Effective sample size

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 \kappa appears in the denominator.)

\frac{1}{n}\,\mathrm{var}_f (h(X)) \left\{1+\dfrac{\mathrm{var}_g(W)}{\kappa^2}\right\}

as

\dfrac{\sum_{i=1}^n \omega_i \left\{ h(x_i) - \delta_h^n \right\}^2 }{n\sum_{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 \omega_i‘s is

\frac{1}{n}\,\sum_{i=1}^n\omega_i^2-\frac{1}{n^2}\left(\sum_{i=1}^n\omega_i\right)^2 \,,

the coefficient 1+\widehat{\mathrm{var}}_g(W)/\kappa^2 is thus estimated by

n\,\sum_{i=1}^n \omega_i^2 \bigg/ \left(\sum_{i=1}^n \omega_i\right)^2\,.

which leads to the definition of the effective sample size

\text{ESS}_n=\left(\sum_{i=1}^n\omega_i\right)^2\bigg/\sum_{i=1}^n\omega_i^2\,.

The confusing part in the current version is whether or not we use normalised W’s and \omega_i‘s. I hope this clarifies the issue!

One Response to “Effective sample size”

  1. davharris Says:

    Just to clarify, W is *not* normalized, but little omega is?

    Thanks

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