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Unformatted text preview: 0.6 0.6
0.4 0.6 0.6 0.4 0.02 0.4
0.4 0.2
0.99
0.8 0.99
0.01 0.99
0.01 0.015 0.99
0.01 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.
Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 page 473 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ State Duration Modeling
If we have multiple parallel states in series, all of which share the
same observation distribution, we can construct much more
interesting (multimodal) distributions.
For example, the following left chain will have distribution as shown
on the right (a mixture of negative binomial distributions).
0.025
0.6 0.6
0.4 0.6 0.6 0.4 0.02 0.4
0.4 0.2
0.99
0.8 0.99
0.01 0.99
0.01 0.015 0.99
0.01 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 4  Jan 23rd, 2013 page 473 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ 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 )
µi µj (Ah )ij πi − = µi π i h −
→ µi µj πj πi −
ij Prof. Jeﬀ Bilmes µi π i i ij i µi π i
i µi π i =0 i EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 page 474 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ 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 4  Jan 23rd, 2013 page 475 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ 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 4  Jan 23rd, 2013 page 476 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ 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 ) = ph (x1:T , q1:T ) (4.76) q1:T what can we say about ph and its accuracy? Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 page 476 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs Generative Accuracy
We can view an HMM as an approximate generative distribution of
t...
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 Winter '14

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