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):