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November 2018 Posteriors, conjugacy, and exponential families for completely random measures
Tamara Broderick, Ashia C. Wilson, Michael I. Jordan
Bernoulli 24(4B): 3181-3221 (November 2018). DOI: 10.3150/16-BEJ855

Abstract

We demonstrate how to calculate posteriors for general Bayesian nonparametric priors and likelihoods based on completely random measures (CRMs). 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.

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Tamara Broderick. Ashia C. Wilson. Michael I. Jordan. "Posteriors, conjugacy, and exponential families for completely random measures." Bernoulli 24 (4B) 3181 - 3221, November 2018. https://doi.org/10.3150/16-BEJ855

Information

Received: 1 October 2014; Revised: 1 May 2016; Published: November 2018
First available in Project Euclid: 18 April 2018

zbMATH: 06869874
MathSciNet: MR3788171
Digital Object Identifier: 10.3150/16-BEJ855

Keywords: Bayesian nonparametrics , beta process , completely random measure , conjugacy , exponential family , Indian buffet process , Posterior , size-biased

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

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Vol.24 • No. 4B • November 2018
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