Expectation Maximization [KF Chapter 19] Incomplete data
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1 Expectation Maximization [KF Chapter 19] CS 786 University of Waterloo Lecture 17: June 28, 2012 Complete data Incomplete data 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 CS786 Lecture Slides (c) 2012 P. Poupart 2 1
2 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! CS786 Lecture Slides (c) 2012 P. Poupart 3 Maximum Likelihood Learning ML learning of Bayes net parameters: For V=true,pa(V)=v = Pr(V=true par(v) = v) 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? CS786 Lecture Slides (c) 2012 P. Poupart 4 2
3 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! CS786 Lecture Slides (c) 2012 P. Poupart 5 Heart disease example Smoking Diet Exercise Smoking Diet Exercise 54 HeartDisease Symptom 1 Symptom 2 Symptom Symptom 1 Symptom 2 Symptom 3 (a) (b) a) simpler (i.e., fewer CPT parameters) b) complex (i.e., lots of CPT parameters) CS786 Lecture Slides (c) 2012 P. Poupart 6 3
4 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 i CPT(V i ) = argmax h log Σ Z i CPT(V i ) Problem: can t push log past sum to linearize product CS786 Lecture Slides (c) 2012 P. Poupart 7 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 CS786 Lecture Slides (c) 2012 P. Poupart 8 4
5 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 CS786 Lecture Slides (c) 2012 P. Poupart 9 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) CS786 Lecture Slides (c) 2012 P. Poupart 10 5
6 Expectation-Maximization (EM) Objective: max h Σ Z P(Z e,h) log P(e,Z h) Iterative approach h i+1 = argmax h Σ Z P(Z e,h i ) log P(e,Z h) Convergence guaranteed h = argmax h Σ Z P(Z e,h) log P(e,Z h) Monotonic improvement of likelihood P(e h i+1 ) P(e h i ) CS786 Lecture Slides (c) 2012 P. Poupart 11 Optimization Step For one data point e: h i+1 = argmax h Σ Z P(Z h i,e) log P(e,Z h) For multiple data points: h i+1 = argmax h Σ e n e Σ Z P(Z h i,e) log P(e,Z h) Where n e is frequency of e in dataset Compare to ML for complete data h* = argmax h Σ d n d log P(d h) CS786 Lecture Slides (c) 2012 P. Poupart 12 6
7 Optimization Solution Since d <z,e> Let n d = n e P(z h i,e) expected frequency Similar to the complete data case, the optimal parameters are obtained by setting the derivative to 0, which yields relative expected frequencies E.g. V,pa(V) = P(V pa(v)) = n V,pa(V) / n pa(v) CS786 Lecture Slides (c) 2012 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? CS786 Lecture Slides (c) 2012 P. Poupart 14 7
8 Candy Example Bag variable is hidden CS786 Lecture Slides (c) 2012 P. Poupart 15 Unsupervised Clustering Class variable is hidden Naïve Bayes model P( Bag= 1) Bag P(F=cherry B) Bag C 1 F1 2 F2 Flavor Wrapper Holes X (a) (b) CS786 Lecture Slides (c) 2012 P. Poupart 16 8
9 Candy Example Unknown Parameters: i = P(Bag=i) Fi = P(Flavour=cherry Bag=i) Wi = P(Wrapper=red Bag=i) Hi = P(Hole=yes Bag=i) When eating a candy: F, W and H are observable B is hidden CS786 Lecture Slides (c) 2012 P. Poupart 17 Candy Example Let true parameters be: =0.5, F1 = W1 = H1 =0.8, F2 = W2 = H2 =0.3 After eating 1000 candies: W=red W=green H=1 H=0 H=1 H=0 F=cherry F=lime CS786 Lecture Slides (c) 2012 P. Poupart 18 9
10 EM algorithm Candy Example Guess h 0 : =0.6, F1 = W1 = H1 =0.6, F2 = W2 = H2 =0.4 Alternate: Expectation: expected # of candies in each bag Maximization: new parameter estimates CS786 Lecture Slides (c) 2012 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 CS786 Lecture Slides (c) 2012 P. Poupart 20 10
11 Candy Example Maximization: relative frequency of each bag 1 = 612/1000 = = 388/1000 = CS786 Lecture Slides (c) 2012 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: F1 = #[B=1,F=cherry] / #[B=1] = F2 = #[B=2,F=cherry] / #[B=2] = CS786 Lecture Slides (c) 2012 P. Poupart 22 11
12 Candy Example Log-likelihood Iteration number CS786 Lecture Slides (c) 2012 P. Poupart 23 Bayesian networks EM algorithm for general Bayes nets Expectation: #[V i =v ij,pa(v i )=pa ik ] = expected frequency Maximization: vij,pa ik = #[V i =v ij,pa(v i )=pa ik ] / #[Pa(V i )=pa ik ] CS786 Lecture Slides (c) 2012 P. Poupart 24 12
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