Alternative effective sample size measures for importance sampling

The Effective Sample Size (ESS) is an important measure of efficiency in the Importance Sampling (IS) technique. A well-known approximation of the theoretical ESS definition, involving the inverse of the sum of the squares of the normalized importance weights, is widely applied in literature. This expression has become an essential piece within Sequential Monte Carlo (SMC) methods, using adaptive resampling procedures. In this work, first we show that this ESS approximation is related to the Euclidean distance between the probability mass function (pmf) described by the normalized weights and the uniform pmf. Then, we derive other possible ESS functions based on different discrepancy measures. In our study, we also include another ESS measure called perplexity, already proposed in literature, that is based on the discrete entropy of the normalized weights. We compare all of them by means of numerical simulations.