Quantifying the uncertainty of partitions for infinite mixture models

Bayesian clustering models, such as Dirichlet process mixture models (DPMMs), are sophisticated flexible models. They induce a posterior distribution on the set of all partitions of a set of observations. Analysing this posterior distribution is of great interest, but it comes with several challenges. First of all, the number of partitions is overwhelmingly large even for moderate values of the number of observations. Consequently the sample space of the posterior distribution of the partitions is not explored well by MCMC samplers. Second, due to the complexity of representing the uncertainty of partitions, usually only maximum a posteriori estimates of the posterior distribution of partitions are provided and discussed in the literature. In this paper we propose a numerical and graphical method for quantifying the uncertainty of the clusters of a given partition of the data and we suggest how this tool can be used to learn about the partition uncertainty.