Effective sample size
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In the previous days I have received several emails asking for clarification of the effective sample size derivation in “Introducing Monte Carlo Methods with R” (Section 4.4, pp. 98-100). Formula (4.3) gives the Monte Carlo estimate of the variance of a self-normalised importance sampling estimator (note the change from the original version in Introducing Monte Carlo Methods with R ! The weight W is unnormalised and hence the normalising constant 
)+%5Cleft%5C%7B1+%5Cdfrac%7B%5Cmathrm%7Bvar%7D_g(W)%7D%7B%5Ckappa%5E2%7D%5Cright%5C%7D&bg=000000&%23038;fg=B0B0B0&%23038;s=0 "frac{1}{n},mathrm{var}_f (h(X)) left{1+dfrac{mathrm{var}_g(W)}{kappa^2}right}")
as
+-+%5Cdelta_h%5En+%5Cright%5C%7D%5E2+%7D%7Bn%5Csum_%7Bi=1%7D%5En+%5Comega_i%7D+%5C,+%5Cleft%5C%7B+1+++n%5E2%5C,%5Cwidehat%7B%5Cmathrm%7Bvar%7D%7D(W)%5CBigg/+%5Cleft(%5Csum_%7Bi=1%7D%5En+%5Comega_i+%5Cright)%5E2+%5Cright%5C%7D%5C,.&bg=000000&%23038;fg=B0B0B0&%23038;s=0 "dfrac{sum_{i=1}^n omega_i left{ h(x_i) - delta_h^n right}^2 }{nsum_{i=1}^n omega_i} , left{ 1 + n^2,widehat{mathrm{var}}(W)Bigg/ left(sum_{i=1}^n omega_i right)^2 right},.")
Now, the front term is somehow obvious so let us concentrate on the bracketed part. The empirical variance of the 
%5E2+%5C,,&bg=000000&%23038;fg=B0B0B0&%23038;s=0 "frac{1}{n},sum_{i=1}^nomega_i^2-frac{1}{n^2}left(sum_{i=1}^nomega_iright)^2 ,,")
the coefficient /%5Ckappa%5E2&bg=000000&%23038;fg=B0B0B0&%23038;s=0 "1+widehat{mathrm{var}}_g(W)/kappa^2")
%5E2%5C,.&bg=000000&%23038;fg=B0B0B0&%23038;s=0 "n,sum_{i=1}^n omega_i^2 bigg/ left(sum_{i=1}^n omega_iright)^2,.")
which leads to the definition of the effective sample size
%5E2%5Cbigg/%5Csum_%7Bi=1%7D%5En%5Comega_i%5E2%5C,.&bg=000000&%23038;fg=B0B0B0&%23038;s=0 "text{ESS}_n=left(sum_{i=1}^nomega_iright)^2bigg/sum_{i=1}^nomega_i^2,.")
The confusing part in the current version is whether or not we use normalised W’s and
Filed under: Books, R, Statistics Tagged: effective sample size, importance sampling, Introducing Monte Carlo Methods with R
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