# How to choose best proposal distribution for importance sampling

From Robert & Casella p95, we know that the choice of proposal distribution $g(x)$ with minimal variance is the $g$ proportional to $|h(x)|f(x)$. If we restrict our proposal distribution to cetain function class, for example, The normal distribution $N(\mu,\sigma^2)$, is there an algorithm to choose best parameter $\mu$ and $\sigma$ that is most "proportional" to $|h(x)|f(x)$? I mean, is there a score for the "proportionality" between 2 functions?

This is definitely an interesting question, but there is no clear answer as far as I can tell. Indeed, first, one has to define a criterion to optimise. For instance, this could be the variance: $$\min_{\mu,\sigma}\,\text{var}_{\mu,\sigma}\, h(X)f(X)\big/g_{\mu,\sigma}(X)$$or equivalently$$\min_{\mu,\sigma}\,\int \frac{h^2(x)f^2(x)}{g_{\mu,\sigma}(x)}\,\text{d}x$$for which there does exist a solution but one that is unlikely to be derived analytically. In a series of papers on population Monte Carlo methods that we wrote between 2005 and 2008, we construct a sequence of $\mu,\sigma$ towards deriving this optimum. Here is another reference aiming at an optimal step function.
Other criteria could be used though, like minimising a functional distance between $|h(\cdot)|f(\cdot)$ and $g_{\mu,\sigma}(\cdot)$: $$\min_{\mu,\sigma}\,\mathcal{H}\{|h(\cdot)|f(\cdot),g_{\mu,\sigma}(\cdot)\}$$ where $\mathcal{H}$ can be the Hellinger distance, the Kullback-Leibler distance, the Wasserstein distance, or something else.