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Graphical Models

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Graphical Models

ML

Duration
45 Hours

Course Description

        A graphical models course provides a comprehensive understanding of probabilistic graphical models (PGMs), which are powerful tools for representing and reasoning about complex systems with interacting variables. The course typically covers the fundamentals of probability theory and graph theory, then delves into directed (Bayesian networks) and undirected (Markov random fields) graphical models. Students will learn about inference (how to answer probabilistic queries) and learning (how to estimate model parameters and structure from data) in these models, as well as their applications in various domains like computer vision, natural language processing, and bioinformatics. 

Course Outline For Graphical Models

Introduction to Graphical Models:

  • What are graphical models and why are they needed?
  • Types of graphical models (e.g., Bayesian Networks, Markov Random Fields, Factor Graphs).
  • Representing conditional independence using graph structures.

Bayesian Networks (BNs):

  • Directed Acyclic Graphs (DAGs) representing causal and probabilistic relationships.
  • Conditional probability distributions (CPDs) associated with each node.
  • Local Markov Property and d-separation.
  • Representing joint probability distributions using BNs.
  • Applications in areas like medical diagnosis and spam classification.
  • BN examples often include scenarios like the Sprinkler problem.

Markov Random Fields (MRFs):

  • Undirected graphs representing dependencies between variables.
  • Potential functions and cliques.
  • Hammersley–Clifford theorem and Gibbs distributions.
  • Markov properties (pairwise, local, global).
  • Applications in image processing, computer vision, and more.

Factor Graphs:

  • A bipartite graph representing the factorization of a function into factors defined on subsets of variables.
  • Relationship between Factor Graphs, BNs, and MRFs.
  • Benefits of using factor graphs for inference algorithms.

Inference in Graphical Models:

  • Exact Inference:
    • Variable Elimination.
    • Sum-Product Algorithm (Belief Propagation).
    • Junction Tree Algorithm.
  • Approximate Inference:
    • Sampling methods (e.g., Markov Chain Monte Carlo, Gibbs Sampling, Metropolis-Hastings).
    • Variational methods (e.g., Mean Field approximation).
    • Loopy Belief Propagation.

Learning in Graphical Models:

  • Parameter Learning:
    • Estimating parameters (CPDs for BNs, potential functions for MRFs) from data.
    • Techniques like Maximum Likelihood Estimation (MLE) and Bayesian Estimation.
    • Expectation-Maximization (EM) algorithm, especially with incomplete data.
  • Structure Learning:
    • Determining the graph structure (dependencies between variables) from data.
    • Constraint-based methods (using conditional independence tests).
    • Score-based methods (evaluating structures with a scoring function).
    • Using integer programming for structure learning.

Specialized Graphical Models:

  • Hidden Markov Models (HMMs): Representing sequential data and understanding dynamic inference, often using the Forward-Backward Algorithm and Viterbi Algorithm.
  • Conditional Random Fields (CRFs): Undirected graphical models for structured prediction, particularly useful for tasks like named entity recognition and image segmentation.
  • Dynamic Bayesian Networks (DBNs): Extending BNs to model time-series data.
  • Kalman Filters.
  • Restricted Boltzmann Machines (RBMs).
  • Applications of Graphical Models:
    • Machine learning and Deep Learning algorithms like Naive Bayes and Neural Networks.
    • Computer Vision (image segmentation, object detection).
    • Natural Language Processing (NLP) (part-of-speech tagging, named entity recognition).
    • Robotics.
    • Bioinformatics.
    • Causal inference.
    • Web/IR, and biology. 
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