Difference between revisions of "importance Sampling June 2 2009"

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In <math>I = \displaystyle\int h(x)f(x)\,dx</math>, Monte Carlo simulation can be used only if it easy to sample from f(x). Otherwise, another method must be applied. If sampling from f(x) is difficult but there exists a probability distribution function g(x) which is easy to sample from, then <math>I</math> can be written as<br>
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:: <math>I = \displaystyle\int h(x)f(x)\,dx </math>
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:: <math>= \displaystyle\int \frac{h(x)f(x)}{g(x)}g(x)\,dx</math>
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:: <math>= \displaystyle E_g(w(x)) \rightarrow</math>the expectation of w(x) with respect to g(x)
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:: <math>= \frac{\displaystyle\sum_{i=1}^{N} w(x_i)}{N}</math> where <math>\displaystyle w(x) = \frac{h(x)f(x)}{g(x)}</math><br><br>
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'''Process'''<br>
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# Choose <math>\displaystyle g(x)</math> such that it's easy to sample from.
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# Compute <math>\displaystyle w(x)=\frac{h(x)f(x)}{g(x)}</math>
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# <math>\displaystyle \hat{I} = \frac{\displaystyle\sum_{i=1}^{N} w(x_i)}{N}</math><br><br>
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'''"Weighted" average'''<br>
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:The term "importance sampling" is used to describe this method because a higher 'importance' or 'weighting' is given to the values sampled from <math>\displaystyle g(x)</math> that are closer to <math>\displaystyle f(x)</math>, the original distribution we would ideally like to sample from (but cannot because it is too difficult).<br>
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:<math>\displaystyle I = \int\frac{h(x)f(x)}{g(x)}g(x)\,dx</math>
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:<math>=\displaystyle \int \frac{f(x)}{g(x)}h(x)g(x)\,dx</math>
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:<math>=\displaystyle \int \frac{f(x)}{g(x)}E_g(h(x))\,dx</math> which is the same thing as saying that we are applying a regular Monte Carlo Simulation method to <math>\displaystyle\int h(x)g(x)\,dx </math>, taking each result from this process and weighting the more accurate ones (i.e. the ones for which <math>\displaystyle \frac{f(x)}{g(x)}</math> is high) higher.
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One can view <math> \frac{f(x)}{g(x)}\ = B(x)</math> as a weight.
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Then <math>\displaystyle \hat{I} = \frac{\displaystyle\sum_{i=1}^{N} w(x_i)}{N} = \frac{\displaystyle\sum_{i=1}^{N} B(x_i)*h(x_i)}{N}</math><br><br>
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i.e. we are computing a weighted sum of <math> h(x_i) </math> instead of a sum.

Latest revision as of 08:45, 30 August 2017

In [math]I = \displaystyle\int h(x)f(x)\,dx[/math], Monte Carlo simulation can be used only if it easy to sample from f(x). Otherwise, another method must be applied. If sampling from f(x) is difficult but there exists a probability distribution function g(x) which is easy to sample from, then [math]I[/math] can be written as

[math]I = \displaystyle\int h(x)f(x)\,dx [/math]
[math]= \displaystyle\int \frac{h(x)f(x)}{g(x)}g(x)\,dx[/math]
[math]= \displaystyle E_g(w(x)) \rightarrow[/math]the expectation of w(x) with respect to g(x)
[math]= \frac{\displaystyle\sum_{i=1}^{N} w(x_i)}{N}[/math] where [math]\displaystyle w(x) = \frac{h(x)f(x)}{g(x)}[/math]

Process

  1. Choose [math]\displaystyle g(x)[/math] such that it's easy to sample from.
  2. Compute [math]\displaystyle w(x)=\frac{h(x)f(x)}{g(x)}[/math]
  3. [math]\displaystyle \hat{I} = \frac{\displaystyle\sum_{i=1}^{N} w(x_i)}{N}[/math]

"Weighted" average

The term "importance sampling" is used to describe this method because a higher 'importance' or 'weighting' is given to the values sampled from [math]\displaystyle g(x)[/math] that are closer to [math]\displaystyle f(x)[/math], the original distribution we would ideally like to sample from (but cannot because it is too difficult).
[math]\displaystyle I = \int\frac{h(x)f(x)}{g(x)}g(x)\,dx[/math]
[math]=\displaystyle \int \frac{f(x)}{g(x)}h(x)g(x)\,dx[/math]
[math]=\displaystyle \int \frac{f(x)}{g(x)}E_g(h(x))\,dx[/math] which is the same thing as saying that we are applying a regular Monte Carlo Simulation method to [math]\displaystyle\int h(x)g(x)\,dx [/math], taking each result from this process and weighting the more accurate ones (i.e. the ones for which [math]\displaystyle \frac{f(x)}{g(x)}[/math] is high) higher.

One can view [math] \frac{f(x)}{g(x)}\ = B(x)[/math] as a weight.

Then [math]\displaystyle \hat{I} = \frac{\displaystyle\sum_{i=1}^{N} w(x_i)}{N} = \frac{\displaystyle\sum_{i=1}^{N} B(x_i)*h(x_i)}{N}[/math]

i.e. we are computing a weighted sum of [math] h(x_i) [/math] instead of a sum.