Let $p(m)=P(m\mid\mathbf d)$ and draw independently from an importance density $q$. The unbiased estimator
$$
\widehat I=\frac1n\sum_{j=1}^n\frac{m_jp(m_j)}{q(m_j)}
$$
of $I=\int mp(m)\,dm$ has one-sample second moment
$$
\int\frac{m^2p(m)^2}{q(m)}\,dm.
$$
By the <Cauchy-Schwarz inequality>,
$$
\left(\int |m|p(m)\,dm\right)^2
\leq
\left(\int\frac{m^2p(m)^2}{q(m)}\,dm\right)
\left(\int q(m)\,dm\right).
$$
Equality holds precisely when $q(m)\propto|m|p(m)$, giving the <optimal importance density for a single integral>
$$
\boxed{q^*(m)=\frac{|m|p(m)}{\int|u|p(u)\,du}.}
$$
This is circular in practice: constructing and normalizing $q^*$ requires detailed knowledge of the posterior and the expectation of $|m|$. Here log masses are positive, so the unknown normalizer is the posterior mean being estimated. It is also optimal only for this one integral, not for general posterior summaries.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 219 / 2 / e / Solution

For estimating $I=\int g(x)f(x)\,dx$ by the unbiased importance estimator $g(Y)f(Y)/q(Y)$, the proposal minimizing variance is
$$
q^*(x)=\frac{|g(x)|f(x)}{\int |g(u)|f(u)\,du}.
$$
This follows from the <Cauchy-Schwarz inequality>. Its normalizing constant and dependence on the particular integrand often make it impractical.

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