# Explain VAEs, ELBO, KL, and Reparameterization Explain a variational autoencoder, derive the evidence lower bound, and show why the objective contains a reconstruction term and a KL term. Explain the reparameterization trick and why it enables gradient-based training. ### Constraints & Assumptions - Use a latent-variable model p(x, z), encoder q(z | x), and decoder p(x | z). - State the prior over z rather than assuming it silently. - Distinguish maximizing the ELBO from exactly maximizing log evidence. ### Clarifying Questions to Ask - Which likelihood model defines reconstruction quality? - What posterior family does the encoder produce? - How is the KL term weighted in the stated variant? ```hint Add and subtract the posterior Relate log p(x) to the ELBO plus a nonnegative divergence between the approximate and true posterior. ``` ### What a Strong Answer Covers - Encoder, decoder, prior, and generative story. - ELBO derivation and the roles of reconstruction and regularization. - Closed-form or estimated KL behavior and posterior collapse risk. - Reparameterization as deterministic noise transformation and gradient path. ### Follow-up Questions 1. What changes in a beta-VAE? 2. Why can a powerful decoder ignore the latent variable?