9 pqt qt1 pxt qt 410 t 412 prof je bilmes ee596awinter

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Unformatted text preview: p ) = Ep(x1:T ,q1:T |λp ) [log p(x1:T , q1:T |λ)] = Ep [log (4.9) p(qt |qt−1 , λ)p(xt |qt , λ)] (4.10) t (4.12) Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-21 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM - learning with EM The EM algorithm then repeatedly optimizes the following objective: f (λ) = Q(λ, λp ) = Ep(x1:T ,q1:T |λp ) [log p(x1:T , q1:T |λ)] = Ep [log (4.9) p(qt |qt−1 , λ)p(xt |qt , λ)] (4.10) t = Ep [ log p(qt |qt−1 , λ) + t log p(xt |qt , λ)] (4.11) t (4.12) Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-21 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM - learning with EM The EM algorithm then repeatedly optimizes the following objective: f (λ) = Q(λ, λp ) = Ep(x1:T ,q1:T |λp ) [log p(x1:T , q1:T |λ)] = Ep [log (4.9) p(qt |qt−1 , λ)p(xt |qt , λ)] (4.10) t = Ep [ log p(qt |qt−1 , λ) + t log p(xt |qt , λ)] (4.11) t p(Qt = j, Qt−1 = i|x1:T , λp ) log p(Qt = j |Qt−1 = i, λ) = t ij (4.12) Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-21 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM - learning with EM The EM algorithm then repeatedly optimizes the following objective: f (λ) = Q(λ, λp ) = Ep(x1:T ,q1:T |λp ) [log p(x1:T , q1:T |λ)] = Ep [log (4.9) p(qt |qt−1 , λ)p(xt |qt , λ)] (4.10) t = Ep [ log p(qt |qt−1 , λ) + t (4.11) t p(Qt = j, Qt−1 = i|x1:T , λp ) log p(Qt = j |Qt−1 = i, λ) = t ij p(Qt = i|x1:T , λp ) log p(xt |Qt = i, λ) + t Prof. Jeﬀ Bilmes log p(xt |qt , λ)] (4.12) i EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-21 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM - learning with EM The EM algorithm then repeatedly optimizes the following objective: f (λ) = Q(λ, λp ) = Ep(x1:T ,q1:T |λp ) [log p(x1:T , q1:T |λ)] = Ep [log (4.9) p(qt |qt−1 , λ)p(xt |qt , λ)] (4.10) t = Ep [ log p(qt |qt−1 , λ) + t log p(xt |qt , λ)] (4.11) t p(Qt = j, Qt−1 = i|x1:T , λp ) log p(Qt = j |Qt−1 = i, λ) = t ij p(Qt = i|x1:T , λp ) log p(xt |Qt = i, λ) + t (4.12) i So this means that for EM learning, we need for all t , the queries p(Qt = i|x1:T ) and p(Qt = j, Qt−1 = i|x1:T ) in an HMM. Note again that these are clique posteriors. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-21 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM - learning with gradient descent EM isn’t the only way to learn parameters. Suppose we wanted to use a gradient descent like algorithm on f (λ) = log p(x1:T |λ), as in ∂ ∂ ∂ f (λ) = log p(x1:T |λ) = log ∂λ ∂λ ∂λ q p(x1:T , q1:T |λ) (4.13) 1:T = ∂ ∂λ q1:T q1:T p(x1:T , q1:T |λ) p(x1:T , q1:T |λ) = ∂ ∂λ q1:T p(x1:T , q1:T |λ) p(x1:T |λ) (4.14) Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-22 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM - learning with gra...
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