# An Illustration of Importance Sampling

##### Posted on July 16, 2017 0 Comments

We want to evaluate the quantity

where $\pi(x)$ is pdf and $h(x)$ is the target function.

## An Example

Consider a target function given by

$f(x,y)=0.5exp(-90(x-0.5)^2-45(y+0.1)^4)+exp(-45(x+0.4)^2-60(y-0.5)^2)$ where $(x, y)\in [-1, 1]\times [-1 1]$.

We can plot the following function image

For mean, we have

By taking $m$ random samplers, $(x^{(1)}, y^{(1)}), (x^{(2)}, y^{(2)}), \ldots, (x^{(m)}, y^{(m)})$, uniformly in $[-1,1]\times [-1, 1]$

then

The implement code is as follows

## Basic Idea

Suppose one is interested in evaluating $\mu = E_{\pi}(h(\mathbf x))=\int h(\mathbf x)\pi (\mathbf x)d\mathbf x$

The procedure of a simple form of the importance sampling algorithm is as follows.

1. Draw $\mathbf x^{(1)}, \ldots, \mathbf x^{(m)}$ from a trial distribution $g()$
2. Calculate the importance weight $w^{(j)} = \pi (\mathbf x^{(j)})/ g(\mathbf x^{(j)}), \; for \; j=1, \ldots, m$
3. Approximate $\mu$ by $\hat \mu = \frac{\sum \limits_{j=1}^m w^{(j)}h(\mathbf x^{(j)})}{\sum\limits_{j=1}^m w^{j}}$

The implement code is as follows.

Published in categories Monte Carlo