  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.
