On Prior-to-Posterior Stability in the Wasserstein Metric for Bayesian Inverse Problems

Abstract

Priors in Bayesian inverse problems are often approximated through discretization, hyperparameter estimation, or generative modeling. Understanding how prior approximation errors propagate to the posterior and subsequent predictions is therefore important. In this work, we study the stability of the prior-to-posterior map where both prior and posterior perturbations are measured in the same Wasserstein metric $W_p$, $p\geq1$. We identify verifiable conditions on likelihood regularity and admissible prior classes that ensure uniform, Hölder, and Lipschitz stability. For bounded likelihoods that are uniformly continuous on bounded sets, uniform stability holds over prior classes with uniformly integrable $p$-th moments and a common positive evidence lower bound. With global Hölder regularity of the likelihood and uniform bounds on higher prior moments, a coupling argument leads to a Hölder estimate with a sharp exponent. For $p>1$, a Lipschitz likelihood need not give Lipschitz stability, even for priors with bounded support. We establish Lipschitz stability through an interpolation argument under uniform Poincaré bounds and a globally Lipschitz potential with uniformly bounded essential oscillation under the priors. For Gaussian priors with additive Gaussian noise and bounded Lipschitz forward models, these estimates give posterior $W_2$ bounds even for mutually singular prior perturbations. Numerical experiments for a Darcy inverse problem illustrate the predicted Hölder and Lipschitz rates and the resulting control of errors in the posterior mean and standard deviation of a Lipschitz quantity of interest.