\begin{align}\label{eqn:BDMH_ratio}
    r[(\boldsymbol{\Theta}, \boldsymbol{\Phi}), (\boldsymbol{\Theta}, (\mathbf{c}, \boldsymbol{U}, \Sigma))] &= \frac{p_d\pi(\boldsymbol{\Theta}, \boldsymbol{\Phi} \cup \{(\mathbf{c}, \boldsymbol{U}, \Sigma)\} \mid \mathbf{D})|S|}{p_b\pi(\boldsymbol{\Theta}, \boldsymbol{\Phi} \mid \mathbf{D})(|\boldsymbol{\Phi}|+1)q_b^u(\boldsymbol{U})q_b^s(\Sigma)}.
\end{align}
