A mixture of normal distributions


There is a simple theoretical result involving a mixture of normal distributions. If you select a normal distribution with a mean of zero and let the precision of that normal distribution vary according to a gamma distribution, you end up with a t-distribution.

Let x have a conditional distribution

$f(x|\tau) = \sqrt{\frac{\tau}{2 \pi}} e^{-0.5\tau x^2}$

and let $\tau$ have an unconditional distribution gamma distribution with any positive value for $\alpha$ and $\beta$.

$g(\tau) = \frac {\beta^\alpha} {\Gamma(\alpha)} \tau^{\alpha-1} e^{-\beta\tau}$

To get the unconditional distribution of x, you need to multiply the two distributions to get a joint distribution and integrate out the $\tau$ parameter.

$f(x)= \int \sqrt{\frac{\tau}{2 \pi}} e^{-0.5\tau x^2} \frac{\beta^{\alpha}}{\Gamma(\alpha)} \tau^{\alpha-1} e^{-\beta\tau} d\tau$

Pull out any multiplicative terms not involving $\tau$.

$f(x)= \frac {\beta^{\alpha}} {\sqrt{2 \pi}\Gamma(\alpha)} \int \tau^{\alpha-0.5} e^{-\tau\left(\beta+\frac{x^2}{2}\right)} d\tau$

Recall that the gamma distribution with parameters $\alpha$ and $\beta$ has the form

$h(y) = \frac{\beta^\alpha}{\Gamma(\alpha)} y^{\alpha-1} e^{-\beta y}$

What is inside the integral looks like a gamma distribution with

$\alpha^* = \alpha+0.5$,

$\beta^* = \beta+\frac{x^2}{2}$

Multiply inside by the normalizing constant

$\frac {\left(\beta+\frac{x^2}{2}\right)^{\alpha+0.5}} {\Gamma(\alpha+0.5)}$

and outside by the reciprocal. With the normalizing constant in place, the integral equals 1 and you are left with

$f(x) = \frac {\beta^{\alpha}} {\sqrt{2\pi}\Gamma(\alpha)} \frac {\Gamma(\alpha+0.5)} {\left(\beta+\frac{x^2}{2}\right)^{\alpha+0.5}}$

which simplifies to

$f(x) = \frac {\Gamma(\alpha+0.5)} {\sqrt{2\beta\pi}\Gamma(\alpha)} \left(1+\frac{x^2}{2\beta}\right)^{-(\alpha+0.5)}$

This can also be shown to relate to a t-distribution

$\frac {\Gamma\left(\frac{\nu+1}{2}\right)} {\sqrt{\nu\pi},\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}$

if you let