\begin{equation}
\label{factorization}
    p(\boldsymbol\beta, \boldsymbol\mu , \text{\bf{c}}, \text{\bf{x}}, \text{\bf{y}})= \prod_{k=1}^K p(\beta_k) \prod_{k=1}^K p(\mu_k) \prod_{t=1}^T \left[ p(c_t) p(x_t|\boldsymbol\mu, c_t) p(y_{t+1} | x_t, \boldsymbol\beta, c_t) \right]
\end{equation}
