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Unformatted text preview: p://j.ee.washington.edu/~bilmes/ classes/ee596a_winter_2013/). uid is this class name (lower case) and pwd are the quarter/year of the class. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-4 (of 232) Logistics Review Relevant Readings: Readings/handouts Readings are in a sub-directory “reading drafts” directly below our web page (http://j.ee.washington.edu/~bilmes/ classes/ee596a_winter_2013/). uid is this class name (lower case) and pwd are the quarter/year of the class. Note, the PDF file is password protected. Send me email if you have trouble (adobe reader should have no problems reading it). Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-4 (of 232) Logistics Review Cumulative Outstanding Reading Read 8.1 - 8.3 in “doc.pdf” Read HMM section in readings. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-5 (of 232) Logistics Review HMM Probabilistic Queries Queries associated with HMMs The quantities we typically wish to compute for an HMM include: Compute p(qt |x1:t ), or the filtering problem. Compute p(qt |x1:s ), with t > s, or the prediction problem. Compute p(qt |x1:u ), with t < u, or the smoothing problem. Above three named from linear systems literature in EE (e.g., Kalman filters). Note: above includes p(qt |x1:T ) for t ∈ {1, 2, . . . , T }. Also needed query is p(qt , qt+1 |xr:s ) (often r = 1 and s = T ). In all above cases, we need to sum out hidden variables from joint distributions. E.g., p(qt |x1:T ) = p(qt , x1:T )/p(x1:T ), so also need p(x1:T ). I.e., we compute both the numerator and denominator in each of the above queries. Next few slides show how this relates to clique potentials in the HMM graph. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-6 (of 232) Logistics Review HMMs and elimination - forward recursion From the last line of the elimination (when we sum out qr−1 , we see α-recursion, which is: i αt (i)p(Qt+1 = j |Qt = i)p(xt+1 |Qt+1 = j ) (5.22) α1 (j ) = p(Q1 = j )p(¯1 |Q1 = j ) x αt+1 (j ) = (5.23) and We have that α1 (Q1 = j ) = p(Q1 = j )p(¯1 |Q1 = j ) = p(¯1 , Q1 = j ), and x x α1 (q1 ) = p(¯1 , q1 ). x Also, α2 (q2 ) = q1 p(q2 |q1 )p(¯2 |q2 )α(q1 ) = x x x ¯¯ x¯ q1 p(q1 , q2 , x1 , x2 ) = p(¯1 , x2 , q2 ) q1 p(q2 |q1 )p(¯2 |q2 )p(¯1 , q1 ) = In general, the forward (α) recursion has meaning αt (j ) = p(x1:t , Qt = j ) So α (forward) recursion is just an instance of the elimination algorithm run on the GM for the HMM graph. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-7 (of 232) Logistics Review HMMs and elimination - backward recursion T −1 ... xT −2 qT t=1 p(xt |qt )p(qt |qt−1 )δ (xt , xt ) p(¯T |qT )p(qT |qT −1 ) ¯ x T −1 = ... qT −1 xT −1 t=1 p(xt |qt )p(qt |qt−1 )δ (xt , xt ) ¯ qT 1 βT (qT ) p(¯T |qT )p(qT |qT −1 )βT (qT ) x βT −1 (qT −1 ) This corresponds to the β recursion βt (qt ) = qt+1 βt+1 (qt+1 )p(qt+1 |qt )p(¯t+1 |qt+1 ) x (5.22) (5.23) β T ( qT ) = 1 Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-8 (of 232) Logistics Review HMMs and elimin...
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This document was uploaded on 04/05/2014.

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