  The numerical approximation of posterior expected quantities of interest is
considered. A novel control variate technique is proposed for post-processing
of Markov chain Monte Carlo output, based both on Stein's method and an
approach to numerical integration due to Sard. The resulting estimators are
proven to be polynomially exact in the Gaussian context, while empirical
results suggest 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 are presented across a selection of
Bayesian inference tasks. All methods used in this paper are available in the R
package ZVCV.
