Bayesian analysis of product feature allocation models

TL;DR

This paper develops a comprehensive theoretical framework for Bayesian feature allocation models, including predictive and posterior analysis, with applications to ecology and species richness estimation.

Contribution

It introduces a general theory for product-form feature models, deriving closed-form predictive and posterior distributions, and explores novel examples like mixtures of Indian buffet processes.

Findings

Closed-form expressions for predictive distributions and posterior laws.

Application to ecological data for estimating species richness.

Analysis of models with finite and infinite features.

Abstract

Feature allocation models are an extension of Bayesian nonparametric clustering models, where individuals can share multiple features. We study a broad class of models whose probability distribution has a product form, which includes the popular Indian buffet process. This class plays a prominent role among existing priors, and it shares structural characteristics with Gibbs-type priors in the species sampling framework. We develop a general theory for the entire class, obtaining closed form expressions for the predictive structure and the posterior law of the underlying stochastic process. Additionally, we describe the distribution for the number of features and the number of hitherto unseen features in a future sample, leading to the $\alpha$-diversity for feature models. We also examine notable novel examples, such as mixtures of Indian buffet processes and beta Bernoulli models, where the latter entails a finite random number of features. This methodology finds significant applications in ecology, allowing the estimation of species richness for incidence data, as we demonstrate by analyzing plant diversity in Danish forests and trees in Barro Colorado Island.