Need help from someone who knows about importance sampling.

[af_231]

Hello!
My question is about Importance Sampling. I am trying to apply a complex sampling method, which combine two sampling techniques. To do this, the importance sampling is used ONLY to divide the probability distribution into two main parts: Part 1: from 0 to p*, and Part 2: from p* to 1. Then, the method continues with the application of stratified sampling.

I would appreciate if someone can help me or explain me Importance Sampling... how to use this technique only to divide the probability? How to know which is the p* value, where the distribution should be divided?

I really appreciate your help...Any information will helpful.

I found in internet the following text:
"Importance Sampling attempts to do more samples at the areas of the function that are more
important. The way it does this is by bringing in a probability distribution function (pdf). All this is, is a function that attempts to say which areas of the function in the interval should get more samples. It does this by having a higher probability in that area."

Does this mean that p* is the CDF value that corresponds to the largest PDF value? ... I don't know, I am confused.

 

[mathman]

http://en.wikipedia.org/wiki/Importance_sampling
http://ib.berkeley.edu/labs/slatkin/eriq/classes/guest_lect/mc_lecture_notes.pdf

I suggest you start with above. There is no short answer to your question. Your comment about p* is incorrect.

 

[af_231]

Thanks!... and I'm sorry about my mistake, I am new in statistics area, but I'm trying to learn.

 

[mathman]

Thanks!... and I'm sorry about my mistake, I am new in statistics area, but I'm trying to learn.

No need to apologize.

A very short answer to your original question. Assume you have a random variable X and you want to estimate the average of f(X) using Monte Carlo. Then you would oversample X where f(X) is high and undersample where f(X) is low and compensate for the biased sapling by weghts (low for oversample and high for undersample). If done properly, the weighted average has a mean equal to the answer you are looking for, while the standard deviation is reduced in comparison to using unbiased samples.

 

[af_231]

Thanks for your help Mathman!

Can I ask you a favor?... maybe you can help me answering some questions about sampling methods and analysis of risk. I have these doubts and maybe you can help me to answer them.

Question 1) Is this correct?: The basic function of the sampling methods is to generate random numbers with similar characteristics or properties to the original sample. I mean, applying a sampling method, the output is a random sample?

Question 2) My original data is a time series data from which I selected the best fitted distribution through frequency analysis. As part of a risk analysis, I must apply a sampling method... my question is, the random numbers generated by this sampling method should be generated according to the best fitted distribution chosen on the frequency analysis?

Thanks! I really appreciate your help!

 

[mathman]

We need to clarify terminology. You are using the term "random number" in a non-standard fashion. I find it hard to understand what you are trying to do.