Introduction

Hierarchical models have played a central role in Bayesian inference since the time of Good (1965) with Lindley and Smith (1972) a key milestone providing the first comprehensive treatment of hierarchical priors for the parametric Bayes linear model. The Bayesian hierarchies have proved so popular because they provide a natural framework for “borrowing of strength” (a term apparently due to Tukey) or sharing partial information across components through the hierarchical structure. The Bayesian construction also provides a clear distinction from frequentist models in that the dependence structures do not have to be directly related to population random effects.

In this chapter we will look a little more closely at a couple of key issues emerging from the work of Teh and Jordan. First, we briefly step back a little and consider the fundamental notion played by de Finetti's representation theorem, reiterating the operational focus of Bayesian modeling on finite-dimensional specification of joint distributions on observables.