\begin{align}
    q_c^{pred}(c_{t(k)}=1; \phi_t) & \propto \text{exp} \Big( \mathbb{E}_{-c_{t}} \left[ \text{log} \ p(\boldsymbol\beta, \boldsymbol\mu , \text{\bf{c}}, \mathcal{D}_t) \right]   \Big) \nonumber \\
    & \propto \text{exp} \Big( \mathbb{E}_{-c_{t}} \left[ \text{log} \ p(c_{t(k)}=1,x_t | \pi, \boldsymbol\mu ) \right]  + \text{const} \Big) \nonumber \\
    & \propto \text{exp} \Big( \mathbb{E} \left[ \text{log} \ p(c_{t(k)}=1 | \pi ) \right] + \mathbb{E}_\mu \left[ \text{log} \ p(x_t | c_{t(k)}=1, \mu_k ) \right] + \text{const} \Big) \nonumber \\
    & \propto \text{exp} \Big( \text{log} (\pi_k) - \frac{1}{2} x_t'M^{-1} x_t + x_t'M^{-1} \ \mathbb{E}_\mu( \mu_k) - \frac{1}{2} \mathbb{E}_\mu(\mu_k' M^{-1} \mu_k) \ + \ \text{const} \Big) \nonumber \\
    \label{qpred1}
    & \propto \text{exp} \Big( \text{log} (\pi_k) + x_t'M^{-1} \hat{\mu}_k - \frac{1}{2} \text{trace}(M^{-1} (\hat{\mu}_k \hat{\mu}_k'+\hat{R}_k))
     + \ \text{const} \Big)
\end{align}
