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All moments below are under the proper posterior distribution of part (b). Write the known covariance matrix in block formThe score covariance $S$ is positive definite in this model. Indeed, a further integration by parts gives $\mathbb{E}[g(\beta ){\beta }^T]=-I_p$. If $a^{T}Sa=0$, then $a^{T}g=0$ almost surely, because $\mathbb{E}[g]=0$. Multiplying by $a^T\beta$ would give $0=-\Vert a{\Vert }^2$, so $a=0$.

For a fixed coefficient vector $a$, the control variate summand has varianceThus the unique minimum is attained at $a_{\ast }=S^{-1}c$, givingThe coefficient is fixed because the covariance is assumed known, and the posterior score control variate has mean zero, so this estimator remains unbiased.

We can identify exactly when the improvement is strict. Integration by parts with the bounded smooth function $h$ givesThe scalar expectation is strictly positive. Therefore $c\ne 0$ whenever $x_{\mathrm{test}}\ne 0$, and positive definiteness gives a strictly smaller variance in that case. If $x_{\mathrm{test}}=0$, $h\equiv 1/2$ and both estimators already have variance zero. The printed request for a smaller variance therefore needs this nondegeneracy qualification; a non-increasing variance always holds.

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.

For probit regression with $s_i=2Y_i-1$ and prior $N(0,{\sigma }^{2}I)$, the posterior score control variate isFor $h(\beta )=\Phi (x_{\ast }^T\beta )$, its covariance with the score is $-x_{\ast }\mathbb{E}[\varphi (x_{\ast }^T\beta )]$. It is nonzero whenever $x_{\ast }\ne 0$, ensuring strict improvement by the optimal control variate.