The EM Algorithm

# The EM Algorithm - 1 The EM Algorithm Sin-Horng Chen 2 •...

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Unformatted text preview: 1 The EM Algorithm Sin-Horng Chen 2006/12/29 2 • X‚º” incomplete data h model h h ML (maximum likelihood) R observed data sequence • h observed data h independent, i.e., • h incomplete data q data q + mixture component h HMM q state h 1 2 N X x x x = L n So ML: maxlog ( ) max log ( ) n P X P x Θ Θ Θ = Θ max log ( ) P X Θ Θ ( ) ( ) n n P X P x Θ = Θ 3 • 1& data h complete . L optimize L * 6 : i+ * Newton-Raphson method h • h two-step iterative procedure steps h expectation and maximization 4 • Expectation Step: Define an auxiliary function & &&& @ º £ ¡&&&&h 1 1 1 ( , ) ( , )log ( , ) ( is unknown data associated with ) ( , )log[ ( , ) ( )] ( , )log ( , ) lo N old n old n n k n N n old n n n k N n old n n k Q P k x P k x k x P k x P k x P x P k x P k x = = = Θ Θ = Θ Θ = Θ Θ Θ = Θ Θ + 1 1 1 g ( ) ( , ) ( , )log ( , ) log ( ) N n n old n k N N n old n n n k n P x P k x P k x P k x P x = = = Θ Θ = Θ Θ + Θ and old Θ Θ 5 1 1 1 1 1 ( , ) ( , )log ( ,...
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The EM Algorithm - 1 The EM Algorithm Sin-Horng Chen 2 •...

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