MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy
This addresses robustness issues in Bayesian estimation for statisticians and practitioners dealing with misspecified models, representing a novel method for a known bottleneck rather than a foundational shift.
The paper tackles the problem of inconsistent Bayesian estimates under model misspecification by proposing MMD-Bayes, a method that replaces the likelihood with a pseudo-likelihood based on Maximum Mean Discrepancy, showing it yields consistent and robust posterior estimates with numerical simulations indicating improved robustness.
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.