Improved criteria for clustering based on the posterior similarity matrix

In this paper we address the problem of obtaining a single clustering estimate bc based on an MCMC sample of clusterings c (1) ;c (2) :::;c (M) from the posterior distribution of a Bayesian cluster model. Methods to derive b when the number of groups K varies between the clusterings are reviewed and discussed. These include the maximum a posteriori (MAP) estimate and methods based on the posterior similarity matrix, a matrix containing the posterior probabilities that the observations i and j are in the same cluster. The posterior similarity matrix is related to a commonly used loss function by Binder (1978). Minimization of the loss is shown to be equivalent to maximizing the Rand index between esti- mated and true clustering. We propose new criteria for estimating a clustering, which are based on the posterior expected adjusted Rand index. The criteria are shown to possess a shrinkage property and outperform Binder's loss in a simulation study and in an application to gene expression data. They also perform favorably compared to other clustering procedures.