  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 (IS).
The derivation of this approximation, that we will denote as
$\widehat{\text{ESS}}$, is partially available in Kong (1992). 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 $\widehat{\text{ESS}}$, makes it difficult to be considered even
as a reasonable approximation of the ESS. We extend the discussion of the
$\widehat{\text{ESS}}$ in the multiple importance sampling (MIS) setting, we
display numerical examples, and we discuss several avenues for developing
alternative metrics. This paper does not cover the use of ESS for MCMC
algorithms.
