Inner spike and slab Bayesian nonparametric models

Abstract

Discrete Bayesian nonparametric models whose expectation is a convex linear combination of a point mass at some point of the support and a diffuse probability distribution allow to incorporate strong prior information, while still being extremely flexible. Recent contributions in the statistical literature have successfully implemented such a modelling strategy in a variety of applications, including density estimation, nonparametric regression and model-based clustering. A thorough study is presented on a large class of nonparametric models, named inner spike and slab hNRMI models and obtained by considering homogeneous normalized random measures with independent increments (hNRMI) with base measure given by a convex linear combination of a point mass and a diffuse probability distribution. In turn, the distributional properties of these models are investigated, with focus on: i) the exchangeable partition probability function they induce, ii) the distribution of the number of distinct values in an exchangeable sample, iii) the posterior predictive distribution, and iv) the distribution of the number of elements that coincide with the only point of the support with positive probability. These theoretical findings represent the main building block for an actual implementation of Bayesian inner spike and slab hNRMI models by means of a generalized Pólya urn scheme.