Unformatted text preview: MPE Sampling What HMMs can do Summary Scratch HMMs and stationarity (cont.)
. . . continuing
=
q1 p(Qt1 +h = q1 )p(Xt1 +h = x1 Qt1 +h = q1 ) (5.36) n
q2:T i=2 =
q1 p(Xti +h = xi Qti +h = qi )p(Qti +h = qi Qti−1 +h = qi−1 ) p(Qt1 +h = q1 )p(Xt1 +h = x1 Qt1 +h = q1 ) (5.37) n
q2:T i=2 =
q1 Prof. Jeﬀ Bilmes p(Xti = xi Qti = qi )p(Qti = qi Qti−1 = qi−1 ) p(Qt1 +h = q1 )p(Xt1 = x1 Qt1 = q1 )f (x2:n , q1 ) EE596A/Winter 2013/DGMs – Lecture 5  Jan 25th, 2013 (5.38) page 557 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs and stationarity (cont.)
. . . continuing
=
q1 p(Qt1 +h = q1 )p(Xt1 +h = x1 Qt1 +h = q1 ) (5.36) n
q2:T i=2 =
q1 p(Xti +h = xi Qti +h = qi )p(Qti +h = qi Qti−1 +h = qi−1 ) p(Qt1 +h = q1 )p(Xt1 +h = x1 Qt1 +h = q1 ) (5.37) n
q2:T i=2 =
q1 p(Xti = xi Qti = qi )p(Qti = qi Qti−1 = qi−1 ) p(Qt1 +h = q1 )p(Xt1 = x1 Qt1 = q1 )f (x2:n , q1 ) (5.38) where f (x2:n , q1 ) is a function that is independent of the variable h. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5  Jan 25th, 2013 page 557 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs and stationarity (cont.)
. . . continuing
=
q1 p(Qt1 +h = q1 )p(Xt1 +h = x1 Qt1 +h = q1 ) (5.36) n
q2:T i=2 =
q1 p(Xti +h = xi Qti +h = qi )p(Qti +h = qi Qti−1 +h = qi−1 ) p(Qt1 +h = q1 )p(Xt1 +h = x1 Qt1 +h = q1 ) (5.37) n
q2:T i=2 =
q1 p(Xti = xi Qti = qi )p(Qti = qi Qti−1 = qi−1 ) p(Qt1 +h = q1 )p(Xt1 = x1 Qt1 = q1 )f (x2:n , q1 ) (5.38) 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 5  Jan 25th, 2013 page 557 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch 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 5  Jan 25th, 2013 page 558 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch 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 – Lecture 5  Jan 25th, 2013 page 558 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch 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:
Long chains Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5  Jan 25th, 2013 page 558 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch 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:
Long chains
Chains with cycle transition matrices Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 ...
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