Increasing variance of weights in sequential importance sampling

[sisyphuss]

Hi all,

I know about these facts:
1- The variance of importance weights increases in SIR (also know as the degeneracy problem).
2- It's bad (lol), because in practice, there will be a particle with high normalized weight and many particles with insignificant (normalized) weights.

But I can not really understand the meaning of increasing variance of importance weights in sequential importance sampling (SIR) well.

Can you please explain it for me? And why the high variance is bad? Also, is there any intuitive proof for that?

Thanks.

 

[Stephen Tashi]

Hi all,

I know about these facts:
1- The variance of importance weights increases in SIR (also know as the degeneracy problem).
2- It's bad (lol), because in practice, there will be a particle with high normalized weight and many particles with insignificant (normalized) weights.

You may know those facts, but they haven't struck a chord with anyone else, based on those fragmentary descriptions. There are probably people on the forum who know enough probability theory to help you if you describe your question precisely.

But I can not really understand the meaning of increasing variance of importance weights in sequential importance sampling (SIR) well.

I certainly don't know what "increasing variance of importance weights" means. And why do you use the abbreviation "SIR" for "sequential importance sampling"? Did you mean "sequential importance re-sampling"?

 

[hdb]

Hi all,

I know about these facts:
1- The variance of importance weights increases in SIR (also know as the degeneracy problem).
2- It's bad (lol), because in practice, there will be a particle with high normalized weight and many particles with insignificant (normalized) weights.

But I can not really understand the meaning of increasing variance of importance weights in sequential importance sampling (SIR) well.

Thanks.

- The abbreviation of sequential importance sampling is SIS, as far as I know, and this approach does not include resampling, not like the sequential important resampling (SIR)

- basically fact 1 and fact 2 say somewhat the same thing. Variance is the second moment of the normalized weights (informally it gives the sum squared deviation from the mean). The sample weights are normalized. At the initialization of the algorithm they are distributed evenly (low variance), and as time goes by, and we proceed with the algorithm, there will be some particles that perform good, and gain more and more weights, and as variance is sensitive to outliers it will grow.
The solution of SIR is to resample to eliminate samples with low importance weights and multiply samples with high weights, this will lower the variance of the weights, and prevent us to work with particles of low weights, this way we will discover the interesting places of the posterior density (the ones with high weights).
The resampling step is often followed by a Markov Chain Monte-Carlo (MCMC)
step to introduce sample variety without affecting the posterior density.

If you are interested, read the great book about the topic:

A. Doucet, N. De Freitas, and N. Gordon. Sequential
Monte Carlo methods in practice. Springer Verlag, 2001.