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Answering Queries: Problems

CSCI 121 Special Topics: Bayesian Networks Lecture #3: Multiply-Connected Graphs and the Junction Tree Algorithm. A. A. B. C. B. C. D. D. Answering Queries: Problems. Difficult if graph is not singly connected (a.k.a polytree ):. Multiply connnected: more than

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Answering Queries: Problems

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  1. CSCI 121 Special Topics: Bayesian NetworksLecture #3: Multiply-Connected Graphs and the Junction Tree Algorithm

  2. A A B C B C D D Answering Queries: Problems Difficult if graph is not singly connected (a.k.a polytree): Multiply connnected: more than one path from A to D. P(D|A) = ? Singly connnected (polytree): just one path from A to D.

  3. Dealing with Multiply Connected Graphs • Three basic options: • Clustering – group “offending” nodes into “meganodes” • Conditioning – set variables to definite values; then build a polytree for each combo • Stochasticsimulation – Generate a large number of concrete models consistent with the domain. • Clustering seems to be the most popular .

  4. A A C C G G B B D D E H E H F F Clustering with the Junction-Tree Algorithm (Huang & Darwiche 1994) 0) Note that a BN is a directed acyclic graph (DAG) 1) “Moralize” the DAG: For parents A, B of node C, draw an edge between A and B. Then remove arrows (undirected graph).

  5. A A C G C G B B D E H D E H F F Clustering with the Junction-Tree Algorithm 2) Triangulate the moral graph so that every cycle of length ≥ 4 contains an edge between nonadjacent nodes. Use a heuristic based on minimal # of edges added, minimal # possible values.

  6. A ABD ACE C G B ADE CEG D E H DEF EGH F Clustering with the Junction-Tree Algorithm 3) Build cliques from triangulated graph: Put your hands in the air and represent your clique! – 112 , “Peaches and Cream”

  7. ABD ADE ACE CEG AD AE CE EG DE DEF EGH Clustering with the Junction-Tree Algorithm 4) Connect cliques by separation sets to form the junction tree:

  8. Marginalization • At this point, each cluster (clique; meganode) has a joint probability table. • To query a variable, we (heuristically) pick a cluster containing it, and marginalizeover the joint probability Ф from the table: ABDФABD T T T .225 T T F .025 T F T .125 T F F .125 F T T .180 F T F .020 F F T .150 F F F .150 D P(D) Σ T .225 + .125 + .180 + .150 = .680 F .025 + .125 + .020 + .150 = .320

  9. ABD ADE ACE CEG AD AE CE EG DE DEF EGH Message-Passing • Sepset potentials are initialized via marginalization. • When evidence is presented (“John calls”), heuristically pick a “root clique” and pass messages around the tree:

  10. Message-Passing • Messages are passed from clique X to Y through sepset R by multiplication and division of table entries. • Evidence is set by “masking” table entries with a bit vector (or probability distribution). E.g., observe B = F: ABDФABD T T T 0 T T F 0 T F T .125 T F F .125 F T T 0 F T F 0 F F T .150 F F F .150

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