Effective Sample Size for Importance Sampling based on discrepancy measures

TL;DR

This paper explores alternative definitions of Effective Sample Size (ESS) for importance sampling, based on various discrepancy measures between probability distributions, and evaluates their theoretical properties and practical performance.

Contribution

It introduces new ESS functions derived from different discrepancy measures and provides a classification framework and comparison with existing methods.

Findings

Several new ESS functions based on discrepancy measures are proposed.

Theoretical criteria for evaluating ESS functions are established.

Numerical simulations compare the performance of different ESS measures.

Abstract

The Effective Sample Size (ESS) is an important measure of efficiency of Monte Carlo methods such as Markov Chain Monte Carlo (MCMC) and Importance Sampling (IS) techniques. In the IS context, an approximation $\widehat{ESS}$ of the theoretical ESS definition is widely applied, involving the inverse of the sum of the squares of the normalized importance weights. This formula, $\widehat{ESS}$, has become an essential piece within Sequential Monte Carlo (SMC) methods, to assess the convenience of a resampling step. From another perspective, the expression $\widehat{ESS}$ is related to the Euclidean distance between the probability mass described by the normalized weights and the discrete uniform probability mass function (pmf). In this work, we derive other possible ESS functions based on different discrepancy measures between these two pmfs. Several examples are provided involving, for instance, the geometric mean of the weights, the discrete entropy (including theperplexity measure, already proposed in literature) and the Gini coefficient among others. We list five theoretical requirements which a generic ESS function should satisfy, allowing us to classify different ESS measures. We also compare the most promising ones by means of numerical simulations.