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.)

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

$\dfrac{\sum_{i=1}^n \omega_i \left\{ h(x_i) - \delta_h^n \right\}^2 }{n\sum_{i=1}^n \omega_i} \, \left\{ 1 + n^2\,\widehat{\mathrm{var}}(W)\Bigg/ \left(\sum_{i=1}^n \omega_i \right)^2 \right\}\,.$

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!

This entry was posted on September 24, 2010 at 12:09 am and is filed under Books, R, Statistics with tags effective sample size, importance sampling, Introducing Monte Carlo Methods with R. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

One Response to “Effective sample size”

davharris Says:
April 20, 2014 at 4:59 am

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

Thanks

Reply

Xi'an's Og