General Gibbs Distribution Representation Probabilistic Graphical Models Markov Networks General Gibbs Distribution

P(A,B,C,D) A D B C

Consider a fully connected pairwise Markov network over X1,…,Xn where each Xi has d values. How many parameters does the network have? O(dn) O(nd) O(n2d2) O(nd) Not every distribution can be represented as a pairwise Markov network

Gibbs Distribution Parameters: General factors i (Di)  = {i (Di)} 0.25 c2 0.35 b2 0.08 0.16 a2 0.05 0.07 a3 0.15 0.21 0.09 0.18 Parameters: General factors i (Di)  = {i (Di)}

Gibbs Distribution

Induced Markov Network B C Induced Markov network H has an edge Xi―Xj whenever

Factorization P factorizes over H if there exist such that H is the induced graph for 

Which Gibbs distribution would induce the graph H? All of the above Graph structure doesn’t uniquely determine parameterization

Flow of Influence A D B C Influence can flow along any trail, regardless of the form of the factors

Active Trails A trail X1 ─ … ─ Xn is active given Z if no Xi is in Z A D B C

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

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