importance Sampling June 2 2009

Importance Sampling

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.