MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy

TL;DR

This paper introduces MMD-Bayes, a robust Bayesian estimation method using Maximum Mean Discrepancy, which improves robustness and consistency under model misspecification, supported by theoretical proofs and numerical simulations.

Contribution

It proposes a novel MMD-based pseudo-likelihood for Bayesian inference that is both consistent and robust to model misspecification, with practical variational approximation algorithms.

Findings

MMD-Bayes is consistent under misspecification.

The method demonstrates robustness in numerical simulations.

Variational approximations retain robustness and consistency.

Abstract

In some misspecified settings, the posterior distribution in Bayesian statistics may lead to inconsistent estimates. To fix this issue, it has been suggested to replace the likelihood by a pseudo-likelihood, that is the exponential of a loss function enjoying suitable robustness properties. In this paper, we build a pseudo-likelihood based on the Maximum Mean Discrepancy, defined via an embedding of probability distributions into a reproducing kernel Hilbert space. We show that this MMD-Bayes posterior is consistent and robust to model misspecification. As the posterior obtained in this way might be intractable, we also prove that reasonable variational approximations of this posterior enjoy the same properties. We provide details on a stochastic gradient algorithm to compute these variational approximations. Numerical simulations indeed suggest that our estimator is more robust to misspecification than the ones based on the likelihood.