Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 326 / 2 / f / Solution

The total variation distance isThe problem is a well-posed Bayesian inverse problem in total variation when every $v\in {\mathbb{R}}^N$ determines a unique posterior ${\mu }^v$ and

The heat solution operator at positive time is bounded from $L^2[0,1]$ to $C[0,1]$, so the finite sensor map $G:X\to {\mathbb{R}}^N$ is bounded and continuous. Therefore $\Phi (u;v)$ is jointly continuous and $0<e^{-\Phi }\le 1$. The normalizer satisfies $0<Z(v)\le 1$. If $v_n\to v$, the dominated convergence theorem gives bothand convergence in $L^1({\mu }_0)$ of the normalized posterior densities. Since total variation is one half of this $L^1$ distance for absolutely continuous measures, $d_{\mathrm{TV}}({\mu }^{v_n},{\mu }^v)\to 0$. Existence, uniqueness, and continuous dependence all follow.