 Posterior score control variate

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.