A Technique for Estimating Marginal Posterior Densities in Hierarchical Models Using Mixtures of Conditional Densities

Abstract

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 formulas 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.

Key Words:

• Bayesian inference
• Dynamic linear model
• Empirical Bayes
• Kalman filter
• Quantile integration
• State-space model