importance Sampling June 2 2009: Difference between revisions

From statwiki
Jump to navigation Jump to search
(Created page with 'test')
 
m (Conversion script moved page Importance Sampling June 2 2009 to importance Sampling June 2 2009: Converting page titles to lowercase)
 
(2 intermediate revisions by one other user not shown)
Line 1: Line 1:
test
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>
:: <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><br><br>
 
'''Process'''<br>
# Choose <math>\displaystyle g(x)</math> such that it's easy to sample from.
# Compute <math>\displaystyle w(x)=\frac{h(x)f(x)}{g(x)}</math>
# <math>\displaystyle \hat{I} = \frac{\displaystyle\sum_{i=1}^{N} w(x_i)}{N}</math><br><br>
 
'''"Weighted" average'''<br>
: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>
:<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><br><br>
 
i.e. we are computing a weighted sum of <math> h(x_i) </math> instead of a sum.

Latest revision as of 09:45, 30 August 2017

In [math]\displaystyle{ 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]\displaystyle{ I }[/math] can be written as

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

Process

  1. Choose [math]\displaystyle{ \displaystyle g(x) }[/math] such that it's easy to sample from.
  2. Compute [math]\displaystyle{ \displaystyle w(x)=\frac{h(x)f(x)}{g(x)} }[/math]
  3. [math]\displaystyle{ \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{ \displaystyle g(x) }[/math] that are closer to [math]\displaystyle{ \displaystyle f(x) }[/math], the original distribution we would ideally like to sample from (but cannot because it is too difficult).
[math]\displaystyle{ \displaystyle I = \int\frac{h(x)f(x)}{g(x)}g(x)\,dx }[/math]
[math]\displaystyle{ =\displaystyle \int \frac{f(x)}{g(x)}h(x)g(x)\,dx }[/math]
[math]\displaystyle{ =\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{ \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{ \displaystyle \frac{f(x)}{g(x)} }[/math] is high) higher.

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

Then [math]\displaystyle{ \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]\displaystyle{ h(x_i) }[/math] instead of a sum.