Every posterior given by <Bayes theorem> is absolutely continuous with respect to its prior $\mu_0$, so it belongs to $\operatorname{Prob}(X,\mathcal B_X,\mu_0)$. If $y_n\to y$, total-variation well-posedness gives
$$
d_{\mathrm{TV}}(\mu^{y_n},\mu^y)\longrightarrow0.
$$
Part c, now using the common dominating measure $\mu_0$, yields
$$
d_{\mathrm{Hel}}(\mu^{y_n},\mu^y)^2
\leq2d_{\mathrm{TV}}(\mu^{y_n},\mu^y)
\longrightarrow0.
$$
Existence and uniqueness are unchanged. The problem is therefore a <well-posed Bayesian inverse problem in Hellinger distance>.

Solution

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

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.

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