# Gibbs sampling for parameter estimation

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 AX,\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 AX,\Sigma_{\gamma}) \times \prod\limits_{j}N(\alpha_{j}|0,\sigma^{2}_{\alpha}\mathbf{I}_{3}) \propto |\Sigma_{\gamma}|^{-1} exp[(\gamma_{i} - AX) ^{T} |\Sigma_{\gamma}|^{-1}(\gamma_{i} - AX)] \times exp[\alpha^{T}\alpha / \sigma^{2}]\\ here, \gamma_{i} = (\gamma_{1},\gamma_{2},\gamma_{3}), vector(A)= (\alpha_{1},\alpha_{2},\alpha_{3})^{T}, X= (x_{1},x_{2},x_{3}) \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 $\alpha$ ?

My question is how did they get the posterior distribution of $\alpha$ and how the Kronecker product is coming?

• 1. Tagging someone who has not participated in a post won't work. 2. Please don't tag someone to ask them to answer your question. – Glen_b Nov 7 '17 at 0:51
• i) Ignore all the bits that don't have $\alpha$ in them - it looks like it will be MVN by eyeball. ii) I don't see any Kronecker products in your post, so the question isn't self-contained. iii) There are 3 different kinds of indexing for $\alpha$, this is confusing without more information to the extent the question probably isn't answerable - cf ii. – conjectures Jan 16 '18 at 12:35
• @conjectures: i have edited the question. Please have a look. – Benzamin Jan 16 '18 at 17:05
• @Benzamin the paper's expression for the posterior (3) is different than yours. I also notice that your expression places Normal priors on variance components, that you have different means for your $\gamma$ terms, and that there are some discrepancies with your subscripts. Have you tried a problem like this with a simplified model? You might notice a pattern, and be able to answer this question more easily. – Taylor Jan 16 '18 at 17:43
• @Taylor: yes, you are right. in that posterior they assume mean 0 , but if you look at the supplementary document they assumes mean AX. Again,they have considered the multivariate response and I am trying for univariate case. I already derived every other parameter.I am stuck in this one.any kind of help will be highly appreciated. – Benzamin Jan 16 '18 at 18:13