a Deeper Look into Importance Sampling

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]