RELATIVE ENTROPY UNDER MAPPINGS BY STOCHASTIC MATRICES
RELATIVE ENTROPY UNDER MAPPINGS BY STOCHASTIC MATRICES
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DOI：
10.1016/0024-3795(93)90331-h
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发表时间：
1993-01-15
影响因子：
1.1
通讯作者：
ZBAGANU, G
中科院分区：
文献类型：
--
作者：
COHEN, JE;IWASA, Y;ZBAGANU, G
两个有限离散概率分布x=(x1，...，x(N))和y=(y1，...，y(N))的相对g-熵定义为H(G)(x，y)-Sigma(K)x(K)g(y(K)/x(K)-1)，其中g：(-1，无穷)-gt；R是凸的，g(0)=0。当g(T)=-log(1+t)时，则H(G)(x，y)=Sigma(K)x(K)log(x(K)/y(K))，即通常的相对熵。设P(N)={x是R(N)的一个元素：Sigma(I)x(I)=1，x(I)>；0 for-all i)。我们的主要结果是，对于任意m×n列随机矩阵A，定义为ETA(G)(A)=sup{H(G)(Ax，Ay)/H(G)(x，y)：x，y是P(N)的一个元素，x不等于y)满足ETA(G)(A)小于或等于1-α(A)，其中α(A)=min(j，k)Sigma(I)min(a(Ij)，A(Ik))是Dobrushin的遍历系数。因此，ETA(G)(A)<；1当且仅当A是加扰的。建立了ETA(G)(A)的上下界。类似的结果也适用于连续时间的马氏链。
The relative g-entropy of two finite, discrete probability distributions x = (x1, ..., x(n)) and y = (y1, ..., y(n)) is defined as H(g)(x, y) - SIGMA(k)x(k)g(y(k)/x(k) - 1), where g : (- 1, infinity) --> R is convex and g(0) = 0. When g(t) = - log(1 + t), then H(g)(x, y) = SIGMA(k)x(k) log(x(k)/y(k)), the usual relative entropy. Let P(n) = {x is-an-element-of R(n) : SIGMA(i)x(i) = 1, x(i) > 0 for-all i). Our major result is that, for any m X n column-stochastic matrix A, the contraction coefficient defined as eta(g)(A) = sup{H(g)(Ax, Ay)/H(g)(x, y): x, y is-an-element-of P(n), x not-equal y) satisfies eta(g)(A) less-than-or-equal-to 1 - alpha(A), where alpha(A) = min(j, k) SIGMA(i) min(a(ij), a(ik)) is Dobrushin's coefficient of ergodicity. Consequently, eta(g)(A) < 1 if and only if A is scrambling. Upper and lower bounds on eta(g)(A) are established. Analogous results hold for Markov chains in continuous time.