Control variate

In Monte Carlo methods, one or more control variates may be employed to achieve variance reduction by exploiting the correlation between statistics.

Example
Let the parameter of interest be $$\mu$$, and assume we have a statistic $$m$$ such that $$\mathbb{E}\left[m\right]=\mu$$. If we are able to find another statistic $$t$$ such that $$\mathbb{E}\left[t\right]=\tau$$ and $$\rho_{mt}=\textrm{corr}\left[m,t\right]$$ are known values, then


 * $$m^{\star}=m-c\left(t-\tau\right)$$

is also unbiased for $$\mu$$ for any choice of the constant $$c$$. It can be shown that choosing


 * $$c=\frac{\sigma_m}{\sigma_t}\rho_{mt}$$

minimizes the variance of $$m^{\star}$$, and that with this choice,


 * $$\textrm{var}\left[m^{\star}\right]=\left(1-\rho_{mt}^2\right)\textrm{var}\left[m\right]$$;

hence, the term variance reduction. The greater the value of $$\vert\rho_{tm}\vert$$, the greater the variance reduction achieved.

In the case that $$\sigma_m$$, $$\sigma_t$$, and/or $$\rho_{mt}$$ are unknown, they can be estimated across the Monte Carlo replicates. This is equivalent to solving a certain least squares system; therefore this technique is also known as regression sampling.