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

Let $p(m)=P(m\mid \mathbf{d})$ and draw independently from an importance density $q$. The unbiased estimatorof $I=\int mp(m)dm$ has one-sample second momentBy the Cauchy-Schwarz inequality,Equality holds precisely when $q(m)\propto |m|p(m)$, giving the optimal importance density for a single integralThis is circular in practice: constructing and normalizing $q^{\ast }$ 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.