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Unformatted text preview: CSE 6740 Lecture 13 How Do I Make Fancier Models? I (Graphical Models) Alexander Gray (Thanks to Nishant Mehta) [email protected] Georgia Institute of Technology What are graphical models? For a set of random variables, a graphical model efficiently exhibits the dependence relationships of those variables. We define the following graph G ( V , E ) which serves as an alternate representation of our random variables X and their conditional independence relationships: • For each random variable X i ∈ X , we have a node in the vertex set V . • The existence of an edge ( X i , X j ) ∈ E implies that X i and X j are dependent. X 1 X 2 X 3 X 4 X 5 X 6 X 1 2 / 54 Directed and Undirected Conditional independence relationships are in general less obvious from the graphical structure, but some simple rules can be used to discover these. Three graphs are of particular interest • Directed acyclic graphs  Bayesian networks • Undirected graphs  Markov random fields • Factor graphs  bipartite graphs of nodes and factors 3 / 54 Conditional Independence A set of random variables A is conditionally independent of a set of random variables B , given another set of random variables C , if Pr( A  B , C ) = Pr( A  C ) We concisely can express the above relationship as A ⊥⊥ B  C . 4 / 54 3 Types of Graphical Models zombie hungry brains happy gradstudent Directed acyclic graphs (DAGs) Bayesian networks 5 / 54 3 Types of Graphical Models X 1 , 1 X 2 , 1 X 3 , 1 X 1 , 2 X 2 , 2 X 3 , 2 X 1 , 3 X 2 , 3 X 3 , 3 Undirected graphs Markov random fields 6 / 54 3 Types of Graphical Models zombie hungry brains happy gradstudent Factor graphs bipartite graphs Each node represents a random variable Each factor represents a potential function operating on the adjacent random variables 7 / 54 Examples of Bayesian networks Y Gaussian 8 / 54 Examples of Bayesian networks X Y Mixture of Gaussians 9 / 54 Examples of Bayesian networks X Y μ Σ π Y Bayesian Mixture of Gaussians 10 / 54 Examples of Bayesian networks X Y Principal Components Analysis 11 / 54 Examples of Bayesian networks Y t 1 Y t Y t +1 X t 1 X t X t +1 Y t 1 Y t Y t +1 Hidden Markov model 12 / 54 Examples of Bayesian networks X t 1 X t X t +1 Y t 1 Y t Y t +1 Y t 1 Y t Y t +1 Linear dynamic system 13 / 54 Temporal extensions N X Y Plate notation 14 / 54 Conditional independence for each type of graph Inferring conditional independence relationships is different in directed and undirected graphical models. • DAGs  dseparation, Bayes ball algorithm • Undirected graphs  if all paths from A to B pass through C, then A and B are conditionally independent given C • Factor graphs  a variable is conditionally independent of all other variables, given its neighbors 15 / 54 Directed conditional independence The notion of directed separation, or dseparation, comes down to a few canonical graphical submodels of three nodes: • Chain • Vstructure • Single parent structure 16 / 54 Chain  pass X Y Z Y 17 / 54 Chain  blocked X Y Z 18 / 54 Single parent  pass X Y Z Y 19 / 54 Single parent  blocked X Y Z 20 / 54 Vstructure  blocked X Y Z Y 21 / 54 Vstructure  pass X Y Z 22 / 54 Undirected conditional independence...
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 Fall '08
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 Probability theory, X1, graphical models, elimination algorithm

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