Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 219 / 2 / c / Solution

For arbitrary nonnegative raw weights $W_i$, let $\bar{W}=K^{-1}{\sum }_i{W}_i$. The empirical squared coefficient of variation, using variance divisor $K$, isSubstitution into the stated definition gives the usual effective sample size of importance samplingFor normalized weights $w_i=W_i/{\sum }_j{W}_j$, this reduces to $\boxed{\widehat{\mathrm{ESS}}=1/\sum _i{w}_i^2}$.