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 unknown parameter of interest be

μ

{\displaystyle \mu }

, and assume we have a statistic

m

{\displaystyle m}

such that the expected value of m is μ:

E

[

m

]

=

μ

{\displaystyle \mathbb {E} \left[m\right]=\mu }

, i.e. m is an unbiased estimator for μ. Suppose we calculate another statistic

t

{\displaystyle t}

such that

E

[

t

]

=

τ

{\displaystyle \mathbb {E} \left[t\right]=\tau }

is a known value. Then

m

⋆

=

m

+

c

(

t

−

τ

)

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

is also an unbiased estimator for

μ

{\displaystyle \mu }

for any choice of the coefficient

c

{\displaystyle c}

.

The variance of the resulting estimator

m

⋆

{\displaystyle m^{\star }}

is

Var

(

m

⋆

)

=

Var

(

m

)

+

c

2

Var

(

t

)

+

2

c

Cov

(

m

,

t

)

.

{\displaystyle {\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).}

By differentiating the above expression with respect to

c

{\displaystyle c}

, it can be shown that choosing the optimal coefficient

c

⋆

=

−

Cov

(

m

,

t

)

Var

(

t

)

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

minimizes the variance of

m

⋆

{\displaystyle m^{\star }}

. (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

)

]

2

Var

(

t

)

=

(

1

−

ρ

m

,

t

2

)

Var

(

m

)

{\displaystyle {\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

ρ

m

,

t

=

Corr

(

m

,

t

)

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

is the correlation coefficient of

m

{\displaystyle m}

and

t

{\displaystyle t}

. The greater the value of

|

ρ

m

,

t

|

{\displaystyle \vert \rho _{m,t}\vert }

, the greater the variance reduction achieved.