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9.4-PartialDirectedCRF

# 9.4-PartialDirectedCRF - Machine Learning Srihari Partially...

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Machine Learning Srihari 1 Partially Directed Models and Conditional Random Fields Sargur Srihari

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Machine Learning Srihari Topics Unifying directed and undirected graphs Conditional Random Fields – CRF Semantics – CRFs for Text Analysis Chain Graph Models – Independencies Summary and Discussion – Bayesian Networks versus Markov Networks 2
Machine Learning Srihari Partially Directed Models Can unify directed/undirected dependencies Conditional Random Fields have dependency on some subset of variables They can be generalized to chain graphs – Network in which undirected components depend upon each other in a directed fashion 3

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Machine Learning Srihari Conditional Random Fields Use Markov Network formalism (to represent a joint distribution over X) to represent a conditional distribution P(Y|X) Y is a set of target variables X is a set of observed variables In the case of MNs it is called a CRF Has an analog in directed graphical models – Conditional Bayesian Networks 4
Machine Learning Srihari CRF Representation and Semantics CRF nodes correspond to Y U X Y are target variables and X are observed variables Parameterized as ordinary Markov Network – Set of factors Φ 1 (D 1 ),.. Φ m (D m ) Can be encoded as a log-linear model Viewed as encoding a set of factors Instead of P(Y,X) view it as representing P(Y|X) To naturally represent a conditional distribution – Disallow potentials involving only variables in X 5

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Machine Learning Srihari CRF Definition A CRF is an undirected graph H whose nodes correspond to X U Y Network is annotated with a set of factors Network encodes a conditional distribution as Two variables in H are connected by an edge whenever they appear in the scope of a factor P ( Y | X ) = 1 Z ( X ) P ( Y , X ) P ( Y , X ) = φ i ( D i ) i = 1 m Z ( X ) = P ( Y , X ) Y φ 1 ( D 1 ),.. φ m ( D m ) such that D i X Partition function Is now a function of X Where Z(X) is the marginal distribution of X and P(Y,X) is the joint distribution Joint distribution is a product of factors
Machine Learning Srihari CRF vs Gibbs Distribution Different normalization in partition function Z(X) CRF induces a different value in the partition function for every assignment x to X Difference is denoted graphically by grayed out nodes 7

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Machine Learning Srihari Example of CRF CRF over Y={Y 1 ,..Y k } and X={X 1 ,..X k } Edges are Y i —Y i+1 and Y i —X i 8 P ( Y | X ) = 1 Z ( X ) P ( Y , X ) P ( Y , X ) = φ i ( Y i , Y i + 1 ) i = 1 k 1 φ i ( Y i , X i ) i = 1 k Z ( X ) = P ( Y , X ) Y Observed Feature Variables Called sequence labeling
Machine Learning Srihari CRF as Partially Directed Graph CRF defines a conditional distribution of Y on X Can be viewed as one with undirected component over Y which has X as parents 9

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Machine Learning Srihari CRF of binary variables X={X 1 ,..X k } and Y={Y}
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