Output distributions are dependent on state and

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Unformatted text preview: HMM Trellis with Output Distributions The trellis too can be seen with observation distributions: p(q2 |q2 ) bq1 (x1 ) q1 p(q q 1) 2| bq1 (x2 ) q4 p( q 1) 2| p(q 1) q 2) 3| bq2 (x4 ) |q p(q p(q2 |q2 ) bq1 (x3 ) p(q1 |q1 ) 2 bq3 (x4 ) p(q bq2 (x3 ) |q 1) 4 |q 1) p(q q 2) 3| p(q p(q2 |q2 ) p(q1 |q1 ) 1 q 3) 4| |q 2) q4 p( |q 1) p(q bq2 (x2 ) q3 q2 q 2) 3| p(q p(q3 |q3 ) p( p( p(q q3 bq2 (x1 ) bq3 (x3 ) 2) q 3) 4| p(q p(q3 |q3 ) 4 |q 1) p(q3 |q3 ) p(q4 |q4 ) |q 2) bq3 (x2 ) bq4 (x4 ) 4 |q 1) q 3) 4| q4 q3 Prof. Jeff Bilmes p(q p(q4 |q4 ) |q bq3 (x1 ) bq4 (x3 ) q3 p(q4 |q4 ) p( q4 bq4 (x2 ) p( bq4 (x1 ) q 1) 2| p(q bq1 (x4 ) p(q1 |q1 ) 3 EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 4 page 5-29 (of 232) HMM Trellis with and the forward recursion Consider the alpha recursion computation in the context of a trellis: αt (i)p(j |i)p(xt |j ) = p(xt |j ) q2 bq2 (x2 ) p(q q 2) 3| q1 q 1) 2| p(q bq1 (x2 ) p(q q 1) 2| 4 |q 2) p( q bq2 (x3 ) p(q q 2) 3| bq2 (x4 ) p(q2 |q2 ) b q 1 (x 3 ) p(q1 |q1 ) 2 b q 3 (x 4 ) q1 ) p(q3 |q3 ) p(q2 |q2 ) p(q1 |q1 ) 1 q 3) 4| p(q 4| q1 ) p(q3 |q3 ) p(q2 |q2 ) bq1 (x1 ) p(q 3 |q 1) q 2) 3| p(q bq3 (x3 ) 3 |q 1) bq2 (x1 ) p(q 4| p(q3 |q3 ) q 3) 4| p( q p(q 4 |q 2) bq3 (x2 ) bq4 (x4 ) p(q4 |q4 ) q1 ) q 3) 4| p( q q3 p(q 4 |q 2) b q 3 (x 1 ) bq4 (x3 ) p(q4 |q4 ) 3 |q 1) 4 p(q4 |q4 ) p( q q bq4 (x2 ) α1 (j ) = p(j )p(¯1 |j ) x and p( q bq4 (x1 ) αt (i)p(j |i), i p(q 4| i p( q αt (j ) = p(q q 1) 2| bq1 (x4 ) p(q1 |q1 ) 3 4 HMM Trellis with and the backward recursion Consider the beta recursion computation in the context of a trellis: βt+1 (qt+1 )p(qt+1 |qt )p(¯t+1 |qt+1 ) x bq1 (x1 ) 1 p(q bq1 (x2 ) p(q q 1) 2| p(q q 2) 3| 4 |q 2) 4 |q 1) b q 2 (x 4 ) p(q2 |q2 ) bq1 (x3 ) p(q1 |q1 ) 2 bq3 (x4 ) 1) 1) bq2 (x3 ) |q p(q 4 |q 1) p(q3 |q3 ) q3 q 2) 3| p(q q 3) 4| p( q p(q p(q2 |q2 ) p(q1 |q1 ) 1 bq3 (x3 ) p( 1) 4 |q 1) |q bq2 (x2 ) p(q2 |q2 ) q 1) 2| q 3) 4| p(q3 |q3 ) q3 q2 q 2) 3| p(q p(q4 |q4 ) p( q p(q bq4 (x4 ) p(q bq3 (x2 ) p(q p(q3 |q3 ) bq2 (x1 ) q q 3) 4| p( q q3 p(q p(q4 |q4 ) 4 |q 2) bq3 (x1 ) bq4 (x3 ) 4 |q 2) p(q4 |q4 ) p( q4 bq4 (x2 ) β T ( qT ) = 1 |q bq4 (x1 ) and q3 qt+1 p( βt (qt ) = p(q q 1) 2| b q 1 (x 4 ) p(q1 |q1 ) 3 4 HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMM Trellis A few other points: Notice that the computation proceeds synchronously Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-32 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMM Trellis A few other points: Notice that the computation proceeds synchronously Notice that at each node at each time t, we in a sense “pull” values from previous times t (for α computation). Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-32 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMM Trellis A few other points: Notice that the computation proceeds synchronously Notice that at each node at each time t, we in a sense “pull” values from previous times t (for α computation). Notice that the complexity depends on the density of the transition matrix. If strictly left-to-right (i.e., p(Qt = j...
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This document was uploaded on 04/05/2014.

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