a Deeper Look into Importance Sampling: Difference between revisions
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===A Deeper Look into Importance Sampling - June 3, 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>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> | |||
====Importance Sampling==== | ====Importance Sampling==== | ||
Revision as of 21:04, 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], 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]