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February 2017 Distances between nested densities and a measure of the impact of the prior in Bayesian statistics
Christophe Ley, Gesine Reinert, Yvik Swan
Ann. Appl. Probab. 27(1): 216-241 (February 2017). DOI: 10.1214/16-AAP1202

Abstract

In this paper, we propose tight upper and lower bounds for the Wasserstein distance between any two univariate continuous distributions with probability densities $p_{1}$ and $p_{2}$ having nested supports. These explicit bounds are expressed in terms of the derivative of the likelihood ratio $p_{1}/p_{2}$ as well as the Stein kernel $\tau_{1}$ of $p_{1}$. The method of proof relies on a new variant of Stein’s method which manipulates Stein operators.

We give several applications of these bounds. Our main application is in Bayesian statistics: we derive explicit data-driven bounds on the Wasserstein distance between the posterior distribution based on a given prior and the no-prior posterior based uniquely on the sampling distribution. This is the first finite sample result confirming the well-known fact that with well-identified parameters and large sample sizes, reasonable choices of prior distributions will have only minor effects on posterior inferences if the data are benign.

Citation

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Christophe Ley. Gesine Reinert. Yvik Swan. "Distances between nested densities and a measure of the impact of the prior in Bayesian statistics." Ann. Appl. Probab. 27 (1) 216 - 241, February 2017. https://doi.org/10.1214/16-AAP1202

Information

Received: 1 October 2015; Revised: 1 April 2016; Published: February 2017
First available in Project Euclid: 6 March 2017

zbMATH: 1381.60065
MathSciNet: MR3619787
Digital Object Identifier: 10.1214/16-AAP1202

Subjects:
Primary: 60E15
Secondary: 62F15

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.27 • No. 1 • February 2017
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