\begin{equation} \label{eq:diff_log_p_xi}
    \frac{\partial}{\partial \xi^{k,t}_j} \log p(W_{\mathcal{D}(1:G,t)} | \phi^{\cdot,t}, \psi^{\cdot,t}) = \sum_{v=1}^V \frac{\partial}{\partial  \psi^{k,t}_v} \log p(W_{\mathcal{D}(1:G,t)} | \phi^{\cdot,t}, \psi^{\cdot,t}) \frac{\partial \psi^{k,t}_v}{\partial  \xi^{k,t}_j} \text{.}
\end{equation}
