# control variate other than the score

Posted by Yuling Yao on Dec 08, 2022.In Bayesian computation, we use control variate to reduces Monte Carlo (MC) variance. The idea if we want to compute $E_{p} h(x)$ from MC draws $x_{1, \dots, S}$, instead of computing the sample mean of $h(x_i)$ , we seek a mean zero function m(x): $E_{p} m(x)= 0$, such that $h (x)- m(x)$ has lower variance.

As far as I know, most control variate takes the form of **score** function; Either the gradient of the log density, $\nabla_{x} \log p(x)$, or the stein gradient $\nabla \log p(x) g(x) + \nabla_{x} \cdot (g(x))$.

Are there other zero-mean functions? Well, at least another one: $\nabla_{x} p(x)$ because $E_p \nabla_{x} p(x)=0$ for all $p$. I might be ignorant, but I just notice this identity today.

**Edit:**
Actually, $\nabla_{x} p(x)$ is still generated by the score function. Just take $g(x)=p(x)$ in the formula in the second paragraph, then we get the $\nabla_{x} p(x)$.