Probabilistic Graphical Models Representation Markov Networks General Gibbs
Probabilistic Graphical Models Representation Markov Networks General Gibbs Distribution Daphne Koller
P(A, B, C, D) A D B C Daphne Koller
Consider a fully connected pairwise Markov network over X 1, …, Xn where each Xi has d values. How many parameters does the network have? O(dn) O(nd) O(n 2 d 2) O(nd) Not every distribution can be represented as a pairwise Markov network
Gibbs Distribution • Parameters: General factors i (Di) = { i (Di)} a 1 b 1 c 1 0. 25 a 1 b 1 c 2 0. 35 a 1 b 2 c 1 0. 08 a 1 b 2 c 2 0. 16 a 2 b 1 c 1 0. 05 a 2 b 1 c 2 0. 07 a 2 b 2 c 1 0 a 2 b 2 c 2 0 a 3 b 1 c 1 0. 15 a 3 b 1 c 2 0. 21 a 3 b 2 c 1 0. 09 a 3 b 2 c 2 0. 18 Daphne Koller
Gibbs Distribution Daphne Koller
Induced Markov Network A D B C Induced Markov network H has an edge Xi―Xj whenever Daphne Koller
Factorization P factorizes over H if there exist such that H is the induced graph for Daphne Koller
Which Gibbs distribution would induce the graph H? A D B C All of the above
Flow of Influence A D B C • Influence can flow along any trail, regardless of the form of the factors Daphne Koller
Active Trails • A trail X 1 ─ … ─ Xn is active given Z if no Xi is in Z A D B C Daphne Koller
Summary • Gibbs distribution represents distribution as a product of factors • Induced Markov network connects every pair of nodes that are in the same factor • Markov network structure doesn’t fully specify the factorization of P • But active trails depend only on graph structure Daphne Koller
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