Product of Normal–Gamma Distributions

The normal–gamma distribution is
\begin{eqnarray}
NG(\mu,\lambda|\eta,\tau,\alpha,\beta)
& = & N(\mu|\eta,(\tau\lambda)^{-1})\ G(\lambda|\alpha,\beta) \\
& = &
\frac{\beta^{\alpha}\sqrt{\tau}}{\Gamma(\alpha)\sqrt{2\pi}}
\lambda^{\alpha-\frac{1}{2}}
\exp\left( -\beta\lambda - \frac{1}{2}\tau\lambda(\mu-\eta)^2 \right)
\end{eqnarray}
where \(\Gamma(\cdot)\) is the gamma function.

A product of normal–gamma distributions is proportional to another normal–gamma distribution.
\[
\prod_{i=1}^n NG(\mu,\lambda|\eta_i,\tau_i,\alpha_i,\beta_i)
= Z\times NG(\mu,\lambda|\hat{\eta},\hat{\tau},\hat{\alpha},\hat{\beta})
\]
where
\begin{eqnarray}
\hat{\tau} &=& \sum_{i=1}^n \tau_i \\
\hat{\tau}\hat{\eta} &=& \sum_{i=1}^n \tau_i\eta_i \\
2\hat{\beta} + \hat{\tau}\hat{\eta}^2 &=& \sum_{i=1}^n \left[2\beta_i + \tau_i\eta_i^2\right] \\
\hat{\alpha}-\frac{1}{2} &=& \sum_{i=1}^n \left[\alpha_i - \frac{1}{2}\right]
\end{eqnarray}
and
\[
Z =
\frac{\Gamma(\hat{\alpha})\sqrt{2\pi}}{\hat{\beta}^{\hat{\alpha}}\sqrt{\hat{\tau}}}
\prod_{i=1}^n\left[\frac{\beta_i^{\alpha_i}\sqrt{\tau_i}}{\Gamma(\alpha_i)\sqrt{2\pi}}\right]
\]