Both are graphs but display very dierent information

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Unformatted text preview: ate transition diagrams. Both are graphs (but display very different information), the left graph shows the allowable state transitions at each t ∈ {1, 2, . . . , T }. Left (trellis) would depict a time-inhomogeneous Markov chain if edges are different at each time step, or if the transition probabilities for the same pattern changes — edges indicate potential non-zero values, lack of edges indicate a zero transition value. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-26 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMM Transition Diagram with Output Distributions Informally, we can annotate each state with an observation distribution of the form p(x|Q = j ) = bj (x). p(q1 |q1 ) q1 bq 1 (x p(q2 |q2 ) ) p(q2 |q1 ) q2 bq 2 (x p(q3 |q3 ) ) p(q3 |q2 ) p(q3 |q1 ) q3 bq 3 (x ) p(q4 |q3 ) p(q4 |q4 ) bq 4 (x ) q4 p(q4 |q2 ) p(q4 |q1 ) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-27 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMM Trellis with Output Distributions The trellis too can be seen with observation distributions: bq4 (x2 ) q 3) 4| bq1 (x1 ) p(q 1 b q 1 (x 2 ) p(q q4 q 1) 2| bq 1 (x p(q2 |q2 ) ) 4 |q 1) p(q |q bq2 (x4 ) q1 p(q2 |q1 ) q2 b q 1 (x 3 ) p(q bq 2 (x p(q3 |q3 ) ) q3 q 1) 2| p(q3 |q2 ) p(q3 |q1 ) p(q2 |q2 ) p(q1 |q1 ) 2 bq3 (x4 ) p( 4 |q 1) p(q q 2) 3| p(q p(q2 |q2 ) p(q1 |q1 ) p(q1 |q1 ) q 3) 4| p( q4 p( q 1) 2| bq2 (x3 ) |q q 2) 3| q3 p(q p(q2 |q2 ) q1 p(q p(q3 |q3 ) p( p(q 1) bq2 (x2 ) |q q 2) 3| q3 p(q p( bq2 (x1 ) q2 p(q3 |q3 ) 4 |q 1) p(q3 |q3 ) 1) p( q4 q3 b q 3 (x 3 ) |q p(q bq 3 (x ) p(q4 |q4 ) bq 4 (x ) 2) b q 3 (x 2 ) bq4 (x4 ) p(q4 |q4 ) |q q 3) 4| 2) p(q |q bq3 (x1 ) bq4 (x3 ) p(q4 |q4 ) 2) p(q4 |q4 ) 1) bq4 (x1 ) q4 q3 p(q4 |q3 ) q4 p(q4 |q2 ) p(q4 |q1 ) bq1 (x4 ) p(q1 |q1 ) 3 4 Time homogeneity is indicated by transition probabilities being the same. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-28 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMM Trellis with Output Distributions The trellis too can be seen with observation distributions: bq4 (x2 ) q 3) 4| bq1 (x1 ) p(q 1 b q 1 (x 2 ) p(q q4 q 1) 2| bq 1 (x p(q2 |q2 ) ) 4 |q 1) p(q |q bq2 (x4 ) q1 p(q2 |q1 ) q2 b q 1 (x 3 ) p(q bq 2 (x p(q3 |q3 ) ) q3 q 1) 2| p(q3 |q2 ) p(q3 |q1 ) p(q2 |q2 ) p(q1 |q1 ) 2 bq3 (x4 ) p( 4 |q 1) p(q q 2) 3| p(q p(q2 |q2 ) p(q1 |q1 ) p(q1 |q1 ) q 3) 4| p( q4 p( q 1) 2| bq2 (x3 ) |q q 2) 3| q3 p(q p(q2 |q2 ) q1 p(q p(q3 |q3 ) p( p(q 1) bq2 (x2 ) |q q 2) 3| q3 p(q p( bq2 (x1 ) q2 p(q3 |q3 ) 4 |q 1) p(q3 |q3 ) 1) p( q4 q3 b q 3 (x 3 ) |q p(q bq 3 (x ) p(q4 |q4 ) bq 4 (x ) 2) b q 3 (x 2 ) bq4 (x4 ) p(q4 |q4 ) |q q 3) 4| 2) p(q |q bq3 (x1 ) bq4 (x3 ) p(q4 |q4 ) 2) p(q4 |q4 ) 1) bq4 (x1 ) q4 q3 p(q4 |q3 ) q4 p(q4 |q2 ) p(q4 |q1 ) bq1 (x4 ) p(q1 |q1 ) 3 4 Time homogeneity is indicated by transition probabilities being the same. Output distributions are dependent on state, and observation at time t. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-28 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch...
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This document was uploaded on 04/05/2014.

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