Probabilistic Graphical Models

  1. Representation: Types of Graphical Models
    1. Directed Graphical Models: Bayesian Networks
      1. Structure and Basic Properties
        1. Directed Acyclic Graphs (DAGs)
          1. Definition and Properties
            1. Topological Ordering
              1. Parent and Child Relationships
              2. Node Types and Semantics
                1. Root Nodes
                  1. Leaf Nodes
                    1. Internal Nodes
                    2. Local Markov Property
                      1. Definition and Implications
                        1. Local Independence Assumptions
                      2. Factorization and Representation
                        1. The Chain Rule for Bayesian Networks
                          1. Factorization According to Graph Structure
                            1. Ordered Factorization
                            2. Factored Representation of Joint Distribution
                              1. Conditional Probability Tables (CPTs)
                                1. Parameterization of CPTs
                                  1. Continuous Variable Representations
                                2. Conditional Independence in Bayesian Networks
                                  1. D-separation (Directed Separation)
                                    1. Definition and Rules
                                      1. Three Basic Configurations
                                      2. Connection Types
                                        1. Head-to-Head Nodes (V-structures)
                                          1. Head-to-Tail Nodes (Serial Connection)
                                            1. Tail-to-Tail Nodes (Diverging Connection)
                                            2. Markov Blanket
                                              1. Definition and Components
                                                1. Use in Local Inference
                                                  1. Minimal Markov Blanket
                                                2. Common Bayesian Network Architectures
                                                  1. Naive Bayes
                                                    1. Structure and Assumptions
                                                      1. Independence Assumptions
                                                        1. Applications and Limitations
                                                          1. Parameter Estimation
                                                          2. Tree-Augmented Naive Bayes (TAN)
                                                            1. Structure and Motivation
                                                              1. Learning TAN Models
                                                              2. Hidden Markov Models (HMMs)
                                                                1. Structure as a Bayesian Network
                                                                  1. State Variables
                                                                    1. Observation Variables
                                                                      1. Transition and Emission Models
                                                                      2. Kalman Filters
                                                                        1. Linear-Gaussian Assumptions
                                                                          1. State-Space Representation
                                                                            1. Continuous State Variables
                                                                        2. Undirected Graphical Models: Markov Random Fields (MRFs)
                                                                          1. Structure and Basic Properties
                                                                            1. Undirected Graphs
                                                                              1. Definition and Examples
                                                                                1. Neighborhood Relationships
                                                                                2. Markov Properties
                                                                                  1. Local Markov Property
                                                                                    1. Pairwise Markov Property
                                                                                      1. Global Markov Property
                                                                                        1. Equivalence of Markov Properties
                                                                                      2. Factorization and Representation
                                                                                        1. Gibbs Distribution and Potential Functions
                                                                                          1. Definition of Potential Functions
                                                                                            1. Factorization over Cliques
                                                                                              1. Positive Potentials Requirement
                                                                                              2. The Hammersley-Clifford Theorem
                                                                                                1. Statement and Implications
                                                                                                  1. Proof Outline
                                                                                                  2. Factors and Factor Potentials
                                                                                                    1. Definition and Role in MRFs
                                                                                                      1. Clique Potentials
                                                                                                        1. Maximal Clique Factorization
                                                                                                      2. Normalization and Energy Models
                                                                                                        1. Partition Function
                                                                                                          1. Definition and Importance
                                                                                                            1. Computational Challenges
                                                                                                              1. Approximation Methods
                                                                                                              2. Energy-Based Models
                                                                                                                1. Energy Functions
                                                                                                                  1. Relationship to Probabilities
                                                                                                                    1. Log-Linear Models
                                                                                                                  2. Conditional Independence in MRFs
                                                                                                                    1. Graph Separation
                                                                                                                      1. Separation Criteria
                                                                                                                        1. Implications for Inference
                                                                                                                          1. Comparison with D-separation
                                                                                                                        2. Common MRF Architectures
                                                                                                                          1. Ising Model
                                                                                                                            1. Binary Variables
                                                                                                                              1. Pairwise Potentials
                                                                                                                                1. Applications in Physics and Statistics
                                                                                                                                2. Potts Model
                                                                                                                                  1. Multistate Generalization
                                                                                                                                    1. Applications and Extensions
                                                                                                                                    2. Grid-Structured MRFs
                                                                                                                                      1. Image Processing Applications
                                                                                                                                        1. Spatial Dependencies
                                                                                                                                        2. Conditional Random Fields (CRFs)
                                                                                                                                          1. Discriminative Framework
                                                                                                                                            1. Structure and Applications
                                                                                                                                              1. Linear-Chain CRFs
                                                                                                                                          2. Factor Graphs
                                                                                                                                            1. Structure and Representation
                                                                                                                                              1. Bipartite Graph Representation
                                                                                                                                                1. Definition and Properties
                                                                                                                                                  1. Advantages over Other Representations
                                                                                                                                                  2. Variable Nodes
                                                                                                                                                    1. Role and Representation
                                                                                                                                                      1. Connection to Variables
                                                                                                                                                      2. Factor Nodes
                                                                                                                                                        1. Role and Representation
                                                                                                                                                          1. Connection to Factors
                                                                                                                                                        2. Relationships to Other Models
                                                                                                                                                          1. Conversion from Bayesian Networks
                                                                                                                                                            1. Process and Examples
                                                                                                                                                            2. Conversion from Markov Random Fields
                                                                                                                                                              1. Process and Examples
                                                                                                                                                              2. Unified Representation Framework
                                                                                                                                                              3. Applications and Advantages
                                                                                                                                                                1. Message Passing Framework
                                                                                                                                                                  1. Natural Framework for Algorithms
                                                                                                                                                                  2. Algorithmic Benefits
                                                                                                                                                                    1. Clarity in Implementation
                                                                                                                                                                      1. Efficiency Considerations

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                                                                                                                                                                1. Foundations of Probabilistic Graphical Models

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                                                                                                                                                                3. Inference: Answering Queries