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531f10EM2

# 531f10EM2 - STAT 531 EM algorithm HM Kim Department of...

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STAT 531: EM algorithm HM Kim Department of Mathematics and Statistics University of Calgary Fall 2010 1/19

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Applications filling in missing data in samples estimating parameters of finite mixtures discovering the value of latent variables so as to maximize the likelihood of the observed data (distribution). Fall 2010 2/19
EM algorithm Let ˆ θ ( t ) denote the estimate on the t th step. To complete the estimate on the ( t +1) th step do: Expectation Step: Compute Q ( θ | θ ( t ) , x ) = E [log p ( θ | x , z ) | θ ( t ) , x ] . Maximization Step: Let ˆ θ ( t +1) = argmax Q ( θ | ˆ θ ( t ) , x ) Fall 2010 3/19

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, Example : The likelihood associated with the mixture model 1 4 N ( μ 1 , 1)+ 3 4 N ( μ 2 , 1) is bimodal. A simulated sample of 400 observations from this mixture with μ 1 = 0 and μ 2 = 2 . 5, where θ = ( μ 1 , μ 2 ). Introduce a vector ( z 1 , ··· , z n ) ∈ { 0 , 1 } n such that P ( Z i = 1) = 1 4 and P ( Z i = 0) = 3 4 X i | Z i = 1 N ( μ 1 , 1) and X i | Z i = 0 N ( μ 2 , 1) p = i ; z i =1 f 1 ( x i ) i ; z i =0 f 2 ( x i ) log p = z i =1 log f 1 ( x i )+ z i =0 log f 2 ( x i ) Fall 2010 4/19
log p = n i =1 [ z i log f 1 ( x i )+(1 - z i )log f 2 ( x i )] E [log p | θ , x ] = n i =1 [ E ( Z i | θ , x )log f 1 ( x i )+ E (1 - Z i | θ , x )log f 2 ( x i )] where f 1 ( x i ) = φ ( x i - μ 1 ) = 1 2 π exp - ( x i - μ 1 ) 2 2 f 2 ( x 1 ) = φ ( x i - μ 2 ) = 1 2 π exp - ( x i - μ 2 ) 2 2 . Therefore, Q ( θ | ˆ θ ( t ) , x ) = - 1 2 [( x i - μ 1 ) 2 E ( Z i | θ ( t ) , x )+( x i - μ 2 ) 2 E (1 - Z i | θ ( t ) , x )] Fall 2010 5/19

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The joint pdf of Z and X is f ( z i , x i ) = [ z i f 1 ( x )+(1 - z i ) f 2 ( x i )] 1 4 z i 3 4 1 - z i The marginal pdf of X is f ( x i ) = z f ( z i , x i ) = f 1 ( x i ) 1 4 + f 2
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531f10EM2 - STAT 531 EM algorithm HM Kim Department of...

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