# Why is Sampling Importance Resampling (SIR) better than Importance Sampling (IS)?

From what I understand, SIR is a mechanism for sampling from a distribution $p$ that works as follows:

1. Approximate a target distribution $p$ using an importance sample $S$ from a proposal distribution $q$
2. Draw a small sample $S_\text{small}$ from $S$ with replacement according to $p$

The result, $S_\text{small}$, is a sample from $p$.

1. Why use a small subset and not the full sample $S$ in 1?
2. Why is this a variance reduction technique, and why does it matter to reduce the variance?
• The resulting sample $S_\text{small}$ is not a sample from $p$ due to the sampling from self-normalised weights. Nov 25, 2019 at 6:14
• @Xi'an would you please elaborate the reason why $S_{small}$ is not a set of samples from $p$? Thanks
– True
Jul 12, 2021 at 15:31
• @True: dividing the importance weights by the sum of the importance weights modifies or biases the distribution of the resulting sample. Jul 12, 2021 at 17:24
• @Xi'an I see. Thanks a lot! :)
– True
Jul 13, 2021 at 20:56

As I understand it, $S_\rm{small}$ is not smaller than $S$: they have typically the same size.