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Probabilistic Graphical Introduction Models Welcome to Welcome to the PGM Class

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  8. Probabilis#c' Graphical' Introduc#on' Models' Mo#va#on' and'Overview' Daphne Koller

  9. millions of pixels or thousands of superpixels predisposing factors symptoms each needs to be labeled test results {grass, sky, water, cow, horse, …} diseases treatment outcomes Daphne Koller

  10. Probabilistic Graphical Models Daphne Koller

  11. Models domain expert Data Declarative representation Model Learning elicitation Algorithm Algorithm Algorithm Daphne Koller

  12. Uncertainty • Partial knowledge of state of the world • Noisy observations • Phenomena not covered by our model • Inherent stochasticity Daphne Koller

  13. Probability Theory • Declarative representation with clear semantics • Powerful reasoning patterns • Established learning methods Daphne Koller

  14. Complex Systems predisposing factors class labels for symptoms thousands of superpixels test results diseases treatment outcomes Random variables X 1 ,…, X n Joint distribution P(X 1 ,…, X n ) Daphne Koller

  15. Graphical Models Bayesian networks Markov networks A Difficulty Intelligence Grade SAT D B Letter C Daphne Koller

  16. Graphical Models M. Pradhan, G. Provan, B. Middleton, M. Henrion, UAI 94 Daphne Koller

  17. Graphical Representation • Intuitive & compact data structure • Efficient reasoning using general-purpose algorithms • Sparse parameterization – feasible elicitation – learning from data Daphne Koller

  18. Many Applications • Medical diagnosis • Computer vision – Image segmentation • Fault diagnosis – 3D reconstruction • Natural language – Holistic scene analysis processing • Speech recognition • Traffic analysis • Robot localization & • Social network models mapping • Message decoding Daphne Koller

  19. Image Segmentation Daphne Koller

  20. Thanks to: Eric Horvitz, Microsoft Research Medical Diagnosis - Daphne Koller

  21. Textual Information Extraction Mrs. Green spoke today in New York. Green chairs the finance committee. Daphne Koller

  22. Multi-Sensor Integration: Traffic Trained on historical data • Learn to predict current & future road speed, including • Multiple views on unmeasured roads on traffic Dynamic route optimization • Weather Learned Model • I95 corridor experiment: accurate Incident reports to ± 5 MPH in 85% of cases • Fielded in 72 cities Thanks to: Eric Horvitz, Microsoft Research Daphne Koller

  23. This figure may be used for non-commercial and classroom purposes only. Any other uses require the prior written permission from AAAS Biological Network Reconstruction Phospho-Proteins Phospho-Lipids PKC Perturbed in data PKA Known 15/17 Raf Plc γ Supported 2/17 Jnk P38 Reversed 1 Mek Missed 3 PIP3 Erk Subsequently validated in wetlab Akt PIP2 Causal protein-signaling networks derived from multiparameter single-cell data Sachs et al., Science 2005 Daphne Koller

  24. Overview • Representation – Directed and undirected – Temporal and plate models • Inference – Exact and approximate – Decision making • Learning – Parameters and structure – With and without complete data Daphne Koller

  25. Probabilis0c+ Graphical+ Introduc0on+ Models+ Preliminaries:+ Distribu0ons+ Daphne Koller

  26. Joint Distribution I D G Prob. i 0 d 0 g 1 0.126 Intelligence (I) • i 0 d 0 g 2 0.168 i 0 (low), i 1 (high), – i 0 d 0 g 3 0.126 Difficulty (D) • i 0 d 1 g 1 0.009 d 0 (easy), d 1 (hard) – i 0 d 1 g 2 0.045 Grade (G) • i 0 d 1 g 3 0.126 g 1 (A), g 2 (B), g 3 (C) – i 1 d 0 g 1 0.252 i 1 d 0 g 2 0.0224 i 1 d 0 g 3 0.0056 i 1 d 1 g 1 0.06 i 1 d 1 g 2 0.036 i 1 d 1 g 3 0.024 Daphne Koller

  27. Conditioning I D G Prob. condition on g 1 i 0 d 0 g 1 0.126 i 0 d 0 g 2 0.168 i 0 d 0 g 3 0.126 i 0 d 1 g 1 0.009 i 0 d 1 g 2 0.045 i 0 d 1 g 3 0.126 i 1 d 0 g 1 0.252 i 1 d 0 g 2 0.0224 i 1 d 0 g 3 0.0056 i 1 d 1 g 1 0.06 i 1 d 1 g 2 0.036 i 1 d 1 g 3 0.024 Daphne Koller

  28. Conditioning: Reduction I D G Prob. i 0 d 0 g 1 0.126 i 0 d 1 g 1 0.009 i 1 d 0 g 1 0.252 i 1 d 1 g 1 0.06 Daphne Koller

  29. Conditioning: Renormalization I D G Prob. I D Prob. i 0 d 0 g 1 i 0 d 0 0.126 0.282 i 0 d 1 g 1 i 0 d 1 0.009 0.02 i 1 d 0 g 1 i 1 d 0 0.252 0.564 i 1 d 1 g 1 i 1 d 1 0.06 0.134 P(I, D, g 1 ) P(I, D | g 1 ) 0.447 Daphne Koller

  30. Marginalization Marginalize I I D Prob. i 0 d 0 0.282 D Prob. i 0 d 1 0.02 d 0 0.846 i 1 d 0 0.564 d 1 0.154 i 1 d 1 0.134 Daphne Koller

  31. Probabilis1c+ Graphical+ Introduc1on+ Models+ Preliminaries:+ Factors+ Daphne Koller

  32. Factors • A factor φ (X 1 ,…,X k ) φ : Val(X 1 ,…,X k ) → R • Scope = {X 1 ,…,X k } Daphne Koller

  33. Joint Distribution I D G Prob. i 0 d 0 g 1 0.126 i 0 d 0 g 2 0.168 i 0 d 0 g 3 0.126 i 0 d 1 g 1 0.009 i 0 d 1 g 2 0.045 P(I,D,G) i 0 d 1 g 3 0.126 i 1 d 0 g 1 0.252 i 1 d 0 g 2 0.0224 i 1 d 0 g 3 0.0056 i 1 d 1 g 1 0.06 i 1 d 1 g 2 0.036 i 1 d 1 g 3 0.024 Daphne Koller

  34. Unnormalized measure P(I,D,g 1 ) I D G Prob. i 0 d 0 g 1 0.126 P(I,D,g 1 ) i 0 d 1 g 1 0.009 i 1 d 0 g 1 0.252 i 1 d 1 g 1 0.06 Daphne Koller

  35. Conditional Probability Distribution (CPD) g 1 g 2 g 3 i 0 ,d 0 0.3 0.4 0.3 P(G | I,D) i 0 ,d 1 0.05 0.25 0.7 i 1 ,d 0 0.9 0.08 0.02 i 1 ,d 1 0.5 0.3 0.2 Daphne Koller

  36. General factors φ A B a 0 b 0 30 a 0 b 1 5 a 1 b 0 1 a 1 b 1 10 Daphne Koller

  37. Factor Product a 1 b 1 c 1 0.5·0.5 = 0.25 a 1 b 1 c 2 0.5·0.7 = 0.35 a 1 b 2 c 1 0.8·0.1 = 0.08 a 1 b 2 c 2 0.8·0.2 = 0.16 a 1 b 1 0.5 a 2 b 1 c 1 0.1·0.5 = 0.05 a 1 b 2 0.8 b 1 c 1 0.5 a 2 b 1 c 2 0.1·0.7 = 0.07 a 2 b 1 0.1 b 1 c 2 0.7 a 2 b 2 c 1 0·0.1 = 0 b 2 c 1 0.1 a 2 b 2 0 a 2 b 2 c 2 0·0.2 = 0 b 2 c 2 0.2 a 3 b 1 0.3 a 3 b 1 c 1 0.3·0.5 = 0.15 a 3 b 2 0.9 a 3 b 1 c 2 0.3·0.7 = 0.21 a 3 b 2 c 1 0.9·0.1 = 0.09 a 3 b 2 c 2 0.9·0.2 = 0.18 Daphne Koller

  38. Factor Marginalization a 1 b 1 c 1 0.25 a 1 b 1 c 2 0.35 a 1 b 2 c 1 0.08 a 1 c 1 0.33 a 1 b 2 c 2 0.16 a 2 b 1 c 1 0.05 a 1 c 2 0.51 a 2 b 1 c 2 0.07 a 2 c 1 0.05 a 2 b 2 c 1 0 a 2 c 2 0.07 a 2 b 2 c 2 0 a 3 c 1 0.24 a 3 b 1 c 1 0.15 a 3 c 2 0.39 a 3 b 1 c 2 0.21 a 3 b 2 c 1 0.09 a 3 b 2 c 2 0.18 Daphne Koller

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