Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 326 / 3 / 1 / a / Solution

A Bayesian inverse problem consists of a prior distribution ${\mu }_0$ on the unknown $u\in X$, a reference measure on the data space $Y$, and a jointly measurable likelihood $L(u;y)$ such that $L(u;\cdot )$ is a probability density function for ${\mu }_0$-almost every $u$. For observed data $y$, Bayes theorem defines the posterior distribution byprovided $0<Z(y)<\infty$.