Control variates

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.

Underlying principle

Let the parameter of interest be $\mu$, and assume we have a statistic $m$ such that $\mathbb {E} \left[m\right]=\mu$. Suppose we calculate another statistic $t$ such that $\mathbb {E} \left[t\right]=\tau$ is a known value. Then

$m^{\star }=m+c\left(t-\tau \right)\,$

is also an unbiased estimator for $\mu$ for any choice of the coefficient $c$. The variance of the resulting estimator $m^{\star }$ is

${\textrm {Var}}\left(m^{\star }\right)={\textrm {Var}}\left(m\right)+c^{2}\,{\textrm {Var}}\left(t\right)+2c\,{\textrm {Cov}}\left(m,t\right);$

It can be shown that choosing the optimal coefficient

$c^{\star }=-{\frac {{\textrm {Cov}}\left(m,t\right)}{{\textrm {Var}}\left(t\right)}};$

minimizes the variance of $m^{\star }$, and that with this choice,

${\begin{aligned}{\textrm {Var}}\left(m^{\star }\right)&={\textrm {Var}}\left(m\right)-{\frac {\left[{\textrm {Cov}}\left(m,t\right)\right]^{2}}{{\textrm {Var}}\left(t\right)}}\\&=\left(1-\rho _{m,t}^{2}\right){\textrm {Var}}\left(m\right);\end{aligned}}$

where

$\rho _{m,t}={\textrm {Corr}}\left(m,t\right);\,$

hence, the term variance reduction. The greater the value of $\vert \rho _{mt}\vert$, the greater the variance reduction achieved.

In the case that ${\textrm {Cov}}\left(m,t\right)$, ${\textrm {Var}}\left(t\right)$, and/or $\rho _{mt}$ 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.

Example

We would like to estimate

$I=\int _{0}^{1}{\frac {1}{1+x}}\,\mathrm {d} x.$

The exact result is $I=\ln 2\approx 0.69314718$. Using Monte Carlo integration, this integral can be seen as the expected value of $f(U)$, where

$f(x)={\frac {1}{1+x}}$

and U follows a uniform distribution [0, 1]. Using a sample of size n denote the points in the sample as $u_{1},\cdots ,u_{n}$. Then the estimate is given by

$I\approx {\frac {1}{n}}\sum _{i}f(u_{i});$

If we introduce $T=\int _{0}^{1}1+x\,\mathrm {d} x.$ as a control variate with a known expected value ${\textrm {E}}\left[T\left(U\right)\right]={\frac {3}{2}}$

Using $n=1500$ realizations and an estimated optimal coefficient $c^{\star }\approx 0.4773$ we obtain the following results

	Estimate	Variance
Classical estimate	0.69475	0.01947
Control cariates	0.69295	0.00060

The variance was significantly reduced after using the control variates technique.

See also

• Antithetic variates
• Importance sampling

References

• Ross, Sheldon M. (2002) Simulation 3rd edition ISBN 978-0125980531
• Averill M. Law & W. David Kelton (2000), Simulation Modeling and Analysis, 3rd edition. ISBN 0-07-116537-1
• S. P. Meyn (2007) Control Techniques for Complex Networks, Cambridge University Press. ISBN 9780521884419. Downloadable draft (Section 11.4: Control variates and shadow functions)