A novel sequential method for building upper and lower bounds of moments of distributions

Abstract

Approximating integrals is a fundamental task in probability theory and statistical inference, and their applied fields of signal processing, and Bayesian learning, as soon as expectations over probability distributions must be computed efficiently and accurately. When these integrals lack closedform expressions, numerical methods must be used, from the Newton-Cotes formulas and Gaussian quadrature, to Monte Carlo and variational approximation techniques. Despite these numerous tools, few are guaranteed to preserve majoration/minoration inequalities, while this feature is fundamental in certain applications in statistics. In this paper, we focus on the integration problem arising in the estimation of moments of scalar unnormalized distributions. We introduce a sequential method for constructing upper and lower bounds on the sought integral. Our approach leverages the majorization-minimization framework to iteratively refine these bounds using an envelope principle. The method has proven convergence and controlled accuracy under mild conditions. We then generalize the method to the multi-dimensional setting, along with an effective implementation strategy based on power diagrams. We demonstrate the effectiveness of the proposed approach through a detailed numerical example of the estimation of a Monte Carlo sampler variance in a Bayesian inference problem, in one- and two-dimensional cases.