One solution node a separator a 3 clique for a state

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Unformatted text preview: 2013 page 4-40 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and message passing/LBP LBP messages on an HMM are straightforward, and again since an HMM is a tree, we can use any message order that obeys the message-passing protocol. q1 q3 q4 q5 x1 Prof. Jeff Bilmes q2 x2 x3 x4 x5 EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-41 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM, and junction tree - right-state 3-clique Can view as junction tree with cliques/separators. One solution: node a separator, & a 3-clique for a state & observation variable at time t, & another state to the right (& a final 2-clique at the end). Q1 Q2 Q3 Q4 X1 X2 X3 X4 X5 Q 1X1Q 2 Q2 Q 2X2Q 3 Q2 Q3 Q4 Q5 X1 X2 X3 X4 QT XT ... Q3 Q1 Prof. Jeff Bilmes ... Q5 QT ... X5 EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 Q TXT QT XT page 4-42 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM, and junction tree - left-state 3-clique Another solution has each node a separator, and a 3-clique for a state and observation variable at time t, and another state to the left (and an initial 2-clique at the beginning). Q1 Q2 Q3 Q4 X1 X2 X3 X4 X5 Q 1X1 Q 1X2Q 2 Q1 Q 2X3Q 3 Q2 Q1 Q2 Q3 Q4 Q5 X1 Prof. Jeff Bilmes ... Q5 X2 X3 X4 QT XT ... Q T-1 ... X5 EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 Q T-1XTQ T QT XT page 4-43 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM, and junction tree - two 2-cliques Yet another solution has cliques for successive states and for state/observation pair at each t. Q1 Q2 Q3 Q4 Q5 X1 X2 X3 X4 X5 Q 1Q 2 Q 1Q 2 Q2 Q2 Q 1Q 2 Q2 Q 1Q 2 Q2 Q 1X1 Q 3X3 Q 4X4 Q2 Q3 Q4 Q5 X1 X2 X3 X4 ... Q 5X5 Q1 Prof. Jeff Bilmes Q 2X2 ... X5 QT XT Q T-1Q T Q TXT ... EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 QT XT page 4-44 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Discussion of the above three What are the advantages/disadvantages of the above (if any)? Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-45 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Discussion of the above three What are the advantages/disadvantages of the above (if any)? State space is the same. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-45 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Discussion of the above three What are the advantages/disadvantages of the above (if any)? State space is the same. What are implications of summing over states observations if observation distribution is complex (e.g., if p(x|q ) with x high-dimensional real-valued Gaussian mixture). Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-45 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM, and junction tree Using either of these junction trees, we can define a forward and/or backwards recursion, and we’ll again, in some cases, either recover exactly the α, β recursions, or computations that are quite similar. For example, Hugin message passing computes the backward messages based on the forward (if we go right and then left), or computes the forward messages based on the backwards (if we go left and then right). Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23r...
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