Having established the foundations of probabilistic generative models and representation learning, this chapter focuses on the mathematical architecture of Variational Autoencoders (VAEs). A solid understanding of these mathematical principles is essential for effectively working with and extending VAEs.

Here, you will learn the derivation of VAEs from the principles of variational inference. We will examine the Evidence Lower Bound (ELBO), often written as $L_{ELBO}$, which forms the primary objective function for training VAEs. You will understand the reparameterization trick, a critical technique that enables gradient-based optimization through stochastic latent variables. The role and interpretation of the Kullback-Leibler (KL) divergence, typically $D_{KL}(q(z|x) || p(z))$, within the VAE objective will also be detailed. Additionally, we will cover design considerations for encoder and decoder networks, discuss common VAE training difficulties, and analyze different formulations of VAE objective functions. The chapter includes a practical section on implementing a VAE and performing diagnostics to connect theory with application.