To get pq2 q1 x1t we can compute pq2 q1 x1t

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Unformatted text preview: observation. Simple example of “typicality”, flip a P (H ) = 0.9 = 1 − P (T ) biased coin 100 times. Most probable sequence is 100 heads. A typical sequence will be one that has about 90 heads and 10 tails. Much more likely we’ll get one of the 90,10 sequences than the all heads sequence. Viterbi path is potentially giving us an atypical sequence. Two solutions: sample from the posterior distribution p(q1:T |x1:T ), or computing the n-best sequences. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-101 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Posterior sampling We wish to draw samples of the form q1:T ∼ p(q1:T |x1:T ) for ¯ observation sequence x1:T . ¯ In general, to sample from p(a, b) can sample from p(a) and then from p(b|a), for sample a initially drawn. ¯ ¯ We can sample from p(q1 |x1:T ) and then sample from p(q2 |q1 , x1:T ), ¯ ¯¯ and thus from p(qt |q1:t−1 , x1:T ) ¯ ¯ p(q1 |x1:T ) ∝ α1 (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 (4.113) To get p(qt |q1:t−1 , x1:T ), construct p(qt , q1:t−1 , x1:T ) and recursion ¯ ¯ ¯ ¯ for α-like quantity p(qt , q1:t−1 , x1:T ) = p(¯t |qt )βt (qt )p(qt |qt−1 )p(¯1:t−1 , x1:t−1 ) ¯ ¯ x ¯ q ¯ (4.114) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-102 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Posterior sampling is computing p(¯1:t , x1:t ) for various t easy? q¯ Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-103 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Posterior sampling is computing p(¯1:t , x1:t ) for various t easy? q¯ Yes,...
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This document was uploaded on 04/05/2014.

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