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*}

References

• (makalic2016)
• (polson2013)
• (andrews1974)