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August 2020 Convergence rates of variational posterior distributions
Fengshuo Zhang, Chao Gao
Ann. Statist. 48(4): 2180-2207 (August 2020). DOI: 10.1214/19-AOS1883

Abstract

We study convergence rates of variational posterior distributions for nonparametric and high-dimensional inference. We formulate general conditions on prior, likelihood and variational class that characterize the convergence rates. Under similar “prior mass and testing” conditions considered in the literature, the rate is found to be the sum of two terms. The first term stands for the convergence rate of the true posterior distribution, and the second term is contributed by the variational approximation error. For a class of priors that admit the structure of a mixture of product measures, we propose a novel prior mass condition, under which the variational approximation error of the mean-field class is dominated by convergence rate of the true posterior. We demonstrate the applicability of our general results for various models, prior distributions and variational classes by deriving convergence rates of the corresponding variational posteriors.

Citation

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Fengshuo Zhang. Chao Gao. "Convergence rates of variational posterior distributions." Ann. Statist. 48 (4) 2180 - 2207, August 2020. https://doi.org/10.1214/19-AOS1883

Information

Received: 1 February 2018; Revised: 1 June 2019; Published: August 2020
First available in Project Euclid: 14 August 2020

MathSciNet: MR4134791
Digital Object Identifier: 10.1214/19-AOS1883

Subjects:
Primary: 62C10
Secondary: 62F15

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 4 • August 2020
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