9.2-Independencies-MNs

9.2-Independencies-MNs - Machine Learning Srihari Semantics...

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Machine Learning Srihari 1 Semantics of Markov Networks Sargur Srihari [email protected]
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Machine Learning Srihari Topics Independencies in Markov Networks – Global and Local Independencies – From Distributions to Graphs • Parameterization – Factor Graphs – Log-linear Models – Features (Ising, Boltzmann) – Overparameterization & Canonical Parameterization Bayesian Networks and Markov Networks – From BN to MN: Moralized graphs – From MN to BN: Chordal graphs 2
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Machine Learning Srihari Global Independence Definition H is a Markov network X 1 ,..X k is a path in H Z : set of observed variables 1.Path is active given Z if none of the X i is in Z If B and D are observed, path A,C is inactive 2.Set of nodes Z separates sets X and Y If no active path between any node X X and Y Y 3.Global independencies associated with H are I( H )={(X Y|Z): sep H (X,Y|Z)} Independencies in I (H) guaranteed to hold for every distribution P over H I( H )={(A C|B,D), (B D|A,C)}
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Machine Learning Srihari Conditional Independence in Markov Networks Markov network encodes a set of conditional independencies Probabilistic influence flows – in undirected paths Blocked if we condition on intervening nodes Every path from any node in A to B passes through C No explaining away Testing for independence simpler than in directed graphs Alternative view Remove all nodes in set C together with all their connecting links If no paths from A to B then conditional independence holds Markov blanket A node is conditionally independent of all other nodes conditioned only on its neighbors 4 Set of nodes C Separates sets A and B
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Machine Learning Srihari 5 Factorization Properties Factorization rule corresponds to conditional independence test Notion of locality needed Consider two nodes x i and x j not connected by a link They are conditionally independent given all other nodes in graph Because there is no direct path between them and All other paths pass through nodes that are observed and hence those paths are blocked Expressed as – Where denotes set x of all variables with x i and x j removed For conditional independence to hold factorization is such that x i and x j do not appear in the same factor No path between them other than going through others leads to graph concept of clique
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Machine Learning Srihari 6 Graphical Model as Filter UI is set of distributions that are consistent with set of conditional independence statements read from the graph using graph separation UF are set of distributions that can be expressed as factorization of the form Hammersley-Clifford theorem states that UI and UF are identical
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Machine Learning Srihari Gibbs Distribution and Graph • Let P be a distribution over χ ={X 1 ,..X n } and H is a Markov structure over χ • If P is a Gibbs distribution that factorizes over H , then H is an I-map for P – Factorize means every D i in
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