The sampled latent representations encode enough information to allow
our decoder to reconstruct the input. The first goal ensures that we learn to decode meaningful graphs from the encoded latent representations, when we have training graphs as input.

Hamilton College

The key intuition of ELBO is that we want to generate a distribution over latent representations so that the two (conflicting) goals are satisfied.

The Two Goals of ELBO

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

Graph Representation Learning by William Hamilton

Enough Information for ELBO

The latent distribution is as close as possible to the prior. The second goal acts as a regularizer and ensures that we can decode meaningful graphs even when we sample latent representations from the prior p(Z). This second goal is critically important if we want to generate new graphs after training: we can generate new graphs by sampling from the prior and feeding these latent embeddings to the decoder, and this process will only work if this second goal is satisfied.

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