Eg here is a t j pqt j x1t backwards recursion

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Unformatted text preview: Summary Scratch 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. γ ( qt ) = qt+1 = qt+1 = qt+1 Prof. Jeff Bilmes p(qt , qt+1 |x1:T ) = qt+1 p(qt |qt+1 , x1:T )p(qt+1 |x1:T ) p(qt |qt+1 , x1:T )γ (qt+1 ) = qt+1 p(qt |qt+1 , x1:t )γ (qt+1 ) p(qt , qt+1 , x1:t ) γ (qt+1 ) q p(qt , qt+1 , x1:t ) t EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-21 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch 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. γ ( qt ) = qt+1 = qt+1 p(qt , qt+1 |x1:T ) = qt+1 p(qt |qt+1 , x1:T )p(qt+1 |x1:T ) p(qt |qt+1 , x1:T )γ (qt+1 ) = qt+1 p(qt |qt+1 , x1:t )γ (qt+1 ) qt+1 p(qt , qt+1 , x1:t ) γ (qt+1 ) q p(qt , qt+1 , x1:t ) qt+1 p(qt+1 |qt )p(qt , x1:t ) γ (qt+1 ) q p(qt+1 |qt )p(qt , x1:t ) = = Prof. Jeff Bilmes t t EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-21 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch 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. γ ( qt ) = qt+1 = qt+1 p(qt , qt+1 |x1:T ) = qt+1 p(qt |qt+1 , x1:T )p(qt+1 |x1:T ) p(qt |qt+1 , x1:T )γ (qt+1 ) = qt+1 p(qt |qt+1 , x1:t )γ (qt+1 ) qt+1 p(qt , qt+1 , x1:t ) γ (qt+1 ) q p(qt , qt+1 , x1:t ) qt+1 p(qt+1 |qt )p(qt , x1:t ) γ (qt+1 ) q p(qt+1 |qt )p(qt , x1:t ) qt+1 p(qt+1 |qt )αt (qt ) γ (qt+1 ) q p(qt+1 |qt )αt (qt ) = = = Prof. Jeff Bilmes t t t EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-21 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch 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. γ ( qt ) = qt+1 = qt+1 p(qt , qt+1 |x1:T ) = qt+1 p(qt |qt+1 , x1:T )p(qt+1 |x1:T ) p(qt |qt+1 , x1:T )γ (qt+1 ) = qt+1 p(qt |qt+1 , x1:t )γ (qt+1 ) qt+1 p(qt , qt+1 , x1:t ) γ (qt+1 ) q p(qt , qt+1 , x1:t ) qt+1 p(qt+1 |qt )p(qt , x1:t ) γ (qt+1 ) q p(qt+1 |qt )p(qt , x1:t ) qt+1 p(qt+1 |qt )αt (qt ) γ (qt+1 ) q p(qt+1 |qt )αt (qt ) = = = t t t 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 5 - Jan 25th, 2013 page 5-21 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMM Trellis/Lattice/Grid We’ll see that it is very useful to view the HMM state-space as a trellis. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-22 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMM Trellis/Lattice/Grid We’ll see that it is very useful to view the HMM state-space as a trellis. It is sometimes called an HMM grid, or even a “...
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