3 1 10 q1 x1 9 5 2 q2 x2 8 7 7 4 q3 x3 6 9 5 6 q4 x4

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Unformatted text preview: ation So we have the forward recursion as an elimination order and the backwards recursion as an elimination order. 3 1 10 Q1 X1 9 5 2 Q2 X2 8 7 7 4 Q3 X3 6 9 5 6 Q4 X4 4 10 3 8 Q5 X5 2 1 Green order is α-recursion, and blue order is β -recursion. Since HMM is a tree, there are no additional fill-in edges via the elimination orders we have chosen. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-9 (of 232) Logistics Review HMM, posteriors But as mentioned before, we want more than just p(x1:T ). We need clique posteriors γt (i) = p(Qt = i|x) and ξt (i, j ) = p(Qt−1 = i, Qt = j |x). We can also get these from α and β. γt (qt ) = p(qt |x1:T ) = p(x1:T |qt )p(qt )/p(x1:T ) = p(x1:t , xt+1:T |qt )p(qt )/p(x1:T ) = p(xt+1:T |qt , x1:t )p(x1:t |qt )p(qt )/p(x1:T ) = p(x1:t , qt )p(xt+1:T |qt )/p(x) = α(qt )β (qt )/p(x) α(qt )β (qt ) = α(qt )β (qt )/ qt Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-10 (of 232) Logistics Review Scratch Paper Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-11 (of 232) Logistics Review Scratch Paper Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-12 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Junction Trees - Review If a graph is not a tree, nodes can be clustered in such a way that the clusters (seen themselves as nodes) from “super nodes” in a tree-structure (junction tree), and then inference is run on a tree as normal. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-13 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Junction Trees - Review If a graph is not a tree, nodes can be clustered in such a way that the clusters (seen themselves as nodes) from “super nodes” in a tree-structure (junction tree), and then inference is run on a tree as normal. For this to work, the tree-structure just satisfy the “running intersection property” — all cluster nodes C ∈ P on the unique path P between any two cluster nodes A, B must contain the the intersection between A ∩ B . X3 , X4 , X5 x3 x3 x5 x4 x1 x6 x7 x2 x1 X4 , X5 x5 x4 x6 X4 , X5 , X6, X7 x2 x7 X6 , X7 X6 , X7 X1 , X6 , X7 Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 X2 , X6 , X7 page 5-13 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Junction Trees - Review Hugin inference in Junction trees. ∗∗ ψW = ∗∗ ψU φ∗∗ ∗ S ψW φ∗ S cop y ly ∗ ψU ∗ φS = ψU U Prof. Jeff Bilmes div ide ψU U \S alize margin φ∗∗ S φ∗ S φS =1 ∗∗ φS = ∗ ψW ∗∗ ψW W\ S margin alize ly multip py co multip multiply ∗ ψW ∗ ψW = φ∗ S ψW φS multiply divid e S EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 ψW W page 5-14 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Junction Trees - Review Hugin inference in Junction trees. ∗∗ ψW = ∗∗ ψU φ∗∗ ∗ S ψW φ∗ S cop y ly div ∗ ψU ∗ φS...
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