Correct renormalization of the product of two probabilistic models I came across these two papers combining an energy-based model (EBM) with a probabilistic generative model: [Xiao et al., 2020] and [Xiao et al., 2021].
In [Xiao et al., 2020], the new model is defined as:
\begin{equation}
p_{\phi, \theta}(x) = \frac{p_{\phi}(x)e^{-E_{\theta}(x)}}{\int p_{\phi}(x)e^{-E_{\theta}(x)} dx} = \frac{p_{\phi}(x) e^{-E_{\theta}(x)}}{Z_{\phi, \theta}}
\end{equation}
where $p_{\phi}(x)$ is a probabilistic generative model and $p_{\theta}(x) = \frac{e^{-E_{\theta}(x)}}{\int e^{-E_{\theta}(x)} dx}$ is an EBM.
In [Xiao et al., 2021], the authors use a VAE generator $p_{\phi}(x, z) = p_{\phi}(z)p_{\phi}(x|z)$ and define the new model as:
\begin{equation}
    h_{\phi, \theta}(x,z) = \frac{p_{\phi}(x, z) e^{-E_{\theta}(x)}}{\int p_{\phi}(x)e^{-E_{\theta}(x)}dx} = \frac{p_{\phi}(x, z) e^{-E_{\theta}(x)}}{Z_{\phi, \theta}}
\end{equation}
My question is: shouldn't the renormalization in the second case be $Z_{\phi, \theta} = \int p_{\phi}(x, z)e^{-E_{\theta}(x)}dx$?

I didn't look at the references but the paper's formula is correct. Indeed (in your notation), the normalization constant of $h_{\phi,\theta}(x,z)$ is
\begin{align*}
Z_{\phi,\theta} &= \iint p_{\phi}(x,z)e^{-E_\theta(x)}\,dz dx\\ 
& = \int\left(\int p_{\phi}(x,z) dz\right) e^{-E_\theta(x)}\,dx\\ 
&= \int p_\phi(x) e^{-E_\theta(x)}\,dx,
\end{align*}
where $p_\phi(x) = \int p_{\phi}(x,z) dz$ is the marginal distribution of $X$ obtained from the joint $p_{\phi}(x,z)$.
So $\int p_{\phi}(x,z)e^{-E_\theta(x)}\,dx$ can't be the norming constant of the joint since it depends on $z$.
