Antithetic Variates

The method of antithetic variates is a variance reduction technique used in Monte Carlo simulation. When we want to estimate the expected value of a function f(U) where U is a uniform random variable on [0,1], we can generate pairs of inputs rather than independent samples. For a monotone function with finite variance, the pair (U, 1-U) gives nonpositive covariance between f(U) and f(1-U). The pair average (f(U) + f(1-U))/2 therefore has no greater variance than the average of two independent evaluations; constant functions can give equality.

Experiment

You can use the Antithetic Variates interactive module to visualize how the mirrored pair is constructed for different functions and observe the effect on the estimator's variance.

1. Set the Function preset to Square.
2. Move the Probe eighths slider to 3 to represent the uniform sample u = 3/8.
3. Observe the canvas where the evaluations f(U) and f(Mirror) are plotted, and the pair average falls exactly in the middle.
4. Examine the Estimator variances panel. Compare the variance of the independent estimator (two independent uniform samples) with the variance of the antithetic pair. Notice that for the Square function, the antithetic variance is significantly lower.
5. Change the Function preset to Folded absolute. You will see that the antithetic estimator now has a higher variance (1/48) compared to the independent estimator (1/96), because this function is not monotonic and its structure defeats the negative correlation strategy.