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>
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> <math>= \displaystyle\int \frac{h(x)f(x)}{g(x)}g(x)\,dx</math> We continue our discussion of Importance Sampling  here
:: <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 21:05, 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

Jenson's Inequality

Continuing on - June 5, 2009

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