MonteCarlo technique

We give a very short summary of the random number technique used here [,]. The method is a combination of the composition and rejection Monte Carlo methods. Suppose we wish to sample x from the distribution $f(x)$ and the (normalised) probability density function can be written as $f(x)\; =\; \sum$_i=1ⁿα_if _i(x) g_i(x)

where $f$_i(x) are normalised density functions, $\alpha$_i> 0 and $0\; \le g$_i(x) ≤1 .

According to this method, x can sampled in the following way:

1. select a random integer i such that $(1\le i\; \le n)$

   with probability proportional to $\alpha$_i

2. select a value $x\text{'}$ from the distribution $f$_i(x)

3. calculate $g$_i(x') and accept $x\; =\; x\text{'}$ with probability $g$_i(x') ;
4. if $x\text{'}$ is rejected restart from step 1.

In practice we have a good method of sampling from the distribution $f(x)$ , if

• all the subdistributions $f$_i(x) can be sampled easily;
• the rejection functions $g$_i(x) can be evaluated easily/quickly;
• the mean number of tries is not too large.