Expectation Maximisation (EM) CS 486/686: Introduction to Artificial Intelligence University of Waterloo
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1 Expectation Maximisation (EM) CS 486/686: Introduction to Artificial Intelligence University of Waterloo 1
2 Incomplete Data So far we have seen problems where - Values of all attributes are known - Learning is relatively easy Many real-world problems have hidden variables - Incomplete data - Missing attribute values 2
3 Maximum Likelihood Learning Learning of Bayes nets parameters - ΘV=true, Par(V)=x = P(V=true Par(V)=x) - ΘV=true, Par(V)=x =(#Insts V=true)/(Total #V=x) Assumes all attributes have values - What if some values are missing? 3
4 Naïve Solutions Ignore examples with missing attribute values - What if all examples have missing attribute values? Ignore hidden variables - Model might become much more complex 4
5 Hidden Variables Heart disease example a) Uses a Hidden Variable, simpler (fewer CPT parameters) b) No Hidden Variable, complex (many CPT parameters) 5
6 Direct ML Maximize likelihood directly where E are the evidence variables and Z are the hidden variables 6
7 Expectation-Maximization (EM) If we knew the missing values computing hml is trivial Guess hml Iterate - Expectation: based on hml compute expectation of (missing) values - Maximization: based on expected (missing) values compute new hml 7
8 Expectation-Maximization (EM) Formally Approximate maximum likelihood Iteratively compute: h i+1 =argmax h Σ Z P(Z h i,e)log P(e,Z hi) Expectation Maximization 8
9 EM Log inside can linearize the product Monotonic improvement of likelihood 9
10 Example Assume we have two coins, A and B The probability of getting heads with A is θa The probability of getting heads with B is θb We want to find θa and θb by performing a number of trials Example from S. Zafeiriohu, Advanced Statistical Machine Learning, Imperial College 10
11 Example Coin A and Coin B H T T T H H T H T H H H H H T H H H H H H T H H H H H T H H H T H T T T H H T T T H H H T H H H T H Coin A Coin B 5 H, 5 T 9 H, 1 T 8 H, 2 T 4 H, 6 T 7 H, 3 T 24 H, 6 T 9 H, 11 T 11
12 Example Coin A Coin B 5 H, 5 T 9 H, 1 T 8 H, 2 T 4 H, 6 T 7 H, 3 T 24 H, 6 T 9 H, 11 T 12
13 Example Now assume we do not know which coin was used in which trial (hidden variable) H T T T H H T H T H H H H H T H H H H H H T H H H H H T H H H T H T T T H H T T T H H H T H H H T H 13
14 Example Initialization: E Step: Compute the Expected counts of Heads and Tails Trial 1: H T T T H H T H T H Coin A 2.2 H, 2.2 T Coin B 2.8 H, 2.8 T 14
15 Example H T T T H H T H T H (0.55 A, 0.45 B) H H H H T H H H H H (0.80 A, 0.20 B) H T H H H H H T H H (0.73 A, 0.27 A) H T H T T T H H T T (0.35 A, 0.65 B) T H H H T H H H T H (0.65 A, 0.35 B) Coin A Coin B 2.2H, 2.2T 2.8H, 2.8T 7.2H, 0.8T 1.8H, 0.2T 5.9H, 1.5T 2.1H, 0.5T 1.4H, 2.1T 2.6H, 3.9T 4.5H, 1.9T 2.5H, 1.1T 21.3H, 8.6T 11.7H, 8.4T 15
16 Example M Step: Compute parameters based on expected counts Coin A Coin B 2.2H, 2.2T 2.8H, 2.8T 7.2H, 0.8T 1.8H, 0.2T 5.9H, 1.5T 2.1H, 0.5T 1.4H, 2.1T 2.6H, 3.9T 4.5H, 1.9T 2.5H, 1.1T 21.3H, 8.6T 11.7H, 8.4T Repeat 16
17 EM: k-means Algorithm Input Set of examples, E Input features X1,,Xn val(e,x)=value of feature j for example e k classes Output Function class:e-> {1,,k} where class(e)=i means example e belongs to class i Function pval where pval(i,xj) is the predicted value of feature Xj for each example in class i 17
18 k-means Algorithm Sum-of-squares error for class i and pval is Goal: Final class and pval that minimizes sum-of-squares error. 18
19 Minimizing the error Given class, the pval that minimizes sum-ofsquare error is the mean value for that class Given pval, each example can be assigned to the class that minimizes the error for that example 19
20 k-means Algorithm Randomly assign the examples to classes Repeat the following two steps until E step does not change the assignment of any example M: For each class i and feature Xj E: For each example e, assign e to the class that minimizes 20
21 k-means Example Data set: (X,Y) pairs (0.7,5.1) (1.5,6), (2.1, 4.5), (2.4, 5.5), (3, 4.4), (3.5, 5), (4.5, 1.5), (5.2, 0.7), (5.3, 1.8), (6.2, 1.7), (6.7, 2.5), (8.5, 9.2), (9.1, 9.7), (9.5, 8.5) 21
22 Example Data 22
23 Random Assignment to Classes 23
24 Assign Each Example to Closest Mean 24
25 Reassign each example 25
26 Properties of k-means An assignment is stable if both M step and E step do not change the assignment - Algorithm will eventually converge to a stable local minimum - No guarantee that it will converge to a global minimum Increasing k can always decrease error until k is the number of different examples 26
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