Monte Carlo Integration & Variance Reduction

Estimating π and integrals — crude MC vs stratified sampling vs importance sampling

Estimate: True value: π = 3.14159 Error: N total: 0
About: Monte Carlo integration approximates integrals by averaging random function evaluations. Crude MC samples uniformly and converges as 1/√N. Stratified sampling divides the domain into strata and samples each — reduces variance by eliminating sampling imbalance. Importance sampling uses a proposal distribution q(x) to sample more where the integrand is large, reducing variance dramatically for peaked functions.