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December 2020 Robust Bayes-like estimation: Rho-Bayes estimation
Yannick Baraud, Lucien Birgé
Ann. Statist. 48(6): 3699-3720 (December 2020). DOI: 10.1214/20-AOS1948

Abstract

We observe $n$ independent random variables with joint distribution ${\mathbf{P}}$ and pretend that they are i.i.d. with some common density $s$ (with respect to a known measure $\mu $) that we wish to estimate. We consider a density model $\overline{S}$ for $s$ that we endow with a prior distribution $\pi $ (with support in $\overline{S}$) and build a robust alternative to the classical Bayes posterior distribution which possesses similar concentration properties around $s$ whenever the data are truly i.i.d. and their density $s$ belongs to the model $\overline{S}$. Furthermore, in this case, the Hellinger distance between the classical and the robust posterior distributions tends to 0, as the number of observations tends to infinity, under suitable assumptions on the model and the prior. However, unlike what happens with the classical Bayes posterior distribution, we show that the concentration properties of this new posterior distribution are still preserved when the model is misspecified or when the data are not i.i.d. but the marginal densities of their joint distribution are close enough in Hellinger distance to the model $\overline{S}$.

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Yannick Baraud. Lucien Birgé. "Robust Bayes-like estimation: Rho-Bayes estimation." Ann. Statist. 48 (6) 3699 - 3720, December 2020. https://doi.org/10.1214/20-AOS1948

Information

Received: 1 November 2017; Revised: 1 October 2019; Published: December 2020
First available in Project Euclid: 11 December 2020

Digital Object Identifier: 10.1214/20-AOS1948

Subjects:
Primary: 62C20, 62F15, 62F99, 62G05, 62G07, 62G35

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 6 • December 2020
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