Control variates

Short description: Technique for increasing the precision of estimates in Monte Carlo experiments

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]

## Contents

* 1 Underlying principle
* 2 Example
* 3 See also
* 4 Notes
* 5 References

## Underlying principle

Let the unknown parameter of interest be μ, and assume we have a statistic m such that the expected value of m is μ: 𝔼[m]=μ, i.e. m is an unbiased estimator for μ. Suppose we calculate another statistic t such that 𝔼[t]=τ is a known value. Then

m⋆=m+c(t−τ)

is also an unbiased estimator for μ for any choice of the coefficient c. The variance of the resulting estimator m⋆ is

Var(m⋆)=Var(m)+c2Var(t)+2cCov(m,t).

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

c⋆=−Cov(m,t)Var(t)

minimizes the variance of m⋆. (Note that this coefficient is the same as the coefficient obtained from a linear regression.) With this choice,

Var(m⋆)=Var(m)−[Cov(m,t)]2Var(t)=(1−ρm,t2)Var(m)

where

ρm,t=Corr(m,t)

is the correlation coefficient of m and t. The greater the value of |ρm,t|, the greater the variance reduction achieved.

In the case that Cov(m,t), Var(t), and/or ρm,t 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, 𝔼[t]=τ, is not known analytically, it is still possible to increase the precision in estimating μ (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 |ρm,t| is close to unity. [3]

## Example

We would like to estimate

I=∫0111+xdx

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

f(U)=11+U

and U follows a uniform distribution [0, 1]. Using a sample of size n denote the points in the sample as u1,⋯,un. Then the estimate is given by

I≈1n∑if(ui).

Now we introduce g(U)=1+U as a control variate with a known expected value 𝔼[g(U)]=∫01(1+x)dx=32 and combine the two into a new estimate

I≈1n∑if(ui)+c(1n∑ig(ui)−3/2).

Using n=1500 realizations and an estimated optimal coefficient c⋆≈0.4773 we obtain the following results

| Estimate | Variance

|
|

Classical estimate | 0.69475  | 0.01947
Control variates  | 0.69295  | 0.00060

The variance was significantly reduced after using the control variates technique. (The exact result is I=ln⁡2≈0.69314718.)

## See also

* Antithetic variates
* Importance sampling

## Notes

1. ↑ Lemieux, C. (2017). "Control Variates". Wiley StatsRef: Statistics Reference Online. pp. 1–8. doi:10.1002/9781118445112.stat07947. ISBN 9781118445112.
2. ↑ Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. New York: Springer. ISBN 0-387-00451-3 (p. 185)
3. ↑ 3.0 3.1 Botev, Z.; Ridder, A. (2017). "Variance Reduction". Wiley StatsRef: Statistics Reference Online. pp. 1–6. doi:10.1002/9781118445112.stat07975. ISBN 9781118445112.

## References

* 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. Downloadable draft (Section 11.4: Control variates and shadow functions)

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