9 1 p t biased coin 100 times most probable sequence

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Unformatted text preview: or computing the n-best sequences. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-48 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Posterior sampling - recursion We wish to draw samples of the form q1:T ∼ p(q1:T |x1:T ) for ¯ observation sequence x1:T . ¯ Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-49 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Posterior sampling - recursion 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. ¯ ¯ Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-49 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Posterior sampling - recursion 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 ) ¯ ¯ Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-49 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Posterior sampling - recursion 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. ¯ ¯ Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-49 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Posterior sampling - recursion 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 Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 (5.24) page 5-49 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Posterior sampling - recursion 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...
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