Control variates

Let the unknown parameter of interest be $\backslash mu$, and assume we have a statistic $m$ such that the expected value of m is μ: $\backslash mathbb\{E\}\backslash left[m\backslash right]=\backslash mu$, i.e. m is an unbiased estimator for μ. Suppose we calculate another statistic $t$ such that $\backslash mathbb\{E\}\backslash left[t\backslash right]=\backslash tau$ is a known value. Then

:$m^\backslash star\; =\; m\; +\; c\backslash left(t-\backslash tau\backslash right)\; \backslash ,$

is also an unbiased estimator for $\backslash mu$ for any choice of the coefficient $c$.

The variance of the resulting estimator $m^\{\backslash star\}$ is

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

By differentiating the above expression with respect to $c$, it can be shown that choosing the optimal coefficient

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

minimizes the variance of $m^\{\backslash star\}$. (Note that this coefficient is the same as the coefficient obtained from a linear regression.) With this choice,

:$\backslash begin\{align\}$

\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{align}

where

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

is the correlation coefficient of $m$ and $t$. The greater the value of $\backslash vert\backslash rho\_\{m,t\}\backslash vert$, the greater the variance reduction achieved.

In the case that $\backslash textrm\{Cov\}\backslash left(m,t\backslash right)$, $\backslash textrm\{Var\}\backslash left(t\backslash right)$, and/or $\backslash rho\_\{m,t\}\backslash ;$ 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, $\backslash mathbb\{E\}\backslash left[t\backslash right]=\backslash tau$, is not known analytically, it is still possible to increase the precision in estimating $\backslash mu$ (for a given fixed simulation budget), provided that the two conditions are met: 1) evaluating $t$ is significantly cheaper than computing $m$; 2) the magnitude of the correlation coefficient $|\backslash rho\_\{m,t\}|$ is close to unity.

We would like to estimate

:$I\; =\; \backslash int\_0^1\; \backslash frac\{1\}\{1+x\}\; \backslash ,\; \backslash mathrm\{d\}x$

using Monte Carlo integration. This integral is the expected value of $f(U)$, where

:$f(U)\; =\; \backslash frac\{1\}\{1+U\}$

and U follows a uniform distribution [0, 1].

Using a sample of size n denote the points in the sample as $u\_1,\; \backslash cdots,\; u\_n$. Then the estimate is given by

:$I\; \backslash approx\; \backslash frac\{1\}\{n\}\; \backslash sum\_i\; f(u\_i).$

Now we introduce $g(U)\; =\; 1+U$ as a control variate with a known expected value $\backslash mathbb\{E\}\backslash left[g\backslash left(U\backslash right)\backslash right]=\backslash int\_0^1\; (1+x)\; \backslash ,\; \backslash mathrm\{d\}x=\backslash tfrac\{3\}\{2\}$ and combine the two into a new estimate

:$I\; \backslash approx\; \backslash frac\{1\}\{n\}\; \backslash sum\_i\; f(u\_i)+c\backslash left(\backslash frac\{1\}\{n\}\backslash sum\_i\; g(u\_i)\; -3/2\backslash right).$

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

class="wikitable" | | align="right" | Estimate | align="right" | Variance

Classical estimate | align="right" | 0.69475 | align="right" | 0.01947

Control variates | align="right" | 0.69295 | align="right" | 0.00060

The variance was significantly reduced after using the control variates technique. (The exact result is $I=\backslash ln\; 2\; \backslash approx\; 0.69314718$.)

• Antithetic variates
• Importance sampling

{{refimprove|date=August 2011}}

• Ross, Sheldon M. (2002) Simulation 3rd edition {{ISBN|978-0-12-598053-1}}
• 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|978-0-521-88441-9}}. [https://web.archive.org/web/20100619011046/https://netfiles.uiuc.edu/meyn/www/spm_files/CTCN/CTCN.html Downloadable draft] (Section 11.4: Control variates and shadow functions)

Category:Monte Carlo methods

Category:Statistical randomness

Category:Computational statistics

Category:Variance reduction