Importance sampling reduces Monte Carlo variance by drawing samples proportional to the integrand's magnitude

Monte Carlo integration estimates an integral ∫f(x)dx ≈ (1/n)Σf(X_i)/p(X_i) where X_i are drawn from distribution p. When p matches f’s shape (proportional to f), the quotient f/p is constant and variance drops to zero. In practice, p cannot exactly match the unknown integrand, but sampling proportional to known factors (BSDF, cosine term, light source intensity) concentrates samples where the integrand is large, reducing the error for a given sample count. For path tracing, importance sampling the BSDF or light distribution is the primary variance-reduction technique. Multiple importance sampling (MIS) combines several sampling strategies optimally. Without importance sampling, rendering complex scenes requires prohibitively many samples.

Examples

Sampling the BSDF vs. uniform hemisphere sampling: for a glossy material, most hemisphere directions contribute near-zero radiance. BSDF importance sampling places 90%+ of samples in the sharp specular lobe where they matter, reducing noise dramatically.

Assessment

Given a 2D integral with a sharp Gaussian peak, compare (numerically or by sketch) the variance of: (a) uniform sampling, (b) Gaussian importance sampling. Calculate how many times more samples option (a) needs to match option (b)‘s error.