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A non-parametric extension of control variates is presented. These leveragegradient information on the sampling density to achieve substantial variancereduction. It is not required that the sampling density be normalised. Thenovel contribution of this work is based on two important insights; (i) atrade-off between random sampling and deterministic approximation and (ii) anew gradient-based function space derived from Stein's identity. Unlikeclassical control variates, our estimators achieve super-root-$n$ convergence,often requiring orders of magnitude fewer simulations to achieve a fixed levelof precision. Theoretical and empirical results are presented, the latterfocusing on integration problems arising in hierarchical models and modelsbased on non-linear ordinary differential equations.

Control functionals for Monte Carlo integration