Minimum-variance importance distribution
= Minimum-variance importance distribution
{title2=$q_*(x)=|h(x)|f(x)/\int|h|f$}
For $\mu=\int hf$ and $0<A=\int|h|f<\infty$, the proposal $q_*=|h|f/A$ minimizes the <variance> of the ordinary unbiased <importance sampling> estimator. The <Cauchy-Schwarz inequality> gives $\int h^2f^2/q\geq A^2$, so the minimum <variance> for $n$ independent samples is $(A^2-\mu^2)/n$. The proposal need only cover the nonzero integrand.