This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ionary
distribution, this covariance goes to zero exponentially fast.
Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 page 468 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlation over time of simple HMM
Thus, cov(Xt , Xt+h ) is in general not equal to zero.
h But recall, Ah − 1π from lecture 3, and this is a matrix with all
→
rows equal to the stationary distribution.
Therefore, we have
cov(Xt , Xt+h ) (4.70) µi µj (Ah )ij πi − = µi π i h −
→ µi µj πj πi −
ij µi π i i ij µi π i
i (4.71) i µi π i =0 (4.72) i Thus, while the covariance between to observations is not
necessarily zero in an HMM, once we are at a stationary
distribution, this covariance goes to zero exponentially fast.
Exercise: must there be a unqiue stationary distribution here?
Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 page 468 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlation over time of simple HMM
Example of the decay in the mutualinformation correlation from a
realworld HMM. I.e., we see f (τ ) = I (Xt ; Xt+τ ), where I () is the
mutual information function.
Mutual information is stronger than correlation.
This is compared against i.i.d. samples (high peak at τ = 0 is
expected).
−3 14 x 10 HMM MI
IID MI 13
12 MI (bits) 11
10
9
8
7
6
5
−100 Prof. Jeﬀ Bilmes −50 0
time (ms) 50 100 EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 page 469 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ State Duration Modeling
Markov chain, state duration distribution is geometric
Let D be such a random variable, then
P (D = d) = pd−1 (1 − p) (4.73) where d ≥ 1 is an integer and p = aii , if D is the duration random
variable for state i of the chain, giving:
0.25
0.2 0.75
0.15 P(d) 0.25 0.1 0.05
0 10 20 30
d 40 50 60 Many sequential tasks have subsegments that do not follow this
distribution
Prof. Jeﬀ Bilmes
page 470 (of 239)
this seems like EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013
an inherent limitation. HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ State Duration Modeling
Many “tricks” to using an HMM, can alleviate such problems.
“statetying”, where multiple states have the same observation
distribution (parameters are shared).
That is, state q and q are tied if it is the case that
p(xQt = q ) = p(xQt = q ) ∀x ∈ DX (4.74) If n states in a series are strung together, all of which share the
same observation distribution, that observation distribution will be
active for as long as we are in that state. For example:
0.08 0.75 0.75
0.25 0.75
0.25 0.06 0.75
0.25 0.25 0.04 0.02 0 Prof. Jeﬀ Bilmes 10 20 EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 30
d 40 50 page 60 471 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ State Duration Modeling
This corresponds to the sum of random variables. Let {Di }i be a
collection of independent geometrically distributed random variables
with parameter p, and let Wr = r=1 Dr , then
i
k−1 r
p (1 − p)k−r , k = r, r + 1, . . .
r−1 p(Wr = k ) = (4.75) This is a “negati...
View
Full
Document
This document was uploaded on 04/05/2014.
 Winter '14

Click to edit the document details