Gauss-Hermite Quadrature for non-Gaussian Inference via an Importance Sampling Interpretation

Intractable integrals appear in a plethora of problems in science and engineering. Very often, such integrals involve also a targeted distribution which is not even available in a closed form. In both cases, approximations of the integrals must be performed. Monte Carlo (MC) methods are a usual way of tackling the problem by approximating the integral with random samples. Quadrature methods are another alternative, where the integral is approximated with deterministic points and weights. However, the choice of these points and weights is only possible in a selected number of families of distributions. In this paper, we propose a deterministic method inspired in MC for approximating generic integrals. Our method is derived via an importance sampling (IS) interpretation, a MC methodology where the samples are simulated from the so-called proposal density, and weighted properly. We use Gauss-Hermite quadrature rules for Gaussian distributions, transforming them for approximating integrals with respect to generic distributions, even in the case where its normalizing constant is unknown. The novel method allows the use of several proposal distributions, allowing for the incorporation of recent advances in the multiple IS (MIS) literature. We discuss the convergence of the method, and we illustrate its performance with two numerical examples.