The <total variation distance> is
$$
d_{\rm TV}(\mu,\nu)=\sup_{B\in\mathcal B(X)}|\mu(B)-\nu(B)|.
$$
The 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
$$
v_n\to v\quad\Longrightarrow\quad
d_{\rm TV}(\mu^{v_n},\mu^v)\to0.
$$

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 $0<e^{-\Phi}\leq1$. The normalizer satisfies $0<Z(v)\leq1$. If $v_n\to v$, the <dominated convergence theorem> gives both
$$
Z(v_n)\to Z(v)
$$
and 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, $d_{\rm TV}(\mu^{v_n},\mu^v)\to0$. Existence, uniqueness, and continuous dependence all follow.

Solution

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

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>.

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