Inner spike and slab Bayesian nonparametric models

TL;DR

This paper introduces a new class of Bayesian nonparametric models called inner spike and slab hNRMI models, which combine point masses and diffuse distributions to incorporate prior knowledge while maintaining flexibility.

Contribution

It provides a detailed analysis of the properties of inner spike and slab hNRMI models, including their induced partition structures and predictive distributions.

Findings

Derived the exchangeable partition probability function for these models

Characterized the distribution of the number of distinct values in samples

Developed a generalized Pólya urn scheme for implementation

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. We provide a thorough study of a large class of nonparametric models we call inner spike and slab hNRMI models, which are 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 this paper we investigate the distributional properties of these models and our results include: 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. Our findings are the main building block for an actual implementation of Bayesian inner spike and slab hNRMI models by means of a generalized P\'olya urn scheme.