations for the predictive and posterior densities in Kalman filter or state space models, and specific formulae are provided for the special case in which innovations belong to location-scale families. Other applications include a hierarchical empirical Bayes model for Poisson rates and a hierarchical linear model with exchangeable regression parameters and unknown variance components.

Keywords: 

Bayesian inference, quantile integration, Kalman filters, dynamic linear models, state space models, empirical Bayes

Manuscript: 

PDF icon 1991-04.pdf