# 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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