  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.
