M q t prof je bilmes m q 1 pt qt q x pt

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Unformatted text preview: ¯t |Qt = q ) max p(Qt = q |Qt−1 = x r (5.15) m r)αr (t − 1) (5.16) We get the final max can be computed from the final max marginal: Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-41 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Most Probable Explanation We can thus define a modified form of the α-recursion that, rather than uses summation, uses a max operator. m αq (t) m αq (1) = p(¯t |Qt = q ) x = p(¯t |Qt = q ) max p(Qt = q |Qt−1 = x r (5.15) m r)αr (t − 1) (5.16) We get the final max can be computed from the final max marginal: max q1:T ∈DQ1:T Prof. Jeff Bilmes m p(¯1:T , q1:T ) = max αq (T ) x (5.17) q EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-41 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Most Probable Explanation We can thus define a modified form of the α-recursion that, rather than uses summation, uses a max operator. m αq (t) m αq (1) = p(¯t |Qt = q ) x = p(¯t |Qt = q ) max p(Qt = q |Qt−1 = x r (5.15) m r)αr (t − 1) (5.16) We get the final max can be computed from the final max marginal: max q1:T ∈DQ1:T m p(¯1:T , q1:T ) = max αq (T ) x (5.17) q Max operator is similar to sum (a commutative semi-ring), in that it marginalizes out hidden variables. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-41 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Most Probable Explanation Max operator is similar to sum, in that it marginalizes out hidden variables. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-42 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Most Probable Explanation Max operator is similar to sum, in that it marginalizes out hidden variables. Given p(¯, q ) when we form maxq p(¯, q ) we can think of this as a x x marginal, the “max marginal” of the form pm (¯) = max p(¯, q ) x x (5.18) q Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-42 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Most Probable Explanation Max operator is similar to sum, in that it marginalizes out hidden variables. Given p(¯, q ) when we form maxq p(¯, q ) we can think of this as a x x marginal, the “max marginal” of the form pm (¯) = max p(¯, q ) x x (5.18) q m Given this, we can view the αq (t) as the max marginals up to time t m αq (t) = pm (x1:t , Qt = q ) (5.19) so that the above final maximization makes sense. We’ve defined a recursive way to compute the max marginal. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-42 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Most Probable Explanation Max operator is similar to sum, in that it marginalizes out hidden variables. Given p(¯, q ) when we form maxq p(¯, q ) we can think of th...
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This document was uploaded on 04/05/2014.

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