Finite-dimensional discrete random structures and Bayesian clustering
Discrete random probability measures stand out as effective tools for Bayesian clustering. The investigation in the area has been very lively, with a strong emphasis on nonparametric procedures based on either the Dirichlet process or on more flexible generalizations, such as the normalized random measures with independent increments (NRMI). The literature on finite-dimensional discrete priors is much more limited and mostly confined to the standard Dirichlet-multinomial model. While such a specification may be attractive due to conjugacy, it suffers from considerable limitations when it comes to addressing clustering problems. In order to overcome these, we introduce a novel class of priors that arise as the hierarchical compositions of finite-dimensional random discrete structures. Despite the analytical hurdles such a construction entails, we are able to characterize the induced random partition and determine explicit expressions of the associated urn scheme and of the posterior distribution. A detailed comparison with (infinite-dimensional) NRMIs is also provided: indeed, informative bounds for the discrepancy between the partition laws are obtained. Finally, the performance of our proposal over existing methods is assessed on a real application where we study a publicly available dataset from the Italian education system comprising the scores of a mandatory nationwide test.