From robust tests to Bayes-like posterior distributions

TL;DR

This paper introduces a new class of Bayesian-like posterior distributions based on total variation, Hellinger, and L_j distances, demonstrating their robustness and concentration properties under model misspecification and non-i.i.d. data, with applications in nonparametric and high-dimensional density estimation.

Contribution

It proposes a novel posterior construction extending classical Bayes, with proven robustness and concentration properties, applicable to non-i.i.d. data and high-dimensional models.

Findings

New posterior distributions concentrate near the true law with high probability.

Robustness to prior misspecification and non-i.i.d. data is established.

Applications include nonparametric density estimation and high-dimensional sparse models.

Abstract

In the Bayes paradigm and for a given loss function, we propose the construction of a new type of posterior distributions, that extends the classical Bayes one, for estimating the law of an $n$-sample. The loss functions we have in mind are based on the total variation and Hellinger distances as well as some $\mathbb{L}_{j}$-ones. We prove that, with a probability close to one, this new posterior distribution concentrates its mass in a neighbourhood of the law of the data, for the chosen loss function, provided that this law belongs to the support of the prior or, at least, lies close enough to it. We therefore establish that the new posterior distribution enjoys some robustness properties with respect to a possible misspecification of the prior, or more precisely, its support. For the total variation and squared Hellinger losses, we also show that the posterior distribution keeps its concentration properties when the data are only independent, hence not necessarily i.i.d., provided that most of their marginals or the average of these are close enough to some probability distribution around which the prior puts enough mass. The posterior distribution is therefore also stable with respect to the equidistribution assumption. We illustrate these results by several applications. We consider the problems of estimating a location parameter or both the location and the scale of a density in a nonparametric framework. Finally, we also tackle the problem of estimating a density, with the squared Hellinger loss, in a high-dimensional parametric model under some sparsity conditions. The results established in this paper are non-asymptotic and provide, as much as possible, explicit constants.