continuing pqt1 h q1 pxt1 h x1 qt1 h q1

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Unformatted text preview: = qi−1 ) q2:T i=2 (4.54) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-61 (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 = qi−1 ) q2:T i=2 p(Qt1 +h = q1 )p(Xt1 +h = x1 |Qt1 +h = q1 ) = (4.53) q1 n p(Xti = xi |Qti = qi )p(Qti = qi |Qti−1 = qi−1 ) q2:T i=2 (4.54) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-61 (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 = qi−1 ) q2:T i=2 p(Qt1 +h = q1 )p(Xt1 +h = x1 |Qt1 +h = q1 ) = (4.53) q1 n p(Xti = xi |Qti = qi )p(Qti = qi |Qti−1 = qi−1 ) q2:T i=2 p(Qt1 +h = q1 )p(Xt1 = x1 |Qt1 = q1 )f (x2:n , q1 ) = (4.54) q1 Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-61 (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 = qi−1 ) q2:T i=2 p(Qt1 +h = q1 )p(Xt1 +h = x1 |Qt1 +h = q1 ) = (4.53) q1 n p(Xti = xi |Qti = qi )p(Qti = qi |Qti−1 = qi−1 ) q2:T i=2 p(Qt1 +h = q1 )p(Xt1 = x1 |Qt1 = q1 )f (x2:n , q1 ) = (4.54) q1 where f (x2:n , q1 ) is a function that is independent of the variable h. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-61 (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 = qi−1 ) q2:T i=2 p(Qt1 +h = q1 )p(Xt1 +h = x1 |Qt1 +h = q1 ) = (4.53) q1 n p(Xti = xi |Qti = qi )p(Qti = qi |Qti−1 = qi−1 ) q2:T i=2 p(Qt1 +h = q1 )p(Xt1 = x1 |Qt1 = q1 )f (x2:n , q1 ) = (4.54) q1 where f (x2:n , q1 ) is a function that is independent of the variable h. For HMM stationarity to hold, it is required that p(Qt1 +h = q1 ) = p(Qt1 = q1 ) for all h. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-61 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs stationarity depends on MC Therefore, the HMM’s stationarity condition is entirely determined by the stationarity condition of the underlying hidden Markov chain. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-62 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs stationarity depends on MC Therefore, the HMM’s stationarity condition is entirely determined by the stationarity condition of the underlying hidden Markov chain. Consider the way in which HMMs are often used: Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lectu...
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