The control variates method is a variance reduction technique used in Monte Carlo methods.
It exploits information about the errors in estimates of known quantities to reduce the error of an estimate of an unknown quantity.
[1] [2][3] Let the unknown parameter of interest be
, and assume we have a statistic
such that the expected value of m is μ:
, i.e. m is an unbiased estimator for μ.
Suppose we calculate another statistic
The variance of the resulting estimator
is By differentiating the above expression with respect to
, it can be shown that choosing the optimal coefficient minimizes the variance of
(Note that this coefficient is the same as the coefficient obtained from a linear regression.)
With this choice, where is the correlation coefficient of
, the greater the variance reduction achieved.
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.
When the expectation of the control variable,
, is not known analytically, it is still possible to increase the precision in estimating
(for a given fixed simulation budget), provided that the two conditions are met: 1) evaluating
is significantly cheaper than computing
; 2) the magnitude of the correlation coefficient
is close to unity.
[3] We would like to estimate using Monte Carlo integration.
, where and U follows a uniform distribution [0, 1].
Using a sample of size n denote the points in the sample as
as a control variate with a known expected value
realizations and an estimated optimal coefficient
we obtain the following results The variance was significantly reduced after using the control variates technique.