{title2=$g(\beta)=\nabla\log p(\beta\mid Y)$}

If a smooth <posterior density> has vanishing boundary terms, its <log-posterior> gradient satisfies $\mathbb E[g]=0$ and
$$
\mathbb E[g(\beta)h(\beta)]=-\mathbb E[\nabla h(\beta)].
$$
This 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>.


Posterior score control variate

{title2=$\log p(\theta\mid Y)$}

The log-posterior is the logarithm of a positive <posterior density>. Its gradient is the <posterior score control variate> when boundary terms vanish. An additive constant independent of the parameter does not affect optimization or differentiation.


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 Incoming links: Log-posterior