A principled stopping rule for importance sampling

TL;DR

This paper introduces a sequential stopping rule for importance sampling that determines when to stop sampling based on the variability of estimates, improving efficiency and reliability in Monte Carlo integration.

Contribution

A new stopping rule for importance sampling that accounts for multivariate problems and offers asymptotic guarantees, addressing limitations of existing metrics.

Findings

Provides a practical guideline for stopping importance sampling simulations.

Overcomes shortcomings of traditional effective sample size metrics.

Ensures asymptotic validity of the estimation process.

Abstract

Importance sampling (IS) is a Monte Carlo technique that relies on weighted samples, simulated from a proposal distribution, to estimate intractable integrals. The quality of the estimators improves with the number of samples. However, for achieving a desired quality of estimation, the required number of samples is unknown and depends on the quantity of interest, the estimator, and the chosen proposal. We present a sequential stopping rule that terminates simulation when the overall variability in estimation is relatively small. The proposed methodology closely connects to the idea of an effective sample size in IS and overcomes crucial shortcomings of existing metrics, e.g., it acknowledges multivariate estimation problems. Our stopping rule retains asymptotic guarantees and provides users a clear guideline on when to stop the simulation in IS.