t as mixture of Normals

(Based on Rasmus Bååth’s post)

A scaled $t$ distribution, with $\mu$ mean, $s$ scale and $\nu$ degrees of freedom, can be simulated

from a mixture of Normals with $\mu$ mean and precisions following a Gamma distribution:

$$ \begin{aligned} y &\sim \mathcal{N}\left(\mu,\sigma\right) \\ \sigma^2 &\sim \mathcal{IG}\left(\frac{\nu}{2},s^2\frac{\nu}{2}\right) \end{aligned} $$

Since I’ve recently pickep up again the crystal-gsl in my spare time, I’ve decided to replicate the previously mentioned post using a Crystal one-liner.

To simulate 10,000 samples from $t_2\left(0,3\right)$ using the mixture, we can then write:

samples = (0..10000).map { |x|
   Normal.sample 0.0, 1.0/Math.sqrt(Gamma.sample 1.0, 9.0)

We can see the mixture distribution (histogram) converging nicely to the $(t_2(0,3)$ (red):