Posteriors, conjugacy, and exponential families for completely random measures

TL;DR

This paper develops a unified framework for Bayesian nonparametric models using completely random measures, introducing exponential CRMs for conjugacy, and providing new representations and conjugate priors for specific processes.

Contribution

It introduces exponential CRMs for conjugate priors, and provides new size-biased and marginal representations for Bayesian nonparametric models.

Findings

Gamma process is conjugate prior for Poisson likelihood process

Beta prime process is conjugate prior for odds Bernoulli process

Provides explicit size-biased and marginal representations

Abstract

We demonstrate how to calculate posteriors for general CRM-based priors and likelihoods for Bayesian nonparametric models. We further show how to represent Bayesian nonparametric priors as a sequence of finite draws using a size-biasing approach---and how to represent full Bayesian nonparametric models via finite marginals. Motivated by conjugate priors based on exponential family representations of likelihoods, we introduce a notion of exponential families for CRMs, which we call exponential CRMs. This construction allows us to specify automatic Bayesian nonparametric conjugate priors for exponential CRM likelihoods. We demonstrate that our exponential CRMs allow particularly straightforward recipes for size-biased and marginal representations of Bayesian nonparametric models. Along the way, we prove that the gamma process is a conjugate prior for the Poisson likelihood process and the beta prime process is a conjugate prior for a process we call the odds Bernoulli process. We deliver a size-biased representation of the gamma process and a marginal representation of the gamma process coupled with a Poisson likelihood process.