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CS498-EA Reasoning in AI Lecture #15 Instructor: Eyal Amir Fall Semester 2011

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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 –One variable X j eliminated at a time (a) Move unneeded terms (those not involving X j ) outside summation over X j (b) Create a new potential function, f Xj (.) over other variables appearing in the terms of the summation at (a) Works for both BNs and MFs (Markov Fields)

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Today 1.Treewidth methods: 1.Variable elimination 2.Clique tree algorithm 3.Treewidth

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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

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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 ADEABD DEF ADDE Cluster ABD Sepset DE

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Potentials Potentials: –Denoted by Marginalization –, the marginalization of into X Multiplication –, the multiplication of and

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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

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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

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Building Junction Trees DAG Moral GraphTriangulated GraphJunction TreeIdentifying Cliques

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

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

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

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

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Identifying Cliques A clique is a subgraph of an undirected graph that is complete and maximal A B D C E G F H EGH ADEABD ACEDEF CEG

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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 EGH ADEABD ACEDEF CEG ADE ABD ACE AD AE CEG CE DEF DE EGH EG

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

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Example: Create Join Tree X1X2 Y1Y2 HMM with 2 time steps: Junction Tree: X1,X2 X1,Y1 X2,Y2 X1 X2

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Example: Initialization Variable Associated Cluster Potential function X1X1,Y1 Y1X1,Y1 X2X1,X2 Y2X2,Y2 X1,X2 X1,Y1 X2,Y2 X1 X2

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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:

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

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

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

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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:

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

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Next Time Learning BNs and MFs

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THE END

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Example: Naïve Bayesian Model A common model in early diagnosis: –Symptoms are conditionally independent given the disease (or fault) Thus, if –X 1,…,X p denote whether the symptoms exhibited by the patient (headache, high- fever, etc.) and –H denotes the hypothesis about the patients health then, P(X 1,…,X p,H) = P(H)P(X 1 |H)…P(X p |H), This naïve Bayesian model allows compact representation –It does embody strong independence assumptions

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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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