Semi-exact control functionals from Sard’s method

Published in Biometrika, 2022

Recommended citation: South, L.F., Karvonen, T., Nemeth, C., Girolami, M. and Oates, C.J., (2022). "Semi-exact control functionals from Sard’s method." Biometrika, 109(2), pp.351-367.

A novel control variate technique is proposed for the post-processing of Markov chain Monte Carlo output, based on both Stein’s method and an approach to numerical integration due to Sard. The resulting estimators of posterior expected quantities of interest are proven to be polynomially exact in the Gaussian context, while empirical results suggest that the estimators approximate a Gaussian cubature method near the Bernstein–von Mises limit. The main theoretical result establishes a bias-correction property in settings where the Markov chain does not leave the posterior invariant. Empirical results across a selection of Bayesian inference tasks are presented.