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Bayes Nets

Bayes Nets. Bayes Nets Quick Intro. Topic of much current research Models dependence/independence in probability distributions Graph based - aka “graphical models” “belief nets” Two kinds - directed and undirected. Basic idea:.

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Bayes Nets

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  1. Bayes Nets

  2. Bayes Nets Quick Intro • Topic of much current research • Models dependence/independence in probability distributions • Graph based - aka “graphical models” “belief nets” • Two kinds - directed and undirected

  3. Basic idea: • Model a joint distribution as a DAG with attributes (features) at the nodes • Sample distribution from sources Party Apti- tude study Sleep Pass Exam

  4. At each node is a conditional distribution for the attribute depending only on incoming arcs P(yes) = .6 P(no) = .4 Party yes P(yes) = 0.1 P(no)=0.9 Party No P(yes)=0.9 P(no) = 0.1) Party study

  5. Causes and Bayes’ Rule Diagnostic inference: Knowing that the grass is wet, what is the probability that rain is the cause? diagnostic causal

  6. Causal vs Diagnostic Inference Causal inference:If the sprinkler is on, what is the probability that the grass is wet? P(W|S) = P(W|R,S) P(R|S) + P(W|~R,S) P(~R|S) = P(W|R,S) P(R) + P(W|~R,S) P(~R) = 0.95 0.4 + 0.9 0.6 = 0.92 Diagnostic inference: If the grass is wet, what is the probability that the sprinkler is on? P(S|W) = 0.35 > 0.2 P(S) P(S|R,W) = 0.21 Explaining away: Knowing that it has rained decreases the probability that the sprinkler is on.

  7. Bayesian Networks: Causes Causal inference: P(W|C) = P(W|R,S) P(R,S|C) + P(W|~R,S) P(~R,S|C) + P(W|R,~S) P(R,~S|C) + P(W|~R,~S) P(~R,~S|C) and use the fact that P(R,S|C) = P(R|C) P(S|C) Diagnostic: P(C|W ) = ?

  8. Bayesian Nets: Local structure P (F | C) = ?

  9. Bayesian Networks: Inference P (C,S,R,W,F) = P (C) P (S|C) P (R|C) P (W|R,S) P (F|R) P (C,F) = ∑S∑R∑W P (C,S,R,W,F) P (F|C) = P (C,F) / P(C) Not efficient! Belief propagation (Pearl, 1988) Junction trees (Lauritzen and Spiegelhalter, 1988)

  10. Conditional independence • Know values of some nodes, which others are independent? • Nodes depend on ancestors through parents • Node conditionally indep of rest of graph if its parents, children, and children’s parents known • Notion of d-path - two nodes dependent if there is a d-path (or coincidence), independent otherwise

  11. D-paths go through • Linear non-evidence nodes (both ways) • Diverging non-evid. Nodes • converging evidence nodes: • Conveging non-evidence w. evidence descd.

  12. In general, working with Bayes nets hard, (easy for trees, also approximate inference - loopy propagation, sampling) • Concise representation of some distributions • Very flexible - can fix or learn structure, can have unobserved nodes (latent variables) • Can capture conditional dependencies • Generalizes HMMs and other models

  13. Bayes nets problems:

  14. Bayes Nets Applications • Add probabilities to expert systems • Vista system (Horvitz) for space shuttle problem detection/remedies • Used in Microsoft products: Office ‘95 Answer Wizard, Office ‘97 Office Assistant, etc. • Computational Biology • Special cases applied to speech recognition, turbo coding, and many other places. Next: dec. trees

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