Finite-dimensional Discrete Random Structures and Bayesian Clustering
Antonio Lijoi, Igor Prünster, Tommaso Rigon
Abstract
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.