Monte Carlo and variance reduction

Let Y₁,…,Y_N be independent identically distributed copies of a quantity Y with finite variance. Their average is unbiased for E[Y], and its variance is Var(Y)/N because all cross covariances vanish. The central limit theorem describes its centered error after multiplication by √N as approximately normal for large N, with a limiting variance fixed by the model. A standard error estimates the standard deviation of the estimator, not the spread of individual observations and not a deterministic maximum error.

Finite expectation is enough for an IID law of large numbers, but the familiar finite-variance standard error and classical CLT require a finite second moment. Correlated samples from one trajectory require covariance or long-run variance analysis instead of blindly dividing an empirical variance by the number of saved states. A confidence interval also addresses sampling uncertainty conditional on the specified model. It does not include bias from a wrong model, an inaccurate time integrator, or a coding error that every replicate faithfully repeats.

Suppose a correlated auxiliary quantity Z has a known expectation m. Replace Y by Y−c(Z−m). The expected value stays unchanged for any fixed coefficient c, while a well-chosen c cancels fluctuations shared with Z. Minimizing its quadratic variance yields c=Cov(Y,Z)/Var(Z), provided Z has nonzero finite variance and Y also has finite variance. The residual variance becomes Var(Y) times one minus the squared correlation. Large correlation can therefore be valuable even when the auxiliary model is only an approximation.

The coefficient must be handled honestly in implementation. Estimating it from a separate pilot sample and fixing it for the main run preserves the simple unbiasedness argument. Estimating and reusing it on the same sample can introduce finite-sample bias, although consistent procedures may retain desirable asymptotics. Another strategy is conditional averaging: replacing Y by E[Y|G] keeps the mean and cannot increase variance when Y is square integrable. This Rao–Blackwell principle removes randomness that can be integrated analytically instead of simulated.

Importance sampling draws X under a proposal density q instead of target density p, then averages f(X)p(X)/q(X). The proposal must be positive wherever the target integrand contributes, up to the relevant null sets. With absolute integrability the expectation is preserved, but usable error bars also require the weighted quantity to have finite second moment under q. A proposal that misses an important tail can produce an infinite-variance estimator or deceptively quiet runs that have simply not encountered enormous weights yet.

For a rare event, move proposal mass toward the event while retaining the exact likelihood ratio. More frequent event observations are helpful only if the resulting weighted variance is smaller at comparable computational cost. Antithetic pairs and stratification offer other routes, but no variance-reduction label guarantees improvement for every integrand. Compare estimator variance, wall-clock cost, and reproducibility together. Exact small models are particularly valuable test cases because their known answers expose bias that an attractive confidence interval alone cannot detect.