Control Variates
Subtract a centered known-mean control and tune its coefficient to change variance without changing expectation.
Free theory articleSubtract a centered known-mean control and tune its coefficient to change variance without changing expectation.
Concept & interpretation
The method of control variates is a variance reduction technique used in Monte Carlo simulations. When we want to estimate the expectation E[f(U)], we can introduce a known-mean control variable U (where E[U] is known) and form a new estimator h_c(U) = f(U) - c(U - E[U]). Because we subtract a term with expectation zero, the expectation of our new estimator remains exactly the same as the original. However, by carefully selecting the coefficient c, we can significantly reduce the variance of the estimation.
Experimenting with Control Variates
You can interact with these exact scenarios in the module at https://riano.app/module/control-variates.
- Change the Function preset to "Cubic" to study f(u) = u^3.
- Slide Coefficient tenths to 9 to set c = 0.9. Notice that the adjusted variance drops to its minimum point on the parabola.
- Set Probe eighths to 5 to illustrate the selected probe u = 0.625. The adjusted probe becomes exactly 0.131640625.
- Now, try shifting the Coefficient tenths to 0. The adjusted variance returns to the raw variance of 0.080357... because no control is applied.
- If you slide Coefficient tenths to 20 (c = 2.0), the adjusted variance climbs above the raw variance, demonstrating that a poorly chosen coefficient can increase estimation variance.