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

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 $0<Z(y)<\infty$.

The total variation distance isIf both measures have densities $p,q$ with respect to a common dominating measure, then $d_{\mathrm{TV}}(\mu ,\nu )=\frac{1}{2}\int |p-q|$.

The problem is a well-posed Bayesian inverse problem in total variation when a unique posterior ${\mu }^y$ exists for every $y\in Y$ and the posterior map is continuous:Thus the metric supplies the precise meaning of continuous dependence on the observed data in the Bayesian version of Hadamard well-posedness.

The following four assumptions are sufficient, with statements understood for ${\mu }_0$-almost every $u$ and every $y$:

The first two assumptions give $0<Z(y)<\infty$. The last two allow the dominated convergence theorem to prove $L^1({\mu }_0)$ continuity of the normalized posterior density, which is equivalent to continuity in total variation distance.