\begin{align}
\label{qmu1}
q_\mu(\mu_k ; \hat{\mu}_k, \hat{R}_k) & =  \frac{\text{exp}(-\frac{1}{2}(\mu_k-\hat{\mu}_k)'\hat{R}_k^{-1}(\mu_k-\hat{\mu}_k))}{\sqrt{(2 \pi)^n|\hat{R}_{k}|}} \; \; \; \text{for  k } \in \{1,\cdots,K\} \\
\label{qbeta1}
q_\beta(\beta_k ; \hat{\beta}_k, \hat{Q}_k) & =  \frac{\text{exp}(-\frac{1}{2}(\beta_k-\hat{\beta}_k)'\hat{Q}_k^{-1}(\beta_k-\hat{\beta}_k))}{\sqrt{(2 \pi)^n|\hat{Q}_{k}|}} \; \; \; \text{for  k } \in \{1,\cdots,K\} 
\end{align}
