50 q1t t1 this is just a dierent semi ring the

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Unformatted text preview: qt |qt−1 ) (4.49) t=1 Note that this max-marginal can be completely different than the standard summation marginal T pk (xt |qt )pk (qt |qt−1 ) (4.50) q1:T t=1 Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-59 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Observations are not “Viterbi i.i.d.” We can view this as K distributions over just the observations, i.e., ∗ pk (x1:T ) ∝ pk (x1:T , q1:,k ) vit T (4.47) ∝ max pk (x1:T , q1:T ) (4.48) q1:T T ∝ max q1:T pk (xt |qt )pk (qt |qt−1 ) (4.49) t=1 Note that this max-marginal can be completely different than the standard summation marginal T pk (xt |qt )pk (qt |qt−1 ) (4.50) q1:T t=1 This is just a different semi-ring. The resulting distribution over observations does not in general factorize, so no i.i.d. here either. The distribution pk (x1:T ) also does not in general have any vit (marginal) independence assumptions. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-59 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and stationarity HMM is a stochastic process over {Xt }t Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-60 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and stationarity HMM is a stochastic process over {Xt }t An HMM is stationary whenever p(Xt1+h:n+h = x1:n ) = p(Xt1:n = x1:n ) (4.51) When might this be the case? Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-60 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and stationarity HMM is a stochastic process over {Xt }t An HMM is stationary whenever p(Xt1+h:n+h = x1:n ) = p(Xt1:n = x1:n ) (4.51) When might this be the case? We have p(Xt1:n +h = x1:n ) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-60 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and stationarity HMM is a stochastic process over {Xt }t An HMM is stationary whenever p(Xt1+h:n+h = x1:n ) = p(Xt1:n = x1:n ) (4.51) When might this be the case? We have p(Xt1:n +h = x1:n ) p(Xt1:n +h = x1:n , Qt1:n +h = q1:n ) = q1:n Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-60 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and stationarity HMM is a stochastic process over {Xt }t An HMM is stationary whenever p(Xt1+h:n+h = x1:n ) = p(Xt1:n = x1:n ) (4.51) When might this be the case? We have p(Xt1:n +h = x1:n ) p(Xt1:n +h = x1:n , Qt1:n +h = q1:n ) = q1:n = p(Qt1 +h = q1 )p(Xt1 +h = x1 |Qt1 +h = q1 ) q1:n n p(Xti +h = xi |Qti +h = qi )p(Qti +h = qi |Qti−1 +h = qi−1 ) i=2 Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-60 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and stationarity (cont.) . . . continuing p(Qt1 +h = q1 )p(Xt1 +h = x1 |Qt1 +h = q1 ) = (4.52) q1 n p(Xti +h = xi |Qti +h = qi )p(Qti +h = qi |Qti−1 +h...
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This document was uploaded on 04/05/2014.

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