Hierarchical Bayesian Analysis using Monte Carlo Integration: Computing Posterior Distributions when

A Bayesian approach allows the statistician to compute the posterior probability for each model in a set of possible models and therefore to retain consideration of several or many models throughout the analysis rather than to restrict attention to just one 'best' model. This paper illustrates the use of Monte Carlo integration in hierarchical Bayesian analysis when there are many possible models of different dimensions. A hierarchical approach can facilitate the choice of a prior distribution as well as the Monte Carlo computation. The methodology is illustrated by an example in multiple logistic regression involving 256 possible models.