🕹️ 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 📚 Summary Statistics > 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.