Normal-gamma distribution ($\tau \sim \mathrm{Gamma}(\alpha ,\beta ),\mu \mid \tau \sim N({\mu }_0,(K\tau {)}^{-1})$)

A normal-gamma conjugate prior for the mean $\mu$ and precision parameter $\tau$ of a Gaussian distribution specifies a Gamma distribution $\tau \sim \mathrm{Gamma}(\alpha ,\beta )$ in the shape-rate convention and a conditional Gaussian distribution $\mu \mid \tau \sim N({\mu }_0,(K\tau {)}^{-1})$. Its joint probability density function is proportional to ${\tau }^{\alpha -1}\sqrt{\tau }\exp [-\beta \tau -K\tau (\mu -{\mu }_0{)}^2/2]$, with $\alpha ,\beta ,K>0$. For $n$ independent observations with Gaussian distribution $N(\mu ,{\tau }^{-1})$, multiplication by the likelihood and completing the square givesand ${\beta }_n=\beta +{\sum }_i(x_i-\bar{x}{)}^2/2+Kn(\bar{x}-{\mu }_0{)}^2/[2(K+n)]$. The identity behind conjugacy is $K(\mu -{\mu }_0{)}^2+{\sum }_i(x_i-\mu {)}^2=(K+n)(\mu -{\mu }_n{)}^2+{\sum }_i(x_i-\bar{x}{)}^2+Kn(\bar{x}-{\mu }_0{)}^2/(K+n)$.