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Statistical Learning (part II) October 28, 2008 CS 486/686 University of Waterloo Outline Learning from incomplete Data EM algorithm Reading: R&N Ch 20.3 2 CS486/686 Lecture Slides (c) 2008 P. Poupart Incomplete data

  1. Statistical Learning (part II) October 28, 2008 CS 486/686 University of Waterloo

  2. Outline • Learning from incomplete Data – EM algorithm • Reading: R&N Ch 20.3 2 CS486/686 Lecture Slides (c) 2008 P. Poupart

  3. Incomplete data • So far… – Values of all attributes are known – Learning is relatively easy • But many real-world problems have hidden variables (a.k.a latent variables) – Incomplete data – Values of some attributes missing 3 CS486/686 Lecture Slides (c) 2008 P. Poupart

  4. Unsupervised Learning • Incomplete data � unsupervised learning • Examples: – Categorisation of stars by astronomers – Categorisation of species by anthropologists – Market segmentation for marketing – Pattern identification for fraud detection – Research in general! 4 CS486/686 Lecture Slides (c) 2008 P. Poupart

  5. Maximum Likelihood Learning • ML learning of Bayes net parameters: – For θ V=true,pa(V)= v = Pr(V=true|par(V) = v ) – θ V=true,pa(V)= v = #[V=true,pa(V)= v ] #[V=true,pa(V)= v ] + #[V=false,pa(V)= v ] – Assumes all attributes have values… • What if values of some attributes are missing? 5 CS486/686 Lecture Slides (c) 2008 P. Poupart

  6. “Naive” solutions for incomplete data • Solution #1: Ignore records with missing values – But what if all records are missing values (i.e., when a variable is hidden, none of the records have any value for that variable) • Solution #2: Ignore hidden variables – Model may become significantly more complex! 6 CS486/686 Lecture Slides (c) 2008 P. Poupart

  7. Heart disease example 2 2 2 2 2 2 Smoking Diet Exercise Smoking Diet Exercise 54 HeartDisease 6 6 6 54 162 486 Symptom 1 Symptom 2 Symptom 3 Symptom 1 Symptom 2 Symptom 3 (b) (a) • a) simpler (i.e., fewer CPT parameters) • b) complex (i.e., lots of CPT parameters) 7 CS486/686 Lecture Slides (c) 2008 P. Poupart

  8. “Direct” maximum likelihood • Solution 3: maximize likelihood directly – Let Z be hidden and E observable – h ML = argmax h P( e |h) = argmax h Σ Z P( e , Z |h) = argmax h Σ Z Π i CPT(V i ) = argmax h log Σ Z Π i CPT(V i ) – Problem: can’t push log past sum to linearize product 8 CS486/686 Lecture Slides (c) 2008 P. Poupart

  9. Expectation-Maximization (EM) • Solution #4: EM algorithm – Intuition: if we knew the missing values, computing h ML would be trival • Guess h ML • Iterate – Expectation: based on h ML , compute expectation of the missing values – Maximization: based on expected missing values, compute new estimate of h ML 9 CS486/686 Lecture Slides (c) 2008 P. Poupart

  10. Expectation-Maximization (EM) • More formally: – Approximate maximum likelihood – Iteratively compute: h i+1 = argmax h Σ Z P( Z |h i , e ) log P( e , Z |h) Expectation Maximization 10 CS486/686 Lecture Slides (c) 2008 P. Poupart

  11. Expectation-Maximization (EM) • Derivation – log P( e |h) = log [P( e,Z |h) / P( Z | e ,h)] = log P( e,Z |h) – log P( Z | e ,h) = Σ Z P( Z | e ,h) log P( e,Z |h) – Σ Z P( Z | e ,h) log P( Z | e ,h) ≥ Σ Z P( Z | e ,h) log P( e,Z |h) • EM finds a local maximum of Σ Z P(Z|e,h) log P( e,Z |h) which is a lower bound of log P( e |h) 11 CS486/686 Lecture Slides (c) 2008 P. Poupart

  12. Expectation-Maximization (EM) • Log inside sum can linearize product – h i+1 = argmax h Σ Z P( Z |h i , e ) log P( e , Z |h) = argmax h Σ Z P( Z |h i , e ) log Π j CPT j = argmax h Σ Z P( Z |h i , e ) Σ j log CPT j • Monotonic improvement of likelihood – P( e |h i+1 ) ≥ P( e |h i ) 12 CS486/686 Lecture Slides (c) 2008 P. Poupart

  13. Candy Example • Suppose you buy two bags of candies of unknown type (e.g. flavour ratios) • You plan to eat sufficiently many candies of each bag to learn their type • Ignoring your plan, your roommate mixes both bags… • How can you learn the type of each bag despite being mixed? 13 CS486/686 Lecture Slides (c) 2008 P. Poupart

  14. Candy Example • “Bag” variable is hidden 14 CS486/686 Lecture Slides (c) 2008 P. Poupart

  15. Unsupervised Clustering • “Class” variable is hidden • Naïve Bayes model P ( 1) Bag= Bag C P ( F=cherry | B ) Bag 1 F 1 2 F 2 Flavor Wrapper Holes X (a) (b) 15 CS486/686 Lecture Slides (c) 2008 P. Poupart

  16. Candy Example • Unknown Parameters: – θ i = P(Bag=i) – θ Fi = P(Flavour=cherry|Bag=i) – θ Wi = P(Wrapper=red|Bag=i) – θ Hi = P(Hole=yes|Bag=i) • When eating a candy: – F, W and H are observable – B is hidden 16 CS486/686 Lecture Slides (c) 2008 P. Poupart

  17. Candy Example • Let true parameters be: – θ =0.5, θ F1 = θ W1 = θ H1 =0.8, θ F2 = θ W2 = θ H2 =0.3 • After eating 1000 candies: W=red W=green H=1 H=0 H=1 H=0 F=cherry 273 93 104 90 F=lime 79 100 94 167 17 CS486/686 Lecture Slides (c) 2008 P. Poupart

  18. Candy Example • EM algorithm • Guess h 0 : – θ =0.6, θ F1 = θ W1 = θ H1 =0.6, θ F2 = θ W2 = θ H2 =0.4 • Alternate: – Expectation: expected # of candies in each bag – Maximization: new parameter estimates 18 CS486/686 Lecture Slides (c) 2008 P. Poupart

  19. Candy Example • Expectation: expected # of candies in each bag – #[Bag=i] = Σ j P(B=i|f j ,w j ,h j ) – Compute P(B=i|f j ,w j ,h j ) by variable elimination (or any other inference alg.) • Example: – #[Bag=1] = 612 – #[Bag=2] = 388 19 CS486/686 Lecture Slides (c) 2008 P. Poupart

  20. Candy Example • Maximization: relative frequency of each bag – θ 1 = 612/1000 = 0.612 – θ 2 = 388/1000 = 0.388 20 CS486/686 Lecture Slides (c) 2008 P. Poupart

  21. Candy Example • Expectation: expected # of cherry candies in each bag – #[B=i,F=cherry] = Σ j P(B=i|f j =cherry,w j ,h j ) – Compute P(B=i|f j =cherry,w j ,h j ) by variable elimination (or any other inference alg.) • Maximization: – θ F 1 = #[B=1,F=cherry] / #[B=1] = 0.668 – θ F 2 = #[B=2,F=cherry] / #[B=2] = 0.389 21 CS486/686 Lecture Slides (c) 2008 P. Poupart

  22. Candy Example -1975 -1980 -1985 -1990 Log-likelihood -1995 -2000 -2005 -2010 -2015 -2020 -2025 0 20 40 60 80 100 120 Iteration number 22 CS486/686 Lecture Slides (c) 2008 P. Poupart

  23. Bayesian networks • EM algorithm for general Bayes nets • Expectation: – #[V i =v ij ,Pa(V i )=pa ik ] = expected frequency • Maximization: – θ vij,paik = #[V i =v ij ,Pa(V i )=pa ik ] / #[Pa(V i )=pa ik ] 23 CS486/686 Lecture Slides (c) 2008 P. Poupart

  24. Next Class • Next Class: •Ensemble Learning •Russell and Norvig Sect. 18.4 24 CS486/686 Lecture Slides (c) 2008 P. Poupart

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