a Deeper Look into Importance Sampling: Difference between revisions
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====Importance Sampling==== | ====Importance Sampling==== | ||
We can see that the integral <math>\displaystyle\int \frac{h(x)f(x)}{g(x)}g(x)\,dx = \int \frac{f(x)}{g(x)}h(x) g(x)\,dx</math> is just <math>= \displaystyle E_g(h(x)) \rightarrow</math>the expectation of h(x) with respect to g(x), where <math>\displaystyle \frac{f(x)}{g(x)} </math> is a weight <math>\displaystyle\beta(x)</math>. In the case where <math>\displaystyle f > g</math>, a greater weight for <math>\displaystyle\beta(x)</math> will be assigned. Thus, the points with more weight are deemed more | We can see that the integral <math>\displaystyle\int \frac{h(x)f(x)}{g(x)}g(x)\,dx = \int \frac{f(x)}{g(x)}h(x) g(x)\,dx</math> is just <math>= \displaystyle E_g(h(x)) \rightarrow</math>the expectation of h(x) with respect to g(x), where <math>\displaystyle \frac{f(x)}{g(x)} </math> is a weight <math>\displaystyle\beta(x)</math>. In the case where <math>\displaystyle f > g</math>, a greater weight for <math>\displaystyle\beta(x)</math> will be assigned. Thus, the points with more weight are deemed more important, hence "importance sampling". This can be seen as a variance reduction technique. | ||
=====Problem===== | =====Problem===== | ||
Revision as of 22:05, 3 June 2009
A Deeper Look into Importance Sampling - June 3, 2009
From last class, we have determined that an integral can be written in the form [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] We continue our discussion of Importance Sampling here.
Importance Sampling
We can see that the integral [math]\displaystyle{ \displaystyle\int \frac{h(x)f(x)}{g(x)}g(x)\,dx = \int \frac{f(x)}{g(x)}h(x) g(x)\,dx }[/math] is just [math]\displaystyle{ = \displaystyle E_g(h(x)) \rightarrow }[/math]the expectation of h(x) with respect to g(x), where [math]\displaystyle{ \displaystyle \frac{f(x)}{g(x)} }[/math] is a weight [math]\displaystyle{ \displaystyle\beta(x) }[/math]. In the case where [math]\displaystyle{ \displaystyle f \gt g }[/math], a greater weight for [math]\displaystyle{ \displaystyle\beta(x) }[/math] will be assigned. Thus, the points with more weight are deemed more important, hence "importance sampling". This can be seen as a variance reduction technique.
Problem
[math]\displaystyle{ \displaystyle \hat I }[/math]