Variance Reduction

Variance reduction encompasses techniques that improve the precision of Monte Carlo estimators without increasing the sample size. Beyond importance sampling, the toolkit includes stratified sampling (partitioning the domain and sampling proportionally), antithetic variates (using negatively correlated samples to cancel fluctuations), and control variates (exploiting correlation with a tractable quantity to reduce residual variance).

These methods share a common principle: they replace pure randomness with structured randomness that respects known properties of the target. The more structure you can exploit, the less pure Monte Carlo you need — but the more assumptions you make. Variance reduction is therefore a spectrum from pure random sampling to deterministic quadrature, and the optimal point on that spectrum depends on what you know about the problem before you begin. This spectrum mirrors the broader epistemic tension in computational modeling between tractability and fidelity.

Variance reduction is not a technical refinement. It is a philosophical claim: that knowledge about structure can be converted into computational efficiency. The conversion rate is the real measure of what we know.