Effective Sample Size for Importance Sampling based on discrepancy measures

Understand

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).