Question

When performing Monte Carlo integration, what is the primary condition that must be met by a control variate to ensure it successfully reduces the variance of the estimator?

Accepted Answer

To successfully reduce the variance of a Monte Carlo estimator using a control variate, the primary condition is that the control variate must be strongly correlated with the function being integrated. A control variate is a known function whose expected value is easy to calculate and which mimics the behavior of the integrand. In Monte Carlo integration, we estimate the integral of a target function by averaging random samples. By subtracting a scaled version of the control variate from the integrand, we create a new estimator that compensates for random fluctuations in the sample. The variance of this new estimator is reduced if the control variate and the target function share a high absolute correlation. Specifically, if the correlation coefficient between the target function and the control variate is positive, the variance decreases as long as the control variate is subtracted; if the correlation is negative, the variance decreases if the control variate is added. If the correlation is zero, the control variate provides no benefit, and if the correlation is weak, the reduction in variance will be negligible. Mathematically, the variance reduction is proportional to the square of the correlation between the target function and the control variate, meaning that the stronger the linear relationship between them, the more effectively the noise in the estimate is cancelled out.

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When performing Monte Carlo integration, what is the primary condition that must be met by a control variate to ensure it successfully reduces the variance of the estimator?