Semi-Exact Control Functionals From Sard's Method

This paper focuses on the numerical computation of posterior expected quantities of interest, where existing approaches based on ergodic averages are gated by the asymptotic variance of the integrand. To address this challenge, a novel technique is proposed to post-process Markov chain Monte Carlo output, based on Sard's approach to numerical integration and the control functional method. The use of Sard's approach ensures that our control functionals are exact on all polynomials up to a fixed degree in the Bernstein-von-Mises limit, so that the reduced variance estimator approximates the behaviour of a polynomially-exact (e.g. Gaussian) cubature method. The proposed method is shown to combine the robustness of parametric control variates with the flexibility of non-parametric control functionals across a selection of Bayesian inference tasks. All methods used in this paper are available in the R package ZVCV.