In several statistical applications, observations are naturally divided into groups, e.g., multi-arm clinical trials or students’ assessments in different schools. Bayesian nonparametrics offers a flexible framework that can be employed to model the heterogeneity of the groups and, at the same time, make inference more robust thanks to the well known “borrowing of strength”. 

Within this context, I will discuss the Semi-Hierarchical Dirichlet Process (semiHDP), a novel nonparametric prior for partially exchangeable data. Moreover, I will show that embedding the semiHDP in a random partition model allows to formally test homogeneity between distributions when two groups of data are present and, when more than two such groups are under investigation, cluster groups of data into homogeneous populations.

Several aspects concerning the prior, the asymptotics of the Bayes Factors for the homogeneity test, and the MCMC algorithms for posterior inference will be considered.

Finally, I will discuss alternative models that allow to scale inference to large number of groups.