Posteriors, conjugacy, and exponential families for completely random measures
Name
1410.6843.pdf
Description
Submitted version
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425.91 KB
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Adobe PDF
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Author(s)
Broderick, Tamara A
Wilson, Ashia
Jordan, Michael
Date Issued
November 2018
Journal
Bernoulli
Publisher
Bernoulli Society for Mathematical Statistics and Probability
Citation
Broderick, Tamara et al. "Posteriors, conjugacy, and exponential families for completely random measures." Bernoulli 24, 4B (November 2018): 3181-3221 © 2018 ISI/BS
Version
Original manuscript
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.
MIT Department
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.3150/16-bej855
