1. Daphne Koller Markov Networks General Gibbs Distribution Probabilistic Graphical Models Representation

2. Daphne Koller BD C A P(A,B,C,D)

3. Template vertLeftWhite1 Consider a fully connected pairwise Markov network over X 1,…,X n where each X i has d values. How many parameters does the network have? O(d n ) O(n d ) O(n 2 d 2 ) O(nd) Not every distribution can be represented as a pairwise Markov network

4. Daphne Koller Gibbs Distribution Parameters: a1a1 b1b1 c1c1 0.25 a1a1 b1b1 c2c2 0.35 a1a1 b2b2 c1c1 0.08 a1a1 b2b2 c2c2 0.16 a2a2 b1b1 c1c1 0.05 a2a2 b1b1 c2c2 0.07 a2a2 b2b2 c1c1 0 a2a2 b2b2 c2c2 0 a3a3 b1b1 c1c1 0.15 a3a3 b1b1 c2c2 0.21 a3a3 b2b2 c1c1 0.09 a3a3 b2b2 c2c2 0.18 General factors  i (D i )  = {  i (D i )}

5. Daphne Koller Gibbs Distribution

6. Daphne Koller Induced Markov Network Induced Markov network H  has an edge X i ―X j whenever BD C A

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

8. Template vertLeftWhite1 BD C A Which Gibbs distribution would induce the graph H? All of the above

9. Daphne Koller Flow of Influence Influence can flow along any trail, regardless of the form of the factors BD C A

10. Daphne Koller Active Trails A trail X 1 ─ … ─ X n is active given Z if no X i is in Z BD C A

11. 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