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

Suppose an unknown has prior distribution $\Pi$ and its conditional observation law $P_u$ has jointly measurable Radon-Nikodym derivative $L(u,m)$ relative to a fixed observation law $P_0$. If $L>0$ for $\Pi (du)P_0(dm)$-almost every pair, thendefines the posterior distribution for almost every observation. Indeed, $\int L(u,m)P_0(dm)=1$ for almost every $u$, so Tonelli theorem gives $\int Z(m)P_0(dm)=1$. Hence $Z$ is finite almost everywhere, and positivity of $L$ and Fubini's theorem give $Z>0$ almost everywhere. The data law is $Z{P}_0$, and integration of the proposed posterior against this data law recovers the joint law. This proves the conditional distribution property and also shows that the data law is an equivalent probability measure to $P_0$.

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.

A Bayesian inverse problem is well posed in Hellinger distance when every datum determines a unique posterior and the posterior depends continuously on the data in Hellinger distance. Since posterior measures are dominated by the prior and squared Hellinger distance is at most twice total variation distance, total-variation well-posedness implies Hellinger well-posedness.