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CS498-EA Reasoning in AI Lecture #15

CS498-EA Reasoning in AI Lecture #15. Instructor: Eyal Amir Fall Semester 2011. Summary of last time: Inference. We presented the variable elimination algorithm Specifically, VE for finding marginal P(X i ) over one variable, X i from X 1 ,…,X n Order on variables such that

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CS498-EA Reasoning in AI Lecture #15

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  1. CS498-EAReasoning in AILecture #15 Instructor: Eyal Amir Fall Semester 2011

  2. Summary of last time: Inference • We presented the variable elimination algorithm • Specifically, VE for finding marginal P(Xi) over one variable, Xi from X1,…,Xn • Order on variables such that • One variable Xj eliminated at a time (a) Move unneeded terms (those not involving Xj) outside summation over Xj (b) Create a new potential function, fXj(.) over other variables appearing in the terms of the summation at (a) • Works for both BNs and MFs (Markov Fields)

  3. Today • Treewidth methods: • Variable elimination • Clique tree algorithm • Treewidth

  4. Junction Tree • Why junction tree? • Foundations for “Loopy Belief Propagation” approximate inference • More efficient for some tasks than VE • We can avoid cycles if we turn highly-interconnected subsets of the nodes into “supernodes”  cluster • Objective • Compute • is a value of a variable and is evidence for a set of variable

  5. ABD ADE DEF AD DE Cluster ABD SepsetDE Properties of Junction Tree • An undirected tree • Each node is a cluster (nonempty set) of variables • Running intersection property: • Given two clusters and , all clusters on the path between and contain • Separator sets (sepsets): • Intersection of the adjacent cluster

  6. Potentials • Potentials: • Denoted by • Marginalization • , the marginalization of into X • Multiplication • , the multiplication of and

  7. Properties of Junction Tree • Belief potentials: • Map each instantiation of clusters or sepsets into a real number • Constraints: • Consistency: for each cluster and neighboring sepset • The joint distribution

  8. Properties of Junction Tree • If a junction tree satisfies the properties, it follows that: • For each cluster (or sepset) , • The probability distribution of any variable , using any cluster (or sepset) that contains

  9. Moral Graph Triangulated Graph Junction Tree Identifying Cliques Building Junction Trees DAG

  10. A B C G D E H F Constructing the Moral Graph

  11. A B C G D E H F Constructing The Moral Graph • Add undirected edges to all co-parents which are not currently joined –Marrying parents

  12. A B C G D E H F Constructing The Moral Graph • Add undirected edges to all co-parents which are not currently joined –Marrying parents • Drop the directions of the arcs

  13. A B C G D E H F Triangulating • An undirected graph is triangulated iff every cycle of length >3 contains an edge to connects two nonadjacent nodes

  14. EGH CEG A B C G DEF ACE D E H ABD ADE F Identifying Cliques • A clique is a subgraph of an undirected graph that is complete and maximal

  15. EGH CEG ABD ACE CEG ADE AD AE CE DEF ACE DE EG ABD ADE DEF EGH Junction Tree • A junction tree is a subgraph of the clique graph that • is a tree • contains all the cliques • satisfies the running intersection property

  16. DAG Junction Tree Initialization Inconsistent Junction Tree Propagation Consistent Junction Tree Marginalization Principle of Inference

  17. X1 X2 Y1 Y2 X1,Y1 X2,Y2 X1,X2 X1 X2 Example: Create Join Tree HMM with 2 time steps: Junction Tree:

  18. X1,Y1 X2,Y2 X1,X2 X1 X2 Example: Initialization

  19. Example: Collect Evidence • Choose arbitrary clique, e.g. X1,X2, where all potential functions will be collected. • Call recursively neighboring cliques for messages: • 1. Call X1,Y1. • 1. Projection: • 2. Absorption:

  20. X1,Y1 X2,Y2 X1,X2 X1 X2 Example: Collect Evidence (cont.) • 2. Call X2,Y2: • 1. Projection: • 2. Absorption:

  21. Example: Distribute Evidence • Pass messages recursively to neighboring nodes • Pass message from X1,X2 to X1,Y1: • 1. Projection: • 2. Absorption:

  22. X1,Y1 X2,Y2 X1,X2 X1 X2 Example: Distribute Evidence (cont.) • Pass message from X1,X2 to X2,Y2: • 1. Projection: • 2. Absorption:

  23. Example: Inference with evidence • Assume we want to compute: P(X2|Y1=0,Y2=1) (state estimation) • Assign likelihoods to the potential functions during initialization:

  24. Example: Inference with evidence (cont.) • Repeating the same steps as in the previous case, we obtain:

  25. Next Time • Learning BNs and MFs

  26. THE END

  27. Example: Naïve Bayesian Model • A common model in early diagnosis: • Symptoms are conditionally independent given the disease (or fault) • Thus, if • X1,…,Xp denote whether the symptoms exhibited by the patient (headache, high-fever, etc.) and • H denotes the hypothesis about the patients health • then, P(X1,…,Xp,H) = P(H)P(X1|H)…P(Xp|H), • This naïve Bayesian model allows compact representation • It does embody strong independence assumptions

  28. Elimination on Trees • Formally, for any tree, there is an elimination ordering with induced width = 1 Thm • Inference on trees is linear in number of variables

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