CHAPTER 16: Graphical Models

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Presentation transcript:

CHAPTER 16: Graphical Models

Graphical Models Aka Bayesian networks, probabilistic networks Nodes are hypotheses (random vars) and the probabilities corresponds to our belief in the truth of the hypothesis Arcs are direct influences between hypotheses The structure is represented as a directed acyclic graph (DAG) The parameters are the conditional probabilities in the arcs (Pearl, 1988, 2000; Jensen, 1996; Lauritzen, 1996) Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

Conditional Independence X and Y are independent if P(X,Y)=P(X)P(Y) X and Y are conditionally independent given Z if P(X,Y|Z)=P(X|Z)P(Y|Z) or P(X|Y,Z)=P(X|Z) Three canonical cases: Head-to-tail, Tail-to-tail, head-to- head Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

Case 1: Head-to-Tail P(X,Y,Z)=P(X)P(Y|X)P(Z|Y) P(W|C)=P(W|R)P(R|C)+P(W|~R)P(~R|C) Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

Case 2: Tail-to-Tail P(X,Y,Z)=P(X)P(Y|X)P(Z|X) Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

Case 3: Head-to-Head P(X,Y,Z)=P(X)P(Y)P(Z|X,Y) Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

Hidden Markov Model as a Graphical Model Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

Belief Propagation (Pearl, 1988) Chain: Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

Trees Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

Polytrees Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

Undirected Graphs: Markov Random Fields In a Markov random field, dependencies are symmetric, for example, pixels in an image In an undirected graph, A and B are independent if removing C makes them unconnected. Potential function yc(Xc) shows how favorable is the particular configuration X over the clique C The joint is defined in terms of the clique potentials Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

Factor Graphs Define new factor nodes and write the joint in terms of them Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

Learning a Graphical Model Learning the conditional probabilities, either as tables (for discrete case with small number of parents), or as parametric functions Learning the structure of the graph: Doing a state-space search over a score function that uses both goodness of fit to data and some measure of complexity Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)