\begin{align}
    f(y_{i}^{(p)}, z_{i}^{(p)}) &= \sum_{k = 1}^K \theta^{(p)}_{k} \cdot \textrm{TruncNorm}(y_{i}^{(p)}, z_{i}^{(p)} | \boldsymbol{\mu}_k, \boldsymbol{\Sigma}_k) \label{eq:initial_likelihood} \\
    \boldsymbol\theta^{(p)} &\sim \text {Dirichlet}_{K}(\alpha \cdot \boldsymbol\beta) \label{eq:hierarchy}
\end{align}
