Bayesian model selection when the number of components is unknown

In simulation modeling and analysis, there are two situations where there is uncertainty about the number of parameters needed to specify a model. The first is in input modeling where real data is being used to fit a finite mixture model and where there is uncertainty about the number of components in the mixture. Secondly, at the output analysis stage, it may be that a regression model is to be fitted to the simulation output, where the number of terms, and hence the number of parameters, is unknown. In statistical terms, such problems are non-standard and require special handling. One way is to use a Bayesian Markov Chain Monte Carlo (MCMC) analysis. Such a method has been suggested by George and McCulloch (1993) using a hierarchical Bayesian model. This method is flexible, but does introduce many additional parameters. This tends to make the modelling look rather complicated. In this paper we adopt a classical Bayesian approach that is essentially equivalent to the George and McCulloch technique, but that has a much less elaborate structure and which renders model interpretation much simpler. The method is illustrated by a regression metamodel example