Variational autoencoder ELBO

Balances explaining each observation through latent samples with keeping the learned latent distribution close to the model prior.

The variational objective for x equals expected log decoder probability of x given z under q phi z given x, minus K L divergence from the approximate posterior to the prior.

Encode x into plausible latent explanations, reward explanations that reconstruct x, and charge for moving too far from the prior.

Subtraction expresses the tradeoff between reconstruction quality and a usable, prior-aligned latent space.

The paper uses reparameterized Monte Carlo samples to differentiate the expectation efficiently despite latent randomness.