Novel Lower Bounds on the Entropy Rate of Binary Hidden Markov Processes

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by  Or Ordentlich

Novel Lower Bounds on the Entropy Rate of
Binary Hidden Markov Processes
Or Ordentlich
MIT
[email protected]
Abstract—Recently,Samorodnitskyprovedastrengthenedver- [0,1], and a ∗ b , a(1 − b) + b(1 − a). Here, as well as
2
sion of Mrs. Gerber’s Lemma, where the output entropy of a throughouttherestofthepaper,logarithmsaretakentobase2.
binary symmetric channel is bounded in terms of the average
SincetheentropyrateofthesymmetricMarkovprocess{X }
entropyoftheinputprojectedonarandomsubsetofcoordinates. is H¯(X)=h(q), for symmetric hiddenMarkovprocesses tnhe
Here,thisresultisappliedforderivingnovellowerboundsonthe
entropyrateof binaryhiddenMarkovprocesses. Forsymmetric bound (1) takes the simple form
6
underlyingMarkovprocesses,ourboundimprovesuponthebest
1 H¯(Y)≥h(α∗q). (2)
known bound in the very noisy regime. The nonsymmetric case
0
is also considered, and explicit bounds are derived for Markov
2 processes that satisfy the (1,∞)-RLL constraint. Unfortunately, this bound is quite loose for many regimes of
the process parameters α and q.
y
a I. INTRODUCTION Recently, Samorodnitsky [2] proved a strengthened ver-
M sion of MGL, where the normalized input entropy
Let {X }, n = 1,2,..., be a symmetric stationary binary
n H(X ,...,X )/n in the right hand side of (1) is replaced
9 Markov process with transition probability 0 < q < 21, such by th1e averagne normalized entropy of the random vector
that X ∼Bernoulli(1) and for any n>1
] 1 2 (X1,...,Xn) projected on a random subset of coordinates.
T Xn =Xn−1⊕Wn, In this paper we apply the results of [2] to derive a novel
I lower bound on H¯(Y). Despite its simplicity, we show that
s. where{Wn},n=2,3,...,isasequenceofi.i.d.Bernoulli(q) this bound is stronger than the best known lower bounds for
c randomvariables,statisticallyindependentofX1.Weconsider the very noisy regime (α → 1), and recovers the strongest
[ thehiddenMarkovprocess{Yn},n=1,2,...,obtainedatthe bound for the fast transitions2regime (q → 1). For finite
2 output of a binary symmetric channel (BSC) with crossover values of (α,q) it is numericallydemonstratedt2hat the bound
v probability 0 < α < 21, whose input is the process {Xn}. is reasonably close to the true value of H¯(Y), which can
3 Namely, be estimated to an arbitrary precision by various known
5
approximation algorithms.
4 Yn =Xn⊕Zn,
6 We also derive a lower boundon H¯(Y) for the case where
where{Z },n=1,2,...,isasequenceofi.i.d.Bernoulli(α)
0 n the process {X } is a nonsymmetric binary Markov process.
n
. random variables, statistically independentof {Xn}. The task For the special case of Markov processes that satisfy the so-
1
of finding an explicit form for the entropy rate
0 called (1,∞)-RLL constraint, our bound is shown to be tight
6 H(Y ,...,Y ) in the very noisy regime.
1 H¯(Y), lim 1 n
n→∞ n
: II. PRELIMINARIES
v of the process {Y } is a long-standingopen problem,and the
n
Xi main contribution of this paper is in providing novel lower Let X=(X1,...,Xn) be a binary n-dimensional random
bounds for this quantity. vector, [n] , {1,...,n}, and S ⊆ [n] some subset of
r
a A simple lower bound on H¯(Y) can be obtained by in- coordinates. The projection of X onto S is defined as
voking Mrs. Gerber’s Lemma (MGL) [1], which states that if X ,{X :i∈S}.
S i
{X } is the inputto a BSC with crossoverprobabilityα, and
n As before, we assume that Y is the output of a BSC
{Y } is the output, then
n with crossover probability α, whose input is the vector X.
H(Y ,...,Y )≥nh α∗h−1 H(X1,...,Xn) , (1) Samorodnitsky has proved the following result.
1 n
n Theorem 1 ( [2, Theorem 1.11]): Letλ=(1−2α)2andlet
(cid:18) (cid:18) (cid:19)(cid:19)
where h(p) , −plog(p)−(1−p)log(1−p) is the binary B be a random subset uniformly distributed over all subsets
entropyfunction,h−1(·)isitsinverserestrictedtotheinterval of [n] with cardinality ⌈λn⌉. Then
H(X |B)
TheworkofO.OrdentlichwassupportedbytheMIT-TechnionPostdoc- H(Y)≥nh α∗h−1 B −E, (3)
λn
toralFellowship. (cid:18) (cid:18) (cid:19)(cid:19)
where E =O logn ·(n−H(X)). and define the random variable K as the largest k for which
n
A ≤ n. Clearly, the subset S and the subset {A ,...,A }
By Han’s i(cid:18)neqquality(cid:19)[3], the quantity H(XB|B)/λn is hakve the same distribution, and therefore 1 K
monotonicallynonincreasingin λ, and therefore,ignoring the
error term E, it can be seen that the bound (3) is stronger H(X |S) 1 K
S = E H(X |X ,...,X )
than (1). n n Ai Ai−1 A1
Forourpurposes,itwillbeconvenienttoreplaceH(X |B) Xi=1
B n
bwyithindHe(pXenSd|eSn)t,lywshaemreplSingiseaacrhanedleommesnutbiswetitohfp[nro]bgaebnileirtayteλd. = n1E H(XAi|XAi−1,...,XA1)1(i≤K)!
i=1
It is easy to verify that for any distribution PX on {0,1}n Xn
1
holds = E H(X |X )1(i≤K) (7)
n Ai Ai−1
H(X |B)−H(X |S) i=1 !
lim B S =0, X n
and we can thne→re∞fore indeed rλenplace B with S in Theorem 1, = n1E 1(1≤K)+ H(XAi−Ai−1+1|X1)1(i≤K)!
i=2
perhaps with a different convergence rate for E. In fact, X (8)
Polyanskiy and Wu [4] distilled from [2] the inequality
n
1
I(U :Y)≤I(U :XS|S), (4) = n Pr(K ≥1)+ E(H(XGi+1|X1)1(i≤K))!,
i=2
thatholdsforanyrandomvariableU satisfyingtheMarkovre- X (9)
lation U →X→Y. Using(4), the chainrule ofentropy,and
theconvexityoftheMGLfunctionϕ(t),h(α∗h−1(t)), itis where 1(T) is an indicator on the event T, (7) follows since
{X } is a first-order Markov process, and (8) follows from
a simple exercise to prove the following form of Theorem 1. n
the stationarity of {X }. For any 2≤i≤n we have
n
Proposition 1: Let λ = (1−2α)2 and let S be a random
subset of [n] generated by independently sampling each ele- E(H(X |X )1(i≤K))
Gi+1 1
ment i with probability λ. Then =E (H(X |X )Pr(K ≥i|G ))
Gi Gi+1 1 i
H(Y)≥nh α∗h−1 H(XS|S) . (5) =EGi(H(XGi+1|X1)Pr(Binomial(n−Gi,λ)≥i−1))
λn
(cid:18) (cid:18) (cid:19)(cid:19)
By thelawoflargenumbers,foranyǫ>0andfixedg ,there
i
exists some N such that for all n>N holds
0 0
III. MAIN RESULT
[1−ǫ,1) i≤(λ−ǫ)n
In order to apply Proposition 1 for lower bounding H¯(Y), Pr(Binomial(n−gi,λ)≥i−1)∈
we needto evaluatethequantityH(X |S)/λn forsymmetric ((0,ǫ] i≥(λ+ǫ)n
S
Markov processes {X }. We will use the notation
n Combining this with (9) gives that
q∗k ,q∗q∗···∗q,
1
lim H(X |S)=λEH(X |X )
ktimes n→∞n S G+1 1
and note that | {z } =λH(XG+1|X1,G), (10)
q∗k =Pr(W1⊕···⊕Wk =1)= 1−(1−2q)k. (6) andourclaimfollowssince H(XG+1|X1,G)=Eh q∗G for
2 stationary symmetric Markov processes.
(cid:0) (cid:1)
Proposition 2: Let0<λ<1,andletS bearandomsubset Remark 1: Anexpressionsimilarto(10)canalsoberecov-
of [n] generated by independently sampling each element i ered from [5, Corollary II.2].
with probability λ. Then
Our main result now follows directly from combining
H(X |S) Propositions 1 and 2 and using the continuity of the MGL
lim S =Eh q∗G ,
n→∞ λn function ϕ(t).
whereGisageometricrandomvariabl(cid:0)ewith(cid:1) parameterλ,i.e. Theorem 2: The entropy rate of the process obtained by
Pr(G=g)=(1−λ)g−1λ for g =1,2,.... passing a symmetric binary Markov process with transition
probability q through a BSC with crossover probability α
Proof: Let G , i = 1,2,..., be a sequence of i.i.d.
i
satisfies
geometric random variables with parameter λ. Define the
autoregressive process H¯(Y)≥h α∗h−1 Eh q∗G , (11)
k
A = G , k =1,2,..., where G is a geometric r(cid:0)andom v(cid:0)ariab(cid:0)le wi(cid:1)t(cid:1)h(cid:1)parameter λ=
k k
(1−2α)2.
i=1
X
IV. ASYMPTOTICANALYSIS AND NUMERICAL EXAMPLES and below. The upper and lower bounds from [7, Theorem
4.13] on (1−H¯(Y))/ǫ4 are plotted in Figure 1, along with
In this section we evaluate the bound from Theorem 2 in
the upper bound from Theorem 3. It is seen that Theorem 3
the limits of α → 12 and q fixed (very noisy regime), and in improvesupon the best known lower bounds on H¯(Y) in the
the limit of q → 1 and α fixed (fast transitions regime).
2 limit of α → 1. Furthermore, unlike [7, Theorem 4.13] that
Theorem 3: Let q be fixed and α= 1 −ǫ. Then only holds for2q ≥ 1, our result holds for all q.
2 4
∞
log(e) (1−2q)2k
H¯(Y)≥1−16ǫ4 +o(ǫ4).
2k(2k−1)1−(1−2q)2k
k=1 8
X
OW Upper Bound
Proof: By, e.g., [6, Lemma 1], we have that for any 0≤ 7 OThWe oLroewme 3r Bound
α,γ ≤1 holds
6
h(α∗γ)≥h(γ)+(1−h(γ))·4α(1−α)
5
=1−(1−2α)2(1−h(γ)). (12) )
Y
¯H(ǫ44
Setting β ,Eh q∗G , the RHS of (11) reads h(α∗h−1(β)), −1
which, by (12) and the parametrization α = 1 −ǫ, can be 3
(cid:0) (cid:1) 2
bounded as
2
h α∗h−1(β) ≥1−4ǫ2(1−β).
1
It therefore, re(cid:0)mains to ap(cid:1)proximate β. Recall the Taylor
0
expansion of the binary entropy function 0.25 0.3 0.35 0.4 0.45 0.5
q
∞
1 log(e)
h −p =1− (2p)2k. (13)
(cid:18)2 (cid:19) k=12k(2k−1) Fig.1. Comparisonbetween theboundsfrom[7,Theorem4.13]onH¯(Y)
X intheverynoisyregime,andthenewboundfromTheorem3.
Using (6), we have
1−(1−2q)G Next, we move to show that the lower bound from Theo-
β =Eh rem 2 is tight in the extreme regime of fast transitions, i.e.,
2
(cid:18) (cid:19) q → 1 and α fixed. Let q = 1 −ǫ. With this parametriza-
∞ log(e) 2 2
=1− E(1−2q)2kG (14) tion, (15) reads
2k(2k−1)
Xk∞=1 ∞ log(e) λ(2ǫ)2k
log(e) λ(1−2q)2k β =1− (16)
=1− , (15) 2k(2k−1)1−(1−λ)(2ǫ)2k
2k(2k−1)1−(1−λ)(1−2q)2k Xk=1
Xk=1 =1−2log(e)λǫ2+O(ǫ4).
where (14) is justified since the sum ∞ log(e) E(1 −
k=1 2k(2k−1) Now, using (12) and Theorem 2, we have the following
2q)2kG converges, and in (15) we have used the fact that
P proposition.
EtG = λt . To further approximate (15), we write
1−(1−λ)t Proposition 3: Let α be fixed and q = 1 −ǫ. Then
2
λ(1−2q)2k λ(1−2q)2k 1
1−(1−λ)(1−2q)2k = 1−(1−2q)2k · 1+ λ(1−2q)2k 1−H¯(Y)≤λ(1−β)
1−(1−2q)2k =2log(e)λ2ǫ2+O(ǫ4)
λ(1−2q)2k
= ·(1+O(λ)) =2log(e)(1−2α)4ǫ2+O(ǫ4)
1−(1−2q)2k
=4ǫ2 (1−2q)2k +O ǫ4 . In [7, Theorem 4.12] it was proved that for 0 ≤ α ≤ 1,
1−(1−2q)2k q = 1 −ǫ, as ǫ↓0 2
2
(cid:0) (cid:1)
Consequently,
1−H¯(Y)=2log(e)(1−2α)4ǫ2+o(ǫ3).
∞
log(e) (1−2q)2k
β =1−4ǫ2 +O ǫ4 .
2k(k−1)1−(1−2q)2k It therefore follows that the bound from Theorem 2 becomes
Xk=1 (cid:0) (cid:1) tight as q → 21.
which yields the desired result.
Itcanbeshownthatintheregimesofveryhigh-SNR(α→
To date, the best known upper and lower boundson H¯(Y) 0 and q fixed) and rare transitions (q → 0 and α fixed), the
in the very noisy regime were the ones found in [7, Theorem boundfrom Theorem2 is looser than the boundsfoundin [7,
4.13].Inparticular,theratio 1−H¯(Y) wasboundedformabove Theorem 4.11] and in [8], respectively.
ǫ4
1 1
0.95 0.95
0.9 0.9
0.85 0.85
0.8 0.8
) )
Y Y
¯H(0.75 ¯H(0.75
0.7 0.7
0.65 0.65
0.6 0.6
MGL MGL
Lower Bound from Theorem 2 Lower Bound from Theorem 2
0.55 Approximated Value 0.55 Approximated Value
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
q α
(a)α=0.11 (b)q=0.11
Fig.2. Comparisonbetweenthelowerbound(11),MGLbasedbound(2),andtheapproximate value ofH¯(Y)computedusing[7,Algorithm 4.25].
Foranypairoffinitevaluesof(α,q),theentropyrateH¯(Y) probability matrix P, such that X ∼ Bernoulli(π ) and
1 1
canbeapproximatedtoanarbitraryprecision.Forexample,[3, for n = 2,3,... holds Pr(Xn = j|Xn−1 = i) = Pij. For
Theorem 4.5.1] shows that k =1,2,..., we define the quantities
H(Yn|Yn−1...,Y1,X1)≤H¯(Y)≤H(Yn|Yn−1...,Y1) qi#jk , Pk ij =Pr(Xn =j|Xn−k =i).
and the two bounds converge to the same limit as n → ∞. Thefollowing(cid:0)isan(cid:1)extensionof Proposition2 fornonsym-
Unfortunately, the computational complexity of the lower
metric hidden Markov processes.
bound(aswellastheupperbound)above,isexponentialinn.1
Proposition 4: Let{X }beastationaryfirst-orderMarkov
To thatend,variousworksintroduceddifferentalgorithmsfor n
approximatingH¯(Y)[7],[10]–[12],eachalgorithmexhibiting process with transition probability matrix P and stationary
distribution π. Let 0 < λ< 1, and let S be a random subset
a different trade-off between approximation accuracy and
of [n] generated by independently sampling each element i
complexity.
with probability λ. Then
Toobtainabetterappreciationofthetightnessofthebound
from Theorem 2 for finite values of (α,q), we numerically lim H(XS|S) =π Eh q#G +π Eh q#G , (17)
compareittotheoutputofonesuchapproximationalgorithm. n→∞ λn 0 01 1 10
In particular, we use [7, Algorithm 4.25] to approximate whereGisageometricrando(cid:16)mvar(cid:17)iablewith(cid:16)param(cid:17)eterλ,i.e.
H¯(Y), where the algorithm parameters are chosen to ensure Pr(G=g)=(1−λ)g−1λ for g =1,2,....
high enough accuracy, and plot the results alongside with
Proof: The proofis similar to thatof Proposition2 up to
the lower bound of Theorem 2. We also plot the lower
equation(10),wherenowH(X |X ,G)=π Eh q#G +
bound(2)obtainedbysimplyapplyingMrs.Gerber’sLemma. G+1 1 0 01
The results for fixed α = 0.11 and varying q are shown in π Eh q#G . (cid:16) (cid:17)
1 10
Figures 2a, and those for fixed q = 0.11 and varying α, in
Com(cid:16)binin(cid:17)gthiswithProposition1andthecontinuityofthe
Figure 2b.
MGL function ϕ(t) gives the following.
V. NONSYMMETRIC MARKOV CHAINS Theorem 4: The entropy rate of the process obtained by
passinga stationaryfirst-orderMarkovprocesswith transition
In this section we extend ourlower boundfromTheorem2
probability matrix P and stationary distribution π through a
to the case where the input to the BSC is a nonsymmetric
BSC with crossover probability α satisfies
Markov process. Let
P= 1−q01 q01 H¯(Y)≥h α∗h−1 π0Eh q0#1G +π1Eh q1#0G ,
(cid:20) q10 1−q10 (cid:21) where G is a (cid:16)geometric(cid:16)random(cid:16)varia(cid:17)ble with p(cid:16)arame(cid:17)t(cid:17)er(cid:17)λ=
be a transition probability matrix, and π = [π π ] be a (1−2α)2.
0 1
stationary distribution for P, such that πP = π. Let {X }
n
be a stationary first-order Markov process with transition A. Example: Processes Satisfying the (1,∞)-RLL Constraint
In this subsection we lower bound the entropy rate of a
1Although, as shown by Birch [9], the gap between the two bounds also
decreases exponentially (butpossiblywithasmallexponent)inn. nonsymmetric first order Markov process, with q01 = q and
q =1, passed through a BSC with crossover probability α. Proof: Clearly, H¯(Y) ≤ H(Y ) = h(α∗π ). From [6,
10 n 1
ThisunderlyingMarkovprocesssatisfiestheso-called(1,∞)- equation (11)] combined with (13) we have
RLL constraint, where no consecutive ones are allowed to ∞
log(e)
appear in a sequence. It is not difficult to verify that for this h(α∗π )=1− (2ǫ(1−2π ))2k (22)
1 1
2k(2k−1)
choice of q and q we have
01 10 k=1
X
=1−2log(e)(1−2π )2ǫ2+O(ǫ4),
1 q 1
π = , π = ,
0 1+q 1 1+q whichestablishesourupperbound.For thelower bound,note
q0#1k = q+1(−+qq)k+1, q1#0k = 1−1(+−qq)k. tthheatcγon≥cahv(iπty1)o−f hcǫ(2·)fowresohmaveeutnhiavterfsoarlacloln0sta≤ntxc>≤01.Fhoroldms
h(x)≤h(π )+h′(π )(x−π ). Thus,using themonotonicity
1 1 1
In this case we have H¯(Y)≥h(α∗h−1(β)), where of h−1(·) and the fact that h′(π ) > 0 for all 0≤ q < 1, we
1
have
β ,E 1 h 1−(−q)G+1 + q h 1−(−q)G . h−1(γ)≥h−1(h(π )−cǫ2)≥π − c ǫ2. (23)
1+q 1+q 1+q 1+q 1 1 h′(π )
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) 1
(18)
Now, by Theorem 5, (23) and (22), we have
By the concavity of h(·), for any natural number g hold cǫ2
H¯(Y)≥h(α∗h−1(γ))≥h α∗ π −
1 h′(π )
1−(−q)g+1 (1−qg)+qg(1−q(−1)g+1) (cid:18) (cid:18) 1 (cid:19)(cid:19)
h =h =1−2log(e)(1−2π )2ǫ2+O(ǫ4).
1+q 1+q 1
(cid:18) (cid:19) (cid:18) (cid:19)
1 1−q(−1)g+1
≥(1−qg)h +qgh . (19)
1+q 1+q
(cid:18) (cid:19) (cid:18) (cid:19) ACKNOWLEDGMENT
and
The author is grateful to Alex Samorodnitsky for many
1−(−q)g (1−qg−1)+qg−1(1−q(−1)g) valuablediscussionsandobservations,andtoYuryPolyanskiy
h =h
and Yihong Wu for sharing an early draft of [4].
1+q 1+q
(cid:18) (cid:19) (cid:18) (cid:19)
1 1−q(−1)g
≥(1−qg−1)h +qg−1h . (20) REFERENCES
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