MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy

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.

References in corpus (3)

• Generative Moment Matching Networks
• Statistical Inference for Generative Models with Maximum Mean Discrepancy
• Convergence Rates of Variational Inference in Sparse Deep Learning

Cited by in corpus (7)

• Finite sample properties of parametric MMD estimation: robustness to misspecification and dependence
• Bayesian Neural Networks With Maximum Mean Discrepancy Regularization
• Learning under Model Misspecification: Applications to Variational and Ensemble methods
• Generalized Bayesian Likelihood-Free Inference
• Estimation of copulas via Maximum Mean Discrepancy
• Direct loss minimization algorithms for sparse Gaussian processes
• On Robust Pseudo-Bayes Estimation for the Independent Non-homogeneous Set-up