# On the Local Lipschitz Stability of Bayesian Inverse Problems

## TL;DR

This paper establishes the local Lipschitz stability of Bayesian inverse problems, showing how small perturbations in prior and likelihood affect the posterior across various probability metrics, thus extending well-posedness results.

## Contribution

It provides a general stability analysis of Bayesian posteriors under perturbations, extending well-posedness to multiple probability distances including Wasserstein and Kullback-Leibler.

## Key findings

- Posterior stability depends Lipschitz continuously on prior and likelihood perturbations.
- Results apply to total variation, Hellinger, Wasserstein distances, and KL divergence.
- Sensitivity increases as the posterior concentrates with more data.

## Abstract

In this note we consider the stability of posterior measures occuring in Bayesian inference w.r.t. perturbations of the prior measure and the log-likelihood function. This extends the well-posedness analysis of Bayesian inverse problems. In particular, we prove a general local Lipschitz continuous dependence of the posterior on the prior and the log-likelihood w.r.t. various common distances of probability measures. These include the total variation, Hellinger, and Wasserstein distance and the Kullback-Leibler divergence. We only assume the boundedness of the likelihoods and measure their perturbations in an $L^p$-norm w.r.t. the prior. The obtained stability yields under mild assumptions the well-posedness of Bayesian inverse problems, in particular, a well-posedness w.r.t. the Wasserstein distance. Moreover, our results indicate an increasing sensitivity of Bayesian inference as the posterior becomes more concentrated, e.g., due to more or more accurate data. This confirms and extends previous observations made in the sensitivity analysis of Bayesian inference.