Post on 18-Jan-2018
description
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Entropy of Hidden Markov Processes
Or Zuk1 Ido Kanter2 Eytan Domany1 Weizmann Inst.1 Bar-Ilan Univ.2
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Overview Introduction Problem Definition Statistical Mechanics approach Cover&Thomas Upper-Bounds Radius of Convergence Related subjects Future Directions
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HMP - Definitions
Markov Process: X – Markov Process M – Transition Matrix Mij = Pr(Xn+1 = j| Xn = i)
Hidden Markov Process :Y – Noisy Observation of XN – Noise/Emission Matrix Nij = Pr(Yn = j| Xn = i)
M
NN
Xn Xn+1
Yn+1Yn
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Example: Binary HMP
0 1
p(1|0)
p(0|1)
p(1|1)
p(0|0)
)1|1()1|0()0|1()0|0(
pppp
0 1
q(0|0) q(1|0)q(0|1)
q(1|1)
)1|1()1|0()0|1()0|0(
qqqq
Transition Emission
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Example: Binary HMP (Cont.) For simplicity, we will concentrate on
Symmetric Binary HMP :
M = N =
So all properties of the process depend on two parameters, p and . Assume (w.l.o.g.) p, < ½
pppp
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1
1
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HMP Entropy Rate Definition :
H is difficult to compute, given as a Lyaponov Exponent (which is hard to compute generally.) [Jacquet et al 04]
What to do ? Calculate H in different Regimes.
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Different Regimes p -> 0 , p -> ½ ( fixed) -> 0 , -> ½ (p fixed) [Ordentlich&Weissman 04] study several regimes. We concentrate on the ‘small noise regime’ -> 0.Solution can be given as a power-series in :
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Statistical Mechanics First, observe the Markovian Property :
Perform Change of Variables :
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Statistical Mechanics (cont.) Ising Model :
, {-1,1} Spin Glasses
+ + + + - + - -
+ + - - - + + -
1
1
2
2
K
J
K
J
n
n
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Statistical Mechanics (cont.) Summing, we get :
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Statistical Mechanics (cont.) Computing the Entropy (low-temperature/high-field
expansion) :
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Cover&Thomas BoundsIt is known (Cover & Thomas 1991) :
We will use the upper-bounds C(n), and derive their orders :
Qu : Do the orders ‘saturate’ ?
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Cover&Thomas Bounds (cont.)
0 0.1 0.2 0.3 0.4 0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
eps
p
Upperbound / Lowerbound Average
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
eps
p
Upperbound Minus Lowerbound
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 0.1 0.2 0.3 0.4 0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
eps
p
Relative Error Upperbound Minus Lowerbound / Average
0.02
0.04
0.06
0.08
0.1
0.12
0 0.1 0.2 0.3 0.4 0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
eps
p
Relative Error Upperbound Minus Lowerbound / (1-Average)
0
0.5
1
1.5
2
2.5
3
n=4
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Cover&Thomas Bounds (cont.) Ans : Yes. In fact they ‘saturate’ sooner than would have
been expected ! For n (K+3)/2 they become constant. We therefore have :
Conjecture 1 : (proven for k=1)
How do the orders look ? Their expression is simpler when expressed using = 1-2p, which is the 2nd eigenvalue of P.
Conjecture 2 :
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First Few Orders :
Note : H0-H2 proven. The rest are conjectures from the upper-bounds.
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First Few Orders (Cont.) :
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First Few Orders (Cont.) :
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Radius of Convergence : When is our approximation good ? Instructive : Compare to the I.I.D. model
For HMP, the limit is unknown. We used the fit :
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Radius of Convergence (cont.) :
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Radius of Convergence (cont.) :
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Relative Entropy Rate
Relative entropy rate :
We get :
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Index of Coincidence Take two realizations Y,Y’ (of length n) of the same HMP. What is the probability that they
are equal ? Exponentially decaying with n.
We get :
Similarly, we can solve for three and four (but not five) realizations. Can give bounds on the entropy rate.
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Future Directions Proving conjectures Generalizations (e.g. any alphabets, continuous case) Other regimes Relative Entropy of two HMPs
Thank You