Increasing variance of weights in sequential importance sampling

In summary, the variance of importance weights increases in sequential importance sampling (SIS), also known as the degeneracy problem. This means that as the algorithm progresses, some particles will have high normalized weights while others will have insignificant weights, leading to a high variance. This is problematic because it can result in working with particles of low weights and missing out on important areas of the posterior density. To address this, the algorithm often includes a resampling step followed by a Markov Chain Monte-Carlo (MCMC) step to introduce sample variety without affecting the posterior density. A recommended resource on this topic is the book "Sequential Monte Carlo methods in practice" by A. Doucet, N. De Freitas, and N.
  • #1
sisyphuss
1
0
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.
 
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  • #2
sisyphuss said:
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"?
 
  • #3
sisyphuss said:
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)

- Basicly 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.
 

What is sequential importance sampling?

Sequential importance sampling is a method used in statistics and machine learning to estimate the probability distribution of a sequence of random variables. It involves repeatedly sampling from an initial distribution and then updating the weights of the samples based on new information, in order to ultimately obtain a more accurate representation of the true distribution.

Why do we need to increase variance of weights in sequential importance sampling?

The variance of weights in sequential importance sampling can be thought of as a measure of how spread out the samples are around the true distribution. If the variance is too low, the samples may all be concentrated in a small area and not accurately represent the entire distribution. By increasing the variance, we can ensure that the samples cover a wider range and provide a more accurate representation of the true distribution.

How is the variance of weights increased in sequential importance sampling?

There are a few ways to increase the variance of weights in sequential importance sampling. One common method is to use resampling, where the samples with higher weights are duplicated and those with lower weights are discarded, thus increasing the overall spread of weights. Another method is to adjust the importance function used in the sampling process, which can also influence the variance of weights.

What are the potential consequences of increasing the variance of weights in sequential importance sampling?

Increasing the variance of weights can lead to a trade-off between accuracy and computational efficiency. While a higher variance may result in a more accurate representation of the true distribution, it can also require more computational resources and time. Additionally, if the variance is too high, the samples may become too spread out and lose their ability to accurately represent the true distribution.

What are some applications of increasing variance of weights in sequential importance sampling?

Sequential importance sampling is commonly used in fields such as statistics, machine learning, and physics. Some specific applications where increasing the variance of weights may be beneficial include parameter estimation, Bayesian inference, and state estimation in control systems.

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