This module investigates deep latent variable models, focusing on variational autoencoders (VAEs) and related probabilistic methods. You will analyze the mathematics behind sampling strategies, evidence lower bound (ELBO), variational inference, reparameterization tricks, and amortized inference, developing an advanced toolkit for probabilistic generative modeling.

In this module, you'll explore normalizing flows as precise tools for modeling complex probability distributions through invertible neural networks. You’ll examine the underpinnings, including determinants, geometry, invertibility constraints, and specific flow architectures like Real-NVP and autoregressive models. You'll also investigate practical applications and synthesis of complex densities using normalizing flows.

This module provides a deep exploration of Generative Adversarial Networks (GANs), focusing on their formulation as likelihood-free generative models. You'll analyze GAN training dynamics, including optimization challenges, mode collapse, and divergence minimization strategies. The module also covers advanced GAN variants such as f-GAN and Wasserstein GAN (WGAN).

In this module, you will explore energy-based generative models and score-based modeling frameworks from a mathematical and implementation perspective. You'll dive deeply into the details of training via score functions, contrastive divergence, and various forms of score matching including denoising techniques, highlighting their theoretical and practical implications.

You'll delve deeply into diffusion models, understanding them mathematically as stochastic processes and connecting them explicitly to score-based models. The module examines forward and reverse diffusion processes, training objectives, SDEs, predictor-corrector methods, and latent diffusion architectures, providing robust foundations for modern generative modeling.

In this module, you'll study annealed importance sampling (AIS) methods for estimating complex probability distributions with rigorous mathematical treatment. You will mathematically analyze AIS step-by-step processes, intermediate distributions, and normalization constants, applying these techniques effectively to probabilistic models, to wrap up the course. You will also assess the evolution of generative models.