Cluster Analysis, Model Selection, and Prior Distributions on Models

Clustering is an important and challenging statistical problem for which there is an extensive literature. Modeling approaches include mixture models and product partition models. Here we develop a product partition model and a model selection procedure based on Bayes factors from intrinsic priors. We also find that the choice of the prior on model space is of utmost importance, almost overshadowing the other parts of the clustering problem, and we examine the behavior of posterior odds based on different model space priors. We find, somewhat surprisingly, that procedures based on the often-used uniform prior (in which all models are given the same prior probability) lead to inconsistent model selection procedures. We examine other priors, and find that a new prior, the hierarchical uniform prior leads to consistent model selection procedures and has other desirable properties. Lastly, we examine our procedures, and competitors, on a range of examples.