$$p_\theta(\mathbf{A}|\mathbf{Z})$$ corresponds to the binary cross-entropy loss over the edge probabilities when we assume independence between edges: $$p_\theta(\mathbf{A}|\mathbf{Z}) = \prod_{(u,v) \in V^2} p_\theta(A[u,v]=1|z_u,z_v)$$

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Given a matrix of sampled node embeddings $$Z \in \mathbb{R}^{|V| \times d}$$, the goal of the decoder model in a Variational Graph Autoencoder (VGAE) is to predict the likelihood of all the edges in the graph. More formally, the decoder must specify $$p_\theta(\mathbf{A}|\mathbf{Z})$$, which is the posterior probability of the adjacency matrix given the node embeddings.

Decoder for Node-Level Latent Representations in VGAEs

Hamilton, W.L. Graph Representation Learning. *Synthesis Lectures on Artificial Intelligence and Machine Learning*. 2020, 14(3), 1–159. https://doi.org/10.2200/S01045ED1V01Y202009AIM046

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