control variate other than the score

Posted by Yuling Yao on Dec 08, 2022.       Tag: computing  

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 $\nabla_{x} p(x)$.