Effective sample size for importance sampling based on discrepancy measures Articles uri icon



publication date

  • February 2017

start page

  • 386

end page

  • 401


  • 131

International Standard Serial Number (ISSN)

  • 0165-1684

Electronic International Standard Serial Number (EISSN)

  • 1872-7557


  • 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 (ESS) over cap of the theoretical ESS definition is widely applied, involving the inverse of the sum of the squares of the normalized importance weights. This formula, (ESS) over cap, has become an essential piece within Sequential Monte Carlo (SMC) methods, to assess the convenience of a resampling step. From another perspective, the expression (ESS) over cap 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 the perplexity 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. (C) 2016 Elsevier B.V. All rights reserved.


  • effective sample size; perplexity; importance sampling; sequential monte carlo; particle filtering; bayesian inference; monte-carlo