Bayesian inverse problem

A Bayesian inverse problem combines a prior measure ${\mu }_0$ on an unknown $u$ with a likelihood for observed data $y$. When the likelihood is proportional to $e^{-\Phi (u;y)}$, Bayes' formula gives

A Bayesian inverse problem is well posed in total variation when a unique posterior exists for every datum and the map from data to posterior is continuous in total variation distance.