Conditional linear Bayes rules for hierarchical models

Abstract

This paper proposes a new general approach to hierarchical linear Bayesian estimation. The approach is based on the notion of conditioning on hyperparameters and subsequently deriving ‘conditional’ linear Bayes rules. It is shown that the conditional linear Bayes rule is superior in terms of Bayes risk to the linear Bayes rule. A Monte-Carlo study involving Poisson counts, representing, for example, claim numbers in a motor car insurance portfolio, was undertaken. In all cases the risk function of an empirical conditional linear Bayes rule is less than the risk function of a corresponding empirical linear Bayes rule in the region of the parameter space close to zero.