Control Variate Method

It is a variance reduction technique used to improve the efficiency of Monte Carlo simulations when computing a mean value $\mathbb{E}[\mathcal{M}(X)]$.

Core Idea

It uses a simpler, correlated model $\tilde{\mathcal{M}}(X)$, called the control variate, for which the exact mean value $\mathbb{E}[\tilde{\mathcal{M}}(X)]$ is known.

Instead of directly estimating $\mathbb{E}[\mathcal{M}(X)]$ we use the identity:

$$\mathbb{E}[\mathcal{M}(X)]=\mathbb{E}[\mathcal{M}(X)-\tilde{\mathcal{M}}(X)]+\mathbb{E}[\tilde{\mathcal{M}}(X)]$$

A Monte Carlo simulation is then performed on the difference term, $\mathcal{M}(X)-\tilde{\mathcal{M}}(X)$.

Note: $\tilde{\mathcal{M}}(X)$ is can be a surrogate model (Eg. gPC)

Why it’s effective

If the control variate $\tilde{\mathcal{M}}$ is a good approximation of $\mathcal{M}$

$$\mathbb{V}[\mathcal{M}(X)-\tilde{\mathcal{M}}(X)]\ll \mathbb{V}[\mathcal{M}(X)]$$

This leads to a more accurate estimate for the same number of samples.

Estimator

A generalized form of the estimator is:

$${\tilde{\mu }}_K^{CV}:=\frac{1}{K}\sum _{i=1}^K(\mathcal{M}(x^{(i)})-\lambda \tilde{\mathcal{M}}(x^{(i)}))+\lambda \mathbb{E}[\tilde{\mathcal{M}}(X)]$$

Here, $\lambda$ is a coefficient chosen to minimize the variance of the estimator. The basic method simply uses $\lambda =1$.