With the same $h$, the squared <Hellinger distance> is
$$
d_H(f,\phi)^2=\frac12\int(h-1)^2d\gamma
=1-\int h\,d\gamma.
$$
Since $0\leq\int h\,d\gamma\leq1$,
$$
1-\int h\,d\gamma
\leq1-\left(\int h\,d\gamma\right)^2
=\operatorname{Var}_\gamma h.
$$
The <Gaussian Poincaré inequality> and the derivative calculation in part (b) yield
$$
d_H(f,\phi)^2\leq\frac14J(f\Vert\phi).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 3 / c / Solution

{title2=$d_H(P,Q)$}
{wiki}

The Hellinger distance is the square root of a constant multiple of $\int(\sqrt p-\sqrt q)^2d\lambda$. Common conventions use either the factor $1$ or $1/2$.


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 Incoming links: Hellinger distance