Importance Sampling

Importance sampling estimates an expectation under a target distribution by drawing from a different, more convenient proposal distribution and correcting each draw with a weight equal to the ratio of target to proposal density.

Definition

Importance sampling is a Monte Carlo technique that estimates the expectation of a function under a target density by averaging the function times an importance weight over samples drawn from a proposal density.

Scope

This topic covers the importance sampling identity and the resulting weighted estimator, the choice of proposal and its effect on weight variance, the self-normalized estimator used when densities are known only up to a constant, the diagnostic role of effective sample size, and the pathologies of heavy-tailed weights. It connects to variance reduction and to particle methods.

Core questions

• How does reweighting samples from a proposal recover an expectation under the target?
• What choice of proposal minimizes the variance of the importance sampling estimator?
• How is self-normalized importance sampling used when only unnormalized densities are available?
• How does effective sample size diagnose weight degeneracy?

Key concepts

• Importance weights
• Proposal distribution
• Self-normalized estimator
• Effective sample size
• Weight degeneracy

Key theories

Importance sampling identity

An expectation under the target equals the expectation under the proposal of the function times the density ratio, so the weighted sample mean estimates the target expectation without bias when the proposal covers the target's support.

Optimal proposal and weight variance

The variance of the estimator is minimized by a proposal proportional to the absolute integrand, and in practice mismatched or light-tailed proposals produce a few dominant weights that inflate variance, measured by a low effective sample size.

Clinical relevance

Importance sampling enables estimation of rare-event probabilities, marginal likelihoods and posterior expectations, lets one reuse a single sample to evaluate many related targets, and underlies sequential Monte Carlo (particle) filters used in state-space and time-series models.

History

Importance sampling arose within mid-twentieth-century Monte Carlo practice as a variance-reduction device and was later developed by statisticians into self-normalized and adaptive forms, becoming a core component of sequential Monte Carlo and modern Bayesian computation.

Key figures

• Christian P. Robert
• George Casella
• Geneva Hoeting

Seminal works

• robert2004
• givens2013

Frequently asked questions

Why use a proposal distribution instead of sampling the target directly?

Often the target cannot be sampled easily, or one wants to concentrate draws in an important region, such as a rare-event tail. Importance sampling lets you sample a convenient distribution and correct for the mismatch with weights.

What goes wrong if the proposal is poorly chosen?

If the proposal has lighter tails than the target, a few samples receive enormous weights and dominate the estimate, giving high or even infinite variance. The effective sample size diagnostic flags this degeneracy.