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**Unformatted text preview: **Machine Learning Srihari 1 Undirected Graphical Models Sargur Srihari [email protected] Machine Learning Srihari Topics • Directed versus Undirected graphical models • Components of a Markov Network • Independence Properties • Parameterization • Gibbs Distributions and Markov Networks • Reduced Markov Networks 2 Machine Learning Srihari Directed vs Undirected • Bayesian Networks= Directed Graphical Models – Useful in many real-world domains • Undirected Graphical Models – When no natural directionality exists betw. variables – Offer a simpler perspective on directed graphs • Independence structure • Inference task – Also called Markov Networks or MRFs • Can combine directed and undirected graphs • Attention restricted to discrete state spaces 3 Machine Learning Srihari Example to Motivate Undirected Graphs B D A C Alice and Bob are friends Bob and Charles study together Charles and Debbie argue with each other Debbie and Alice study together Alice and Charles can ’ t stand each other Bob and Debbie had relationship ended badly 2. Professor may have mis-spoken e.g., on a machine learning topic 3. Students may have figured out the problem e.g., by thinking about issue or by studying textbook 4. Students transmit this understanding to his/her study partner Alice Debbie Bob Charles 1. Four students study in pairs to work on homework A Social Network Machine Learning Srihari Modeling the Misconception Problem Four binary random variables representing whether or not student has misconception X ∈ { A , B , C , D } x 1 : student has the misconception x : Student does not have a misconcpetion a a 1 0.3 0.7 b b 1 0.2 0.8 a 0 = has misconception a 1 = no misconception Prior Probabilities Machine Learning Srihari Modeling Influences Using a BN Probability of misconception of one person depends on whether their study partner has a misconception Alice and Charles never speak directly Thus A and C should be conditionally independent given B and D We need to model (A ⊥ C|{B,D}) Consider this Proposed Bayesian Network It does model (A ⊥ C|{B,D}) since the path between A and C is blocked when B ,D are known But also means B and D are independent given only A since V-structure through C implies blockage when C is not known But dependent given both A and C since V-structure through C implies no blockage when C is known B D A C Machine Learning Srihari Lack of Perfect Map in BN C A D B B D A C B D A C 7 In both (b) and (c) (A ⊥ C|B,D) holds but (B \ ⊥ D|A,C ) due to v-structure D C B in d-separation We need to model (A ⊥ C|B,D),(B ⊥ D|A,C ) First attempt at BN Second attempt at BN No perfect map since independences imposed by BN are inappropriate for the distribution In a perfect map the graph precisely captures the independencies in the given distribution Misconception Example Alice Debbie Bob Charles Machine Learning Srihari Drawbacks of BN in Example • Independences imposed by BN are inappropriate for the distribution • Interaction between the variables are symmetric 8 Alice Debbie Bob Charles Machine Learning...

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