Transform-scaled process priors for trait allocations in Bayesian nonparametrics

Abstract

Completely random measures (CRMs) provide a broad class of priors, arguably, the most popular, for Bayesian nonparametric (BNP) analysis of trait allocations. As a peculiar property, CRM priors lead to predictive distributions that share the following common structure: for fixed prior's parameters, a new data point exhibits a Poisson (random) number of ``new'' traits, i.e., not appearing in the sample, which depends on the sampling information only through the sample size. While the Poisson posterior distribution is appealing for analytical tractability and ease of interpretation, its independence from the sampling information is a critical drawback, as it makes the posterior distribution of ``new'' traits completely determined by the estimation of the unknown prior's parameters. In this paper, we introduce the class of transform-scaled process (T-SP) priors as a tool to enrich the posterior distribution of ``new'' traits arising from CRM priors, while maintaining the same analytical tractability and ease of interpretation. In particular, we present a framework for posterior analysis of trait allocations under T-SP priors, showing that Stable T-SP priors, i.e., T-SP priors built from Stable CRMs, lead to predictive distributions such that, for fixed prior's parameters, a new data point displays a negative-Binomial (random) number of ``new'' traits, which depends on the sampling information through the number of distinct traits and the sample size. Then, by relying on a hierarchical version of T-SP priors, we extend our analysis to the more general setting of trait allocations with multiple groups of data or subpopulations. The empirical effectiveness of our methods is demonstrated through numerical experiments and applications to real data.