= Hellinger bound for differences of expectations
{c}
If $f$ is a measurable function into a separable <Banach space> and has finite second <moments> under <probability measures> $\mu,\mu'$, then their <Bochner integral> <expected values> satisfy
$$
\|\mathbb E^\mu f-\mathbb E^{\mu'}f\|
\leq2\left(\mathbb E^\mu\|f\|^2+\mathbb E^{\mu'}\|f\|^2\right)^{1/2}d_{\mathrm{Hell}}(\mu,\mu').
$$
Here the <Hellinger distance> includes the factor $1/2$ in its square. For a proof, use $\nu=(\mu+\mu')/2$, $p=d\mu/d\nu$, and $q=d\mu'/d\nu$. Apply the norm bound for a <Bochner integral> and the <Cauchy-Schwarz inequality> to $f(p-q)=f(\sqrt p-\sqrt q)(\sqrt p+\sqrt q)$, then use $(\sqrt p+\sqrt q)^2\leq2(p+q)$. This explains why stability of a <Bayesian posterior> in <Hellinger distance> controls posterior <expected values> when the relevant second <moments> remain bounded.
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