The goal of graph level variational decoder is to define a posterior given a graph level latent embedding. The original graph VAE proposed to combine a multi-layer perceptron (MLP) and Bernoulli distribution assumption to obtain the posterior:

$p_{\theta}(G | \mathbf{z}_G) = \prod_{(u,v)\in \mathcal{V}} \tilde{A}[u,v] A[u,v]+(1-\tilde{A}[u,v]) (1-A[u,v])$

Where $A$ is adjacency matrix and $\tilde{A}=\sigma(MLP(\mathbf{z}_G))$ is the predicted matrix of edge probabilities. The overall log-likelihood objective is equivalent to a set of independent binary cross-entropy loss.