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Assume $\alpha ,\beta ,\kappa >0$, with the given gamma using shape and rate. PutThe joint posterior density before normalization isComplete each square and integrate over all ${\theta }_i$. The Gaussian integral contributes ${\tau }^{-n/2}$, givingThis hierarchical form completely specifies the normal-gamma distribution of the posterior. For comparison, $\tau \mid \theta ,X$ has shape $\alpha +n$ and rate $\beta +\frac{1}{2}{\sum }_i[{\theta }_i^2+\kappa (X_i-{\theta }_i{)}^2]$; confusing this conditional shape with the marginal shape would lose the Gaussian-integration factor.

Marginally $\theta$ has a multivariate Student distribution with $2A=2\alpha +n$ degrees of freedom and scale matrix $B/[A(1+\kappa )]I_n$. The coordinates are conditionally independent given $\tau$, but are generally dependent after integrating their shared precision. This is conjugate shrinkage with uncertainty in the overall noise scale.

Let $\bar{X}=n^{-1}{\sum }_i{X}_i$ and $S={\sum }_i(X_i-\bar{X}{)}^2$. The Gaussian distribution likelihood is proportional to ${\tau }^{n/2}\exp [-\tau {\sum }_i(X_i-\mu {)}^2/2]$. Multiplying by the normal-gamma distribution prior gives the quadratic exponent ${\sum }_i(X_i-\mu {)}^2+K_0(\mu -{\mu }_0{)}^2+2{\beta }_0$. Completing the square,The power of $\tau$ is ${\tau }^{{\alpha }_0+n/2-1}\sqrt{\tau }$, so the posterior distribution has exactly the required family withandFor a proper prior, $K_0,{\alpha }_0,{\beta }_0>0$ and ${\mu }_0\in \mathbb{R}$. The separate $\sqrt{\tau }$ factor is retained in both densities; it does not add $1/2$ to the update of $\alpha$. This is a conjugate prior for the Gaussian mean and precision parameter.