🕹️ Control Variate

Control variates is a method used for variance reduction in 🤔 Monte Carlo Sampling. Consider the mean

Then, define

where is another random variable called the control variate. Observe that

so a mean estimate for is an unbiased estimate for .

However, the variance is

To minimize this value, we differentiate with respect to and get

where is the Correlation between the two random variables. Thus, if we have that’s positively correlated with with a easily-computable mean, we can reduce the variance of our original estimate.