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February 2019 Bayesian fractional posteriors
Anirban Bhattacharya, Debdeep Pati, Yun Yang
Ann. Statist. 47(1): 39-66 (February 2019). DOI: 10.1214/18-AOS1712

Abstract

We consider the fractional posterior distribution that is obtained by updating a prior distribution via Bayes theorem with a fractional likelihood function, a usual likelihood function raised to a fractional power. First, we analyze the contraction property of the fractional posterior in a general misspecified framework. Our contraction results only require a prior mass condition on certain Kullback–Leibler (KL) neighborhood of the true parameter (or the KL divergence minimizer in the misspecified case), and obviate constructions of test functions and sieves commonly used in the literature for analyzing the contraction property of a regular posterior. We show through a counterexample that some condition controlling the complexity of the parameter space is necessary for the regular posterior to contract, rendering additional flexibility on the choice of the prior for the fractional posterior. Second, we derive a novel Bayesian oracle inequality based on a PAC-Bayes inequality in misspecified models. Our derivation reveals several advantages of averaging based Bayesian procedures over optimization based frequentist procedures. As an application of the Bayesian oracle inequality, we derive a sharp oracle inequality in multivariate convex regression problems. We also illustrate the theory in Gaussian process regression and density estimation problems.

Citation

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Anirban Bhattacharya. Debdeep Pati. Yun Yang. "Bayesian fractional posteriors." Ann. Statist. 47 (1) 39 - 66, February 2019. https://doi.org/10.1214/18-AOS1712

Information

Received: 1 November 2016; Revised: 1 April 2018; Published: February 2019
First available in Project Euclid: 30 November 2018

zbMATH: 07036194
MathSciNet: MR3909926
Digital Object Identifier: 10.1214/18-AOS1712

Subjects:
Primary: 62G07, 62G20
Secondary: 60K35

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 1 • February 2019
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