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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