Gaussian scale mixture
The probability density function of a Gaussian scale mixture random variable \(X\) can be written as
\begin{equation*} \pi_{X}(x|\mu,\sigma^{2}) = \int_{0}^{\infty} \mathcal{N}\left(x|\mu, f(\lambda) \sigma^{2}\right)\,\pi_\lambda(\lambda) \mathrm{d}\lambda \end{equation*}where \(\lambda\) is called the mixing parameter, and \(f\) is a positive function of the mixing parameter. Different choices of \(f\) and \(\lambda\) lead to a wide variety of non-gaussian distributions:
Student t distribution
The Student t distribution can be seen as a gaussian scale mixture with \(f(\lambda) = 1 / \lambda\) and \(\lambda\) gamma-distributed:
\begin{align*} X &\sim \operatorname{Normal}(\mu, \frac{\sigma^{2}}{\lambda})\\ \lambda &\sim \operatorname{Gamma}(\delta/2, \delta/2) \end{align*}is equivalent to:
\begin{equation*} X \sim \operatorname{Student}(\delta) \end{equation*}