Random partition models with regression on covariates

Many recent applications of nonparametric Bayesian inference use random partition models, i.e. probability models for clustering a set of experimental units. We review the popular basic constructions. We then focus on an interesting extension of such models. In many applications covariates are available that could be used to a priori inform the clustering. This leads to random clustering models indexed by covariates, i.e., regression models with the outcome being a partition of the experimental units. We discuss some alternative approaches that have been used in the recent literature to implement such models, with an emphasis on a recently proposed extension of product partition models. Several of the reviewed approaches were not originally intended as covariate-based random partition models, but can be used for such inference.