Je bilmes ee596awinter 2013dgms lecture 4 jan 23rd

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Unformatted text preview: M, other recursions It is possible to derive a temporal recursion for quantities other than α and β . E.g., here is a γt (j ) = p(Qt = j |x1:T ) backwards recursion. p(qt , qt+1 |x1:T ) = γ ( qt ) = qt+1 p(qt |qt+1 , x1:T )p(qt+1 |x1:T ) qt+1 p(qt |qt+1 , x1:T )γ (qt+1 ) = = qt+1 p(qt |qt+1 , x1:t )γ (qt+1 ) qt+1 qt+1 p(qt , qt+1 , x1:t ) γ (qt+1 ) qt p(qt , qt+1 , x1:t ) qt+1 p(qt+1 |qt )p(qt , x1:t ) γ (qt+1 ) qt p(qt+1 |qt )p(qt , x1:t ) = = Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-48 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM, other recursions It is possible to derive a temporal recursion for quantities other than α and β . E.g., here is a γt (j ) = p(Qt = j |x1:T ) backwards recursion. p(qt , qt+1 |x1:T ) = γ ( qt ) = qt+1 p(qt |qt+1 , x1:T )p(qt+1 |x1:T ) qt+1 p(qt |qt+1 , x1:T )γ (qt+1 ) = = qt+1 p(qt |qt+1 , x1:t )γ (qt+1 ) qt+1 qt+1 p(qt , qt+1 , x1:t ) γ (qt+1 ) qt p(qt , qt+1 , x1:t ) qt+1 p(qt+1 |qt )p(qt , x1:t ) γ (qt+1 ) qt p(qt+1 |qt )p(qt , x1:t ) qt+1 p(qt+1 |qt )αt (qt ) γ (qt+1 ) qt p(qt+1 |qt )αt (qt ) = = = Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-48 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM, other recursions It is possible to derive a temporal recursion for quantities other than α and β . E.g., here is a γt (j ) = p(Qt = j |x1:T ) backwards recursion. p(qt , qt+1 |x1:T ) = γ ( qt ) = qt+1 p(qt |qt+1 , x1:T )p(qt+1 |x1:T ) qt+1 p(qt |qt+1 , x1:T )γ (qt+1 ) = = qt+1 p(qt |qt+1 , x1:t )γ (qt+1 ) qt+1 qt+1 p(qt , qt+1 , x1:t ) γ (qt+1 ) qt p(qt , qt+1 , x1:t ) qt+1 p(qt+1 |qt )p(qt , x1:t ) γ (qt+1 ) qt p(qt+1 |qt )p(qt , x1:t ) qt+1 p(qt+1 |qt )αt (qt ) γ (qt+1 ) qt p(qt+1 |qt )αt (qt ) = = = Therefore, there is a backwards pass recursion using just the α’s without directly touching the observations again (better memory). Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-48 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Filtering Filtering query is p(qt |x1:t ) If we need this for one particular t, this is identical to one of the posteriors we already have p(qT |x1:T ) More likely, we need recursion from t to t + 1. This is obtained immediately from α-recursion, since αq (t) = p(x1:t , Qt = q ), we have p(qt |x1:t ) = αqt (t) q αqt (t) (4.33) t So, normalized αs are just the filtering operation. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-49 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Prediction p(qt |x1:s ), with t > s. This is also easily obtained from the αs for we have p(qt , x1:s ) = p(qt , qt−1 , . . . , qs+1 , qs , x1:s ) (4.34) qs ,qs+1 ,...,qt−1 p(qt |qt−1 )p(qt−1 |qt−2 ) . . . p(qs+1 |qs )αqs (s) = qs ,qs+1 ,...,qt−1 (4.35) p(qt |qt−1 ) = qt−1 p(qs+1 |qs )αqs (s) (qt−1 |qt−2 ) · · · qt−2 qs (4.36) Prof. Jeff Bilmes EE596A/Win...
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