Do variance reduction techniques change the quantity being estimated?

No. Properly applied, they target the same expectation and remain unbiased or consistent; they only rearrange how randomness enters so that the estimator fluctuates less around the true value.

Can a variance reduction technique ever make things worse?

Yes. A control variate weakly correlated with the integrand, or an importance density mismatched to the target, can increase variance. The benefit depends on matching the technique to the structure of the problem.

Variance Reduction Techniques

Variance reduction techniques are devices that lower the sampling variance of a Monte Carlo estimator, so that a target accuracy is reached with fewer simulated draws than naive sampling would require.

Definition

A variance reduction technique is a modification of a Monte Carlo sampling scheme or estimator that decreases the variance of the resulting estimate while keeping it unbiased or consistent for the same target quantity.

Scope

This topic covers the principal classical schemes: common and antithetic random numbers, control variates, importance sampling viewed as a variance-reduction device, stratified and Latin hypercube sampling, and conditioning (Rao-Blackwellization). The emphasis is on the bias-free reduction of estimator variance and on the conditions under which each scheme helps.

Core questions

How do antithetic and common random numbers exploit correlation to cancel variance?
How does a control variate use a correlated quantity with known mean to adjust an estimator?
Why can reweighting samples through importance sampling reduce variance, and when does it backfire?
How do stratification and conditioning lower variance, and what do they cost?

Key concepts

Antithetic variates
Control variates
Common random numbers
Stratified sampling
Rao-Blackwellization
Effective sample size

Key theories

Correlation-based reduction: Antithetic variates induce negative correlation between paired draws, and control variates subtract a correlated quantity with known expectation; both reduce variance in proportion to the strength of the exploited correlation.
Reweighting and stratification: Importance sampling shifts simulation effort toward influential regions through a change of measure, while stratified and Latin hypercube designs spread draws evenly across the input space; each can sharply cut variance when matched to the integrand.

Clinical relevance

Variance reduction is what makes large simulation studies, rare-event estimation, and expensive Bayesian computations feasible: by cutting the number of draws needed for a given precision it reduces computing time, and techniques like common random numbers sharpen comparisons between competing systems or estimators.

History

The classical variance-reduction toolkit (antithetic variates, control variates, importance sampling, stratification) was developed in the mid-twentieth century alongside the first large-scale Monte Carlo applications, and was later unified with conditioning ideas such as Rao-Blackwellization within the statistical simulation literature.

Key figures

Christian P. Robert
George Casella
John M. Hammersley

Seminal works

robert2004
givens2013

Frequently asked questions

Do variance reduction techniques change the quantity being estimated?: No. Properly applied, they target the same expectation and remain unbiased or consistent; they only rearrange how randomness enters so that the estimator fluctuates less around the true value.
Can a variance reduction technique ever make things worse?: Yes. A control variate weakly correlated with the integrand, or an importance density mismatched to the target, can increase variance. The benefit depends on matching the technique to the structure of the problem.