Finite-dimensional Discrete Random Structures and Bayesian Clustering

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.

KEYWORDS:

• Bayesian nonparametrics
• Normalized random measures
• Random partitions
• Urn schemes

Supplementary Material

Proofs, algorithms, and simulations A manuscript including detailed proofs, the description of the sampling algorithms, additional plots about the INVALSI application, and the results of an extensive simulation study, is available as supplementary material.

Notes

1 The documentation (in Italian) is available at: https://invalsi-serviziostatistico.cineca.it