# Output distributions are dependent on state and

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 04/05/2014.

Ask a homework question - tutors are online