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The Effective Sample Size (ESS) is an important measure of efficiency ofMonte Carlo methods such as Markov Chain Monte Carlo (MCMC) and ImportanceSampling (IS) techniques. In the IS context, an approximation $\widehat{ESS}$of the theoretical ESS definition is widely applied, involving the inverse ofthe 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 anotherperspective, the expression $\widehat{ESS}$ is related to the Euclideandistance between the probability mass described by the normalized weights andthe discrete uniform probability mass function (pmf). In this work, we deriveother possible ESS functions based on different discrepancy measures betweenthese two pmfs. Several examples are provided involving, for instance, thegeometric mean of the weights, the discrete entropy (including theperplexitymeasure, already proposed in literature) and the Gini coefficient among others.We list five theoretical requirements which a generic ESS function shouldsatisfy, allowing us to classify different ESS measures. We also compare themost promising ones by means of numerical simulations.

Effective Sample Size for Importance Sampling based on discrepancy measures