Text: Computational Statistics 2E by Givens and Hoetings
Section: 6.3.2.3 Weight Degeneracy, Rejuvenation, and Effective Sample Size
I am having trouble following another result in the text. Below is a screenshot.
I am not sure on how $X$ is to be generated. I believe there are two options: 1) $X$ is generated using Sampling Importance Resampling, and 2) $X$ is generated using Sequential Importance Sampling.
I have ruled out option 2 because there is no notion of time in the equations (i.e. no subscript for $t$).
So I am assuming that there is a sample $X_1, \ldots, X_n$ ~ (approximately) iid from $f$ (the target). Let $g$ be an envelope for $f$.
Then the standardized importance weights are defined as $w(X) = \frac{f(x)/g(x)}{\sum_{i=1}^n f(x_i)/g(x_i)}$.
The authors are claiming that $E(w(X)) = \frac1n$, but this is not clear to me how to arrive there.
I have managed to put together something that is possibly sensible.
We know, from the definition of $w(X)$, that $\sum_{i=1}^n w(X_i) = 1$.
Then, $E \left( \sum_{i=1}^n w(X_i) \right) = 1$.
Since $E \left( \sum_{i=1}^n w(X_i) \right) = \sum_{i=1}^n E \left( w(X_i) \right)$ and there is and (approximate) iid sample we get that $$1 = \sum_{i=1}^n E \left( w(X_i) \right) = n \cdot E \left( w(X_i) \right)$$
My issue is that the authors don't explicitly state how $X$ is generated. Any insights would be helpful.