Introduction to the EM algorithm - Department of Statistics

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The EM algorithm has two steps: 1 Expectation step (E-step): Q(θ, θ (k) ), the expectation of log p(y, z|θ) conditional on the data and using the current parameter estimates, is calculated. 2 Maximization step (M-step): Find θ (k+1) that maximizes Q(θ, θ (k) ) with respect to θ. Camila Souza (UBC) Introduction to the EM algorithm November 2010 14 / 25
Going back to our example Remember that our goal is to find ˆθ (i.e., the ˆα j ’s and ˆλ j ’s) that maximizes the log-likelihood of the data w.r.t. to θ. log p(y|θ) = log = = n∏ p(y i |θ) i=1 n∑ log p(y i |θ) i=1 ⎡ ⎤ n∑ M∑ log ⎣ α j p j (y i |λ j ) ⎦ i=1 j=1 Camila Souza (UBC) Introduction to the EM algorithm November 2010 15 / 25
Page 1 and 2: Introduction to the EM algorithm Ca
Page 3 and 4: Example: Mixture of 2 normal distri
Page 5 and 6: In order to obtain the parameter es
Page 7 and 8: The EM algorithm The derivation of
Page 9 and 10: Note that log p(y|θ) = log p(y, z|
Page 11 and 12: How can we find a value θ (k+1) so
Page 13: Therefore, we can guarantee the ine
Page 17 and 18: Since L 1 (λ’s) = n∑ M∑ I{z
Page 19 and 20: Since L 2 (α’s) = n∑ M∑ log
Page 21 and 22: M-sptep: Holding α j and σj 2 we
Page 23 and 24: In R I found two packages to work w
Page 25: Thank you !!! Camila Souza (UBC) In

Introduction to the EM algorithm - Department of Statistics

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