Asymptotics of Entropy Rate in Special Families of Hidden Markov Chains

arXiv:0810.2144v1 [cs.IT] 13 Oct 2008 Asymptotics of Entropy Rate in Special Families of Hidden Markov Chains Guangyue Han Brian Marcus University of Hong Kong University of British Columbia email: ghan@maths.hku.hk email: marcus@math.ubc.ca November 4, 2018 Abstract We generalize a result in [8] and derive an asymptotic formula for entropy rate of a hidden Markov chain around a “weak Black Hole”. We also discuss applications of the asymptotic formula to the asymptotic behaviors of certain channels. Index Terms–entropy, entropy rate, hidden Markov chain, hidden Markov model, hidden Markov process 1 Introduction Consider a discrete finite-valued stationary stochastic process Y = Y ∞ −∞:= {Yn : n ∈Z}. The entropy rate of Y is defined to be H(Y ) = lim n→∞H(Y 0 −n)/(n + 1); here, H(Y 0 −n) denotes the joint entropy of Y 0 −n := {Y−n, Y−n+1, · · · , Y0}, and log is taken to mean the natural logarithm. If Y is a Markov chain with alphabet {1, 2, · · · , B} and transition probability matrix ∆, it is well known that H(Y ) can be explicitly expressed with the stationary vector of Y and ∆. A function Z = Z∞ −∞of the Markov chain Y with the form Z = Φ(Y ) is called a hidden Markov chain; here Φ is a function defined on {1, 2, · · · , B}, taking values in A := {1, 2, · · · , A} (alternatively a hidden Markov chain is defined as a Markov chain observed in noise). For a hidden Markov chain, H(Z) turns out (see Equation (1)) to be the integral of a certain function defined on a simplex with respect to a measure due to Blackwell [4]. However Blackwell’s measure is somewhat complicated and the integral formula appears to be difficult to evaluate in most cases. In general it is very difficult to compute H(Z); so far there is no simple and explicit formula for H(Z). Recently, the problem of computing the entropy rate of a hidden Markov chain Z has drawn much interest, and many approaches have been adopted to tackle this problem. For instance, Blackwell’s measure has been used to bound the entropy rate [15] and a variation on the Birch bound [3] was introduced in [5]. An efficient Monte Carlo method for computing the entropy rate of a hidden Markov chain was proposed independently by Arnold and Loeliger [1], Pfister et. al. [17], and Sharma and Singh [19]. The connection between the entropy rate of a hidden Markov chain and the top Lyapunov exponent of a random matrix product has been observed [10, 11, 12, 6]. In [7], it is shown that under mild positivity assumptions the entropy rate of a hidden Markov chain varies analytically as a function of the underlying Markov chain parameters. Another recent approach is based on computing the coefficients of an asymptotic expan- sion of the entropy rate around certain values of the Markov and channel parameters. The first result along these lines was presented in [12], where for a binary symmetric channel with crossover probability ε (denoted by BSC(ε)), the Taylor expansion of H(Z) around ε = 0 is studied for a binary hidden Markov chain of order one. In particular, the first derivative of H(Z) at ε = 0 is expressed very compactly as a Kullback-Liebler divergence between two distributions on binary triplets, derived from the marginal of the input process X. Fur- ther improvements and new methods for the asymptotic expansion approach were obtained in [16], [20], [21] and [8]. In [16] the authors express the entropy rate for a binary hidden Markov chain where one of the transition probabilities is equal to zero as an asymptotic expansion including a O(ε log ε) term. This paper is organized as follows. In Section 2 we give an asymptotic formula (The- orem 2.8) for the entropy rate of a hidden Markov chain around a weak Black Hole. The coefficients in the formula can be computed in principle (although explicit computations may be quite complicated in general). The formula can be viewed as a generalization of the Black Hole condition considered in [8]. The weak Black Hole case is important for hidden Markov chains obtained as output processes of noisy channels, corresponding to input processes, for which certain sequences have probability zero. Examples are given in Section 3. Example 3.1 was already treated in [9] for only the first few coefficients; but in this case, these coefficients were computed quite explicitly. 2 Asymptotic Formula for Entropy Rate Let W be the simplex, comprising the vectors {w = (w1, w2, · · · , wB) ∈RB : wi ≥0, X i wi = 1}, and let Wa be all w ∈W with wi = 0 for Φ(i) ̸= a. For a ∈A, let ∆a denote the B × B matrix such that ∆a(i, j) = ∆(i, j) for j with Φ(j) = a, and ∆a(i, j) = 0 otherwise. For a ∈A, define the scalar-valued and vector-valued functions ra and fa on W by ra(w) = w∆a1, and fa(w) = w∆a/ra(w). Note that fa defines the action of the matrix ∆a on the simplex W. 2 If Y is irreducible, it turns out that H(Z) = − Z X a ra(w) log ra(w)dQ(w), (1) where Q is Blackwell’s measure [4] on W. This measure, which satisfies an integral equation dependent on the parameters of the process,

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