Control variate

A control variate is a random variable with known expected value whose centered value is subtracted from a Monte Carlo estimator to reduce its variance. For a mean-zero vector $G$, put $S=\mathrm{Cov}(G)$ and $c=\mathrm{Cov}(G,H)$. When $S$ is invertible, the optimal estimator averages $H-c^T{S}^{-1}G$ and has variance $(\mathrm{Var}(H)-c^T{S}^{-1}c)/N$ for $N$ independent draws. Improvement is strict exactly when $c\ne 0$.

If a smooth posterior density has vanishing boundary terms, its log-posterior gradient satisfies $\mathbb{E}[g]=0$ andThis is a posterior integration-by-parts identity; it concerns differentiation in the random parameter, distinct from the usual mean-zero score identity for sampling distributions. It makes $g$ a vector control variate. Finite moments and known coefficients preserve unbiasedness of the Monte Carlo estimator.