Editing Normal distribution (section)

==== With known mean ====
For a set of [[i.i.d.]] normally distributed data points '''X''' of size ''n'' where each individual point ''x'' follows <math display=inline>x \sim \mathcal{N}(\mu, \sigma^2)</math> with known mean μ, the [[conjugate prior]] of the [[variance]] has an [[inverse gamma distribution]] or a [[scaled inverse chi-squared distribution]]. The two are equivalent except for having different [[parameter]]izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ<sup>2</sup> is as follows:

<math display=block>p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac{(\sigma_0^2\frac{\nu_0}{2})^{\nu_0/2}}{\Gamma\left(\frac{\nu_0}{2} \right)}~\frac{\exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]}{(\sigma^2)^{1+\frac{\nu_0}{2}}} \propto \frac{\exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]}{(\sigma^2)^{1+\frac{\nu_0}{2}}}</math>

The [[likelihood function]] from above, written in terms of the variance, is:

<math display=block>\begin{align}
p(\mathbf{X}\mid\mu,\sigma^2) &= \left(\frac{1}{2\pi\sigma^2}\right)^{n/2} \exp\left[-\frac{1}{2\sigma^2} \sum_{i=1}^n (x_i-\mu)^2\right] \\
&= \left(\frac{1}{2\pi\sigma^2}\right)^{n/2} \exp\left[-\frac{S}{2\sigma^2}\right]
\end{align}</math>

where

<math display=block>S = \sum_{i=1}^n (x_i-\mu)^2.</math>

Then:

<math display=block>\begin{align}
p(\sigma^2\mid\mathbf{X}) &\propto p(\mathbf{X}\mid\sigma^2) p(\sigma^2) \\
&= \left(\frac{1}{2\pi\sigma^2}\right)^{n/2} \exp\left[-\frac{S}{2\sigma^2}\right] \frac{(\sigma_0^2\frac{\nu_0}{2})^{\frac{\nu_0}{2}}}{\Gamma\left(\frac{\nu_0}{2} \right)}~\frac{\exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]}{(\sigma^2)^{1+\frac{\nu_0}{2}}} \\
&\propto \left(\frac{1}{\sigma^2}\right)^{n/2} \frac{1}{(\sigma^2)^{1+\frac{\nu_0}{2}}} \exp\left[-\frac{S}{2\sigma^2} + \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right] \\
&= \frac{1}{(\sigma^2)^{1+\frac{\nu_0+n}{2}}} \exp\left[-\frac{\nu_0 \sigma_0^2 + S}{2\sigma^2}\right]
\end{align}</math>

The above is also a scaled inverse chi-squared distribution where

<math display=block>\begin{align}
\nu_0' &= \nu_0 + n \\
\nu_0'{\sigma_0^2}' &= \nu_0 \sigma_0^2 + \sum_{i=1}^n (x_i-\mu)^2
\end{align}</math>

or equivalently

<math display=block>\begin{align}
\nu_0' &= \nu_0 + n \\
{\sigma_0^2}' &= \frac{\nu_0 \sigma_0^2 + \sum_{i=1}^n (x_i-\mu)^2}{\nu_0+n}
\end{align}</math>

Reparameterizing in terms of an [[inverse gamma distribution]], the result is:

<math display=block>\begin{align}
\alpha' &= \alpha + \frac{n}{2} \\
\beta' &= \beta + \frac{\sum_{i=1}^n (x_i-\mu)^2}{2}
\end{align}</math>