34 this is just a dierent semi ring the resulting

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Unformatted text preview: 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 5 - Jan 25th, 2013 page 5-55 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs and stationarity HMM is a stochastic process over {Xt }t Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-56 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs and stationarity HMM is a stochastic process over {Xt }t An HMM is stationary whenever (5.35) p(Xt1+h:n+h = x1:n ) = p(Xt1:n = x1:n ) When might this be the case? Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-56 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs and stationarity HMM is a stochastic process over {Xt }t An HMM is stationary whenever (5.35) p(Xt1+h:n+h = x1:n ) = p(Xt1:n = x1:n ) When might this be the case? We have p(Xt1:n +h = x1:n ) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-56 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs and stationarity HMM is a stochastic process over {Xt }t An HMM is stationary whenever (5.35) p(Xt1+h:n+h = x1:n ) = p(Xt1:n = x1:n ) 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 5 - Jan 25th, 2013 page 5-56 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs and stationarity HMM is a stochastic process over {Xt }t An HMM is stationary whenever (5.35) p(Xt1+h:n+h = x1:n ) = p(Xt1:n = x1:n ) 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 i=2 Prof. Jeff Bilmes p(Xti +h = xi |Qti +h = qi )p(Qti +h = qi |Qti−1 +h = qi−1 ) EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-56 (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 p(Xti +h = xi |Qti +h = qi )p(Qti +h = qi |Qti−1 +h = qi−1 ) (5.38) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-57 (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 p(Xti = xi |Qti = qi )p(Qti = qi |Qti−1 = qi−1 ) (5.38) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-57 (of 232) HMMs Trellis Other HMM queries...
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