I am reading the paper by Willemsen et al (2015), "A multivariate Bayesian model for embryonic growth", Statistics in Medicine, 34:8, 1351–1365
where they define the posterior distribution as, \begin{multline} p(\alpha,\beta,\sigma^{2},\gamma,\Sigma_{\gamma}|y) \propto \prod\limits_{ij} N(y_{ij}|\gamma_{i2} + z^{T}_{ij}\beta_{j},\sigma^{2}) \prod\limits_{i}N(\gamma_{i} \mid \alpha X,\Sigma_{\gamma})\\ \times \prod\limits_{j}N(\alpha_{j}|0,\sigma^{2}_{\alpha}\mathbf{I}_{n}) \times \prod\limits_{j}N(\beta_{j}|0,\sigma^{2}_{\beta}\mathbf{I}_{5}) \times \prod\limits_{i}N(\sigma^{2}_{i}|\alpha_{\sigma},\beta_{\sigma}) IW (\Sigma_{\gamma}|\delta,\psi) \end{multline}
where, $y_{ij} = \gamma_{i2} + z^{T}_{ij}\beta + \epsilon_{ij}$
where, $z_{ij} = B(\exp(\gamma_{i3})(t_{ij} + \gamma_{i1} )) , i= 1, \ldots N; j= 1, \ldots n $, $\gamma_{i} = (\gamma_{i1},\gamma_{i2},\gamma_{i3})$, $\gamma_{i} \sim N_{3}(0, \Sigma_{(3*3)})$, $\epsilon_{ij} \sim N(0, \sigma^{2}) $
where, $z^{T}_{ij}$ is a spline function. I am trying to figure out the full conditional distribution(Gibbs sampling) for $\alpha$.
Question:
If I take the parameter associated to alpha then the condition posterior for alpha becomes, \begin{multline} p(\alpha|y,all) \propto \prod\limits_{i}N(\gamma_{i} \mid \alpha X,\Sigma_{\gamma}) \times \prod\limits_{j}N(\alpha_{j}|0,\sigma^{2}_{\alpha}\mathbf{I}_{3}) \propto |\Sigma_{\gamma}|^{-1} exp[(\gamma - \alpha X) ^{T} |\Sigma_{\gamma}|^{-1}(\gamma - \alpha X)] \times exp[\alpha^{T}\alpha / \sigma^{2}] \end{multline}
They said the $\alpha \sim N(\bar \mu_{\alpha},\bar\Sigma_{\alpha})$, where $\bar\mu_{\alpha} = \bar\Sigma_{\alpha}[(X^T \otimes \bar\Sigma_{\gamma}^{-1})\gamma]$ and $\bar\Sigma_{\alpha}=[X^TX \otimes \bar\Sigma_{\gamma}^{-1} + \sigma^{2}I_{n}] ^{-1}$
where, X is the design matrix for $[x_{1},x_{2},...,x_{N}]^{T}$
My question is how did they get the posterior distribution of $\alpha$ ?