This preview shows page 1. Sign up to view the full content.
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 αlike 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 549 (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 550 (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
lefttoright, 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 550 (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 551 (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 551 (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 551 (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 551 (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...
View
Full
Document
This document was uploaded on 04/05/2014.
 Winter '14

Click to edit the document details