In this paper https://arxiv.org/pdf/2007.06075.pdf, the authors give a formula in equations 13 and 14, for the ELBO for a specific VAE (latent variable governed by an SDE) that I have difficulty deriving. The author's result is the following one:
\begin{align} \mathcal{L}(\phi,\psi) & = D_{KL}\left(q_{\psi}\left(Z|X\right)q_D\left(X\right)||p_{\phi}\left(Z|X\right)q_{D}\left(X\right)\right) - E_{q_D(X)}[p_{\phi}(X)] \\ & = E_{q_D(X)}\left[E_{q_{\psi}(Z|X)}[\log q_{\psi}(Z|X)-\log p_{\phi}(X,Z)]\right] \\ \end{align}
My attempt to derive the second line is as follows : \begin{align} \mathcal{L}(\phi,\psi) & = D_{KL}(q_{\psi}(Z|X)q_D(X)||p_{\phi}(Z|X)q_{D}(X)) - E_{q_D(X)}[p_{\phi}(X)] \\ & = E_{q_D(X)q_{\psi}(Z|X)}[\log q_{\psi}(Z|X)-\log p_{\phi}(Z|X)]-E_{q_D(X)}[p_{\phi}(X)] \\ & = E_{q_D(X)}\left[E_{q_{\psi}(Z|X)}\left[\log q_{\psi}(Z|X)-\log p_{\phi}(Z|X)\right]-p_{\phi}(X)\right] \\ & = E_{q_D(X)}\left[E_{q_{\psi}(Z|X)}\left[\log q_{\psi}(Z|X)-\log p_{\phi}(X,Z)+\log p_{\phi}(X)\right]-p_{\phi}(X)\right] \\ & = E_{q_D(X)}\left[E_{q_{\psi}(Z|X)}\left[\log q_{\psi}(Z|X)-\log p_{\phi}(X,Z)\right]+\log p_{\phi}(X)-p_{\phi}(X)\right] \\ \end{align} I don't see a way to further simplify the equation, but the authors seem to have a way to simplify $\log p_{\phi}(X)$ and $p_{\phi}(X)$.
Any suggestions or thoughts? Thanks