Importance sampling (IS) and numerical integration methods are usually
employed for approximating moments of complicated target distributions. In its
basic procedure, the IS methodology randomly draws samples from a proposal
distribution and weights them accordingly, accounting for the mismatch between
the target and proposal. In this work, we present a general framework of
numerical integration techniques inspired by the IS methodology. The framework
can also be seen as an incorporation of deterministic rules into IS methods,
reducing the error of the estimators by several orders of magnitude in several
problems of interest. The proposed approach extends the range of applicability
of the Gaussian quadrature rules. For instance, the IS perspective allows us to
use Gauss-Hermite rules in problems where the integrand is not involving a
Gaussian distribution, and even more, when the integrand can only be evaluated
up to a normalizing constant, as it is usually the case in Bayesian inference.
The novel perspective makes use of recent advances on the multiple IS (MIS) and
adaptive (AIS) literatures, and incorporates it to a wider numerical
integration framework that combines several numerical integration rules that
can be iteratively adapted. We analyze the convergence of the algorithms and
provide some representative examples showing the superiority of the proposed
approach in terms of performance