# In general to sample from pa b can sample from pa and

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Unformatted text preview: (q1 )β1 (q1 ) = p(q1 , x1:T ) so this is readily available. ¯ ¯ To get p(q2 |q1 , x1:T ) we can compute ¯¯ p(q2 , q1 , x1:T ) = p(¯2 |q2 )β2 (q2 )p(q2 |q1 )α1 (¯1 ) ¯¯ x ¯ q (5.24) To get p(qt |q1:t−1 , x1:T ), construct p(qt , q1:t−1 , x1:T ) and recursion ¯ ¯ ¯ ¯ for α-like quantity p(qt , q1:t−1 , x1:T ) = p(¯t |qt )βt (qt )p(qt |qt−1 )p(¯1:t−1 , x1:t−1 ) (5.25) ¯ ¯ x ¯ q ¯ Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-49 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Posterior sampling - recursion Is computing the r.h.s. p(¯1:t , x1:t ) for various t easy? q¯ Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-50 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Posterior sampling - recursion Is computing the r.h.s. p(¯1:t , x1:t ) for various t easy? q¯ Yes, everything is essentially observed and factorizes, so we keep multiplying new observed values for resulting computed sample from left-to-right, as in: p(¯1:t , x1:t ) = p(¯t |qt )p(¯t |qt−1 )p(¯1:t−1 , x1:t−1 ) q¯ x¯ q¯ q ¯ Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 (5.26) page 5-50 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch What HMMs can do HMMs are more powerful than you might think. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-51 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch What HMMs can do HMMs are more powerful than you might think. In fact, many DGMs can be represented by HMMs. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-51 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch What HMMs can do HMMs are more powerful than you might think. In fact, many DGMs can be represented by HMMs. Thus, before studying DGMs, it is worthwhile to understand how ﬂexible and powerful HMMs are (and then as we go through course, we’ll see what the penalties are for making such HMM representations). Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-51 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch What HMMs can do HMMs are more powerful than you might think. In fact, many DGMs can be represented by HMMs. Thus, before studying DGMs, it is worthwhile to understand how ﬂexible and powerful HMMs are (and then as we go through course, we’ll see what the penalties are for making such HMM representations). We next visit a set of properties about HMMs that should be remembered. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-51 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Observations are not i.i.d. Joint probability under an HMM: p(Xt:t+h = xt:t+h ) t+h = qt:t+h j =t Prof. Jeﬀ Bilmes p(Xj = xj |Qj = qj )aqj −1 qj . EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 p...
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## This document was uploaded on 04/05/2014.

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