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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

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

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
  3. Approximate $\mu$ by

The implement code is as follows.


Published in categories Monte Carlo