Rethinking the effective sample size

The effective sample size (ESS) is widely used in sample-based simulation methods for assessing the quality of a Monte Carlo approximation of a given distribution and of related integrals. In this paper, we revisit the approximation of the ESS in the specific context of importance sampling. The derivation of this approximation, that we will denote as ESSˆ , is partially available in a 1992 foundational technical report of Augustine Kong. This approximation has been widely used in the last 25 years due to its simplicity as a practical rule of thumb in a wide variety of importance sampling methods. However, we show that the multiple assumptions and approximations in the derivation of ESSˆ make it difficult to be considered even as a reasonable approximation of the ESS. We extend the discussion of the ESSˆ in the multiple importance sampling setting, we display numerical examples and we discuss several avenues for developing alternative metrics. This paper does not cover the use of ESS for Markov chain Monte Carlo algorithms