i πφ θ 2 y i which is difficult to maximize c 2019 The Trustees of the Stevens

# I πφ θ 2 y i which is difficult to maximize c 2019

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i ) + πφ θ 2 ( y i )] which is difficult to maximize c 2019 The Trustees of the Stevens Institute of Technology

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Bootstrap Methods EM Algorithm We can simplify it by introducing a new variable Δ i (an unobserved latent variable) that takes values either 1 or 0. 0 ( θ ; Z , Δ) = N X i = 1 [( 1 - Δ i ) log φ θ 1 ( y i ) + Δ i log φ θ 2 ( y i )] + N X i = 1 [( 1 - Δ i ) log( 1 - π ) + Δ i log π ] c 2019 The Trustees of the Stevens Institute of Technology
Bootstrap Methods EM Algorithm Since the values of the Δ i ’s are unknown, we substitute for each Δ i its expected value: γ i ( θ ) = E i | θ, Z ] = P i = 1 | θ, Z ) c 2019 The Trustees of the Stevens Institute of Technology

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Bootstrap Methods EM Algorithm Algorithm 8.1: EM Algorithm for Two-component Gaussian Mixture 1. Take initial guesses for the parameters ˆ μ 1 , ˆ σ 2 1 , ˆ μ 2 , ˆ σ 2 2 , ˆ π 2. Expectation Step: compute the responsibilites ˆ γ i = ˆ πφ ˆ θ 2 ( y i ) ( 1 - ˆ π ) φ ˆ θ 1 ( y i ) + ˆ πφ ˆ θ 2 ( y i ) , i = 1 , . . . , N 3. Maximization Step: compute the weighted means and variances. ˆ μ 1 = N i = 1 ( 1 - ˆ γ i ) y i N i = 1 ( 1 - ˆ γ i ) , ˆ σ 2 1 = N i = 1 ( 1 - ˆ γ i )( y i - ˆ μ 1 ) 2 N i = 1 ( 1 - ˆ γ i ) ˆ μ 2 = N i = 1 ˆ γ i y i N i = 1 ˆ γ i , ˆ σ 2 1 = N i = 1 ˆ γ i ( y i - ˆ μ 2 ) 2 N i = 1 ˆ γ i and the mixing probability ˆ π = N i = 1 ˆ γ i / N 4. Iterate steps 2 and 3 until convergence.[2] c 2019 The Trustees of the Stevens Institute of Technology
Bootstrap Methods EM Algorithm Mixture of M Normals We introduce new unknown random variables ( Y ) and use them to create a simpler expression of the likelihood. p ( X , Y | Θ) = p ( Y | X , Θ) p ( X , Y | Θ) p ( Y | X , Θ) (1) E-Step: P ( t ) ( y ) = P ( y | x , Θ ( t ) ) M-Step: Θ ( t + 1 ) = argmax Θ ( E P ( t ) [ln P ( y , x | Θ)]) c 2019 The Trustees of the Stevens Institute of Technology

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Bootstrap Methods EM Algorithm For a mixture of normals we have the lower bound: λ ( X , Θ) N X i = 1 M X j = 1 p ( t ) ( j | x i , Θ ( t ) ) ln p j g ( x i ; μ j , σ 2 j ) p ( t ) ( j | x i , Θ ( t ) ) = b t where g x i ; μ ( t ) j , σ 2 ( t ) j denotes the Gaussian pdf. Our Expectation Step is expressed as: p ( t ) ( j | x i , Θ ( t ) ) = p ( t ) j g ( x i ; μ ( t ) j , σ 2 ( t ) j ) M j = 1 p ( t ) j g ( x i ; μ ( t ) j , σ 2 ( t ) j ) c 2019 The Trustees of the Stevens Institute of Technology
Bootstrap Methods EM Algorithm Since b t is a lower bound for the log-likelihood, if we maximize b t we will improve the log-likelihood as well. Looking at b t we can see: b t = N X i = 1 M X j = 1 p ( t ) ( j | x i , Θ ( t ) ) ln p j g ( x i ; μ j , σ 2 j ) - N X i = 1 M X j = 1 p ( t ) ( j | x i , Θ ( t ) ) ln p ( t ) ( j | x i , Θ ( t ) ) ˆ Θ = Θ ( t + 1 ) = argmax Θ N X i = 1 M X j = 1 p ( t ) ( j | x i , Θ ( t ) ) ln p j g ( x i ; μ j , σ 2 j ) (2) c 2019 The Trustees of the Stevens Institute of Technology

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Bootstrap Methods EM Algorithm To ease writing out formulas we will define the function q ( j , i ) = p j g ( x i ; μ j , σ 2 j ) This function has the following partial derivatives with respect to the parameters, q ∂μ j = q ( j , i ) x i - μ j σ 2 j !
• Fall '16
• alec schimdt

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