04-654   Introduction to Probabilistic Graphical Model

Course description

This course provides an introduction to the subject of Probabilistic Graphical Models (PGM). PGM gives a unified view for a wide range of problems arising in several domains such as artificial intelligence, statistics, computer systems, computer vision, natural language processing, and computational biology, among many other fields. They provide a very flexible and powerful framework for capturing statistical dependencies in complex, multivariate data. PGM brings together probability theory and graph theory to enable efficient inference, decision-making, and learning in problems with a very large number of attributes and huge datasets. This introductory course will provide you with a strong foundation necessary for applying graphical models to complex problems.

Learning objectives

We will cover key issues including representation, efficient algorithms, inference, and statistical estimation. The course starts by introducing probabilistic graphical models from the very basics, and by presenting the different PGM algorithms and techniques used for inference and learning with directed and undirected graphical models. Then, it delves into the applications and highlights specific comparisons to deep learning models.

Outcomes

Overall, at the end of this course, students will acquire the background knowledge and skills necessary to apply probabilistic graphical models for the efficient resolution of real problems and gain insights on some advanced Deep Learning models. Upon completion, students will be well-prepared for research roles or advanced studies in fields like machine learning, AI, and data science.

Modeling and Analysis of Complex Systems:

• Develop the ability to model and analyze complex systems across diverse fields, including computer vision, biology, and language processing.
• Gain expertise in both discrete and continuous sequential problem-solving.
• Acquire skills to handle time series data and make informed predictions.

Foundational Knowledge in PGMs:

• Understand the fundamentals of Markov Random Fields (Undirected Graphical Models) and Bayesian Networks (Directed Graphical Models).
• Learn the associations between random variables, conditional independence, and factorization.
• Apply this foundational knowledge to infer latent variables, estimate parameters, and make predictions in complex models, essential for machine learning, computer vision, and natural language processing.

Application of Inference Algorithms:

• Master inference algorithms such as Variable Elimination and Belief Propagation to solve real-world problems.
• Solve parameter estimation challenges using techniques like Maximum Likelihood Estimation and Expectation-Maximization.
• Gain practical experience with Hidden Markov Models (HMMs) and Conditional Random Fields (CRFs) in applications like image segmentation and speech recognition.

Advanced Variational and Sampling Methods:

• Implement variational inference and sampling methods, crucial for approximating complex probability distributions in machine learning and data analysis.
• Apply advanced techniques like Variational Inference, Metropolis-Hastings, Gibbs Sampling, and Hamiltonian Monte Carlo to handle large-scale data and computational challenges.
• Utilize synthetic data generation for inference, enhancing the capacity to model real-world phenomena.

Integration with Deep Learning:

• Explore deep generative models like Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs) for tasks such as image and text generation.
• Build expertise in designing advanced AI systems, with applications in autonomous systems, healthcare, and finance.
• Develop a foundation in structure learning and causal discovery, essential for tasks like policy evaluation, decision-making, and understanding complex systems.
• Learn reinforcement learning as inference and meta-learning, vital for cutting-edge AI research in areas like robotics and gaming.

Preparation for Research and Industry Roles:

• Scale algorithms and systems to handle large datasets, preparing for real-world challenges.
• Implement large-scale algorithms and systems in research labs, contributing to groundbreaking AI and data science advancements.

Content details

Module 1: Motivation and Revision

• Motivational Examples
• Revision Probability
• Revision Graph Theory

Module 2: Basics of Graphical Models

• Directed Graphs
• Factorization
• Markov Properties
• Conditional Independence
• Undirected Graphs
• Factorization
• Markov Properties
• Conditional Independence
• Factor Graphs

Module 3: Exact Inference

• Variable Elimination
• Sum-Product on Trees
• Factor Tree
• Belief Propagation
• Junction Tree
• Mode computation (Max product)

Module 4: Parameter Estimation

• Fully observed: MLE, MAP, Bayesian
• Exponential family distributions, GLMs,
• Partially observed: EM Algorithm

Module 5: Case studies (HMM & CRF)

• HMM Forward-Backward algorithm
• Viterbi Decoding
• Posterior decoding
• Conditional Random Field
• Image segmentation
• Handwritten recognition
• Discriminative Random field
• MCMC sampling(Variational Method)

Module 6: Variational Inference

• Variational methods
• LDA
• Stochastic/Black-box VI
• VI Theory

Module 7: Sampling

• Accept-reject sampling
• Importance sampling
• Metropolis-Hastings
• Gibbs sampling
• Hamiltonian Monte Carlo
• Langevin dynamics
• Sequential Monte Carlo

Module 8: Foundation of Deep Learning

• Insights into deep learning
• Connections to Graphical Models

Module 9: Deep Sequence Models

• Recurrent Neural Network
• Long Short-Term Memory
• Attention Mechanism
• Transformers
• Case Study: Text Generation

Module 10: Deep Generative Models

• Convolutional Neural Networks
• Variational autoencoder
• Generative adversarial networks
• More GANs and variants
• Normalizing flows
• Integrating domain knowledge in deep learning

• Undirected GM: Gaussian GM
• Directed GM: Causal Discovery

Module 12: Causality & RL as Inference

• Causal Model
• Identification
• Identification and Estimation of Causal Effects
• Causal Discovery
• RL as Inference

Module 13: Spectral Graphical Models & Meta Learning

• Spectral Graphical Models
• Large-scale Algorithms and Systems
• Meta-Learning

Prerequisites

Students are expected to have an undergraduate-level background in linear algebra, multivariate calculus, probability theory, statistics, and some basic graph theory, Python Programming, Numpy.

Faculty

Assane Gueye