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Unformatted text preview: 0.01 0.01 0.005 10 20 30
d 40 50 60 Other examples: very long chains, ladders, ﬁxedlength distributions
(histograms), and so on.
HMMs can have ﬂexible distributions, cost of extra states.
Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5  Jan 25th, 2013 page 569 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch What HMMs can do  summary so far
Observations are not i.i.d., but conditioned on state variables, they
are independent.
Observations are not “Viterbi i.i.d.”
HMMs are a stationary process over p(x1:n ) whenever the
underlying hidden Markov chain is a stationary process.
Single Gaussian per state HMM: Covariance decays as:
cov(Xt , Xt+h )
=
ij
h −
→
Prof. Jeﬀ Bilmes ij µi µj (Ah )ij πi −
µi µj πj πi − µi π i µi π i i i µi π i
i µi π i =0 i EE596A/Winter 2013/DGMs – Lecture 5  Jan 25th, 2013 page 570 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch What HMMs can do  summary so far but mutual information (in practice) can apparently extend in time
reasonably far (but also decays).
Parameter sharing means enormous ﬂexibility in state duration
models (e.g., negative binomial, mixtures thereof, ﬁxed histograms). Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5  Jan 25th, 2013 page 571 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs Generative Accuracy
We can view an HMM as an approximate generative distribution of
the observation variables, as in ph (x1:T ) ≈ p(x1:T ) Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5  Jan 25th, 2013 page 572 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs Generative Accuracy
We can view an HMM as an approximate generative distribution of
the observation variables, as in ph (x1:T ) ≈ p(x1:T )
Given that ph is an approximation, one that is a mixture
ph (x1:T ) = (5.60) ph (x1:T , q1:T )
q1:T what can we say about ph and its accuracy? Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5  Jan 25th, 2013 page 572 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs Generative Accuracy
We can view an HMM as an approximate generative distribution of
the observation variables, as in ph (x1:T ) ≈ p(x1:T )
Given that ph is an approximation, one that is a mixture
ph (x1:T ) = (5.60) ph (x1:T , q1:T )
q1:T what can we say about ph and its accuracy?
Accuracy can be measured by KLdivergence
D(p(x1:T )ph (x1:T )) = p(x1:T ) log
x1:T p(x1:T )
ph (x1:T ) (5.61) and if D(p(x1:T )ph (x1:T )) = 0, then the HMM is perfectly
generatively accurate.
Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5  Jan 25th, 2013 page 572 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs Generative Accuracy
For an HMM to be generatively accurate, we can derive necessary
conditions on the HMM, e.g., number of required states. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5  Jan 25th, 2013 page 573 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs Generative Accuracy
For an HMM...
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 Winter '14

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