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Informationtheoretic asymptotics of Bayes methods
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1990
"... In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance D,, between the true density and the Bayesian density and sh ..."
Abstract

Cited by 124 (12 self)
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In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance D,, between the true density and the Bayesian density and show that the asymptotic distance is (d/2Xlogn)+ c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate D,,/n converges to zero at rate (logn)/n. The constant c, which we explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estimation, universal data compression, composite hypothesis testing, and stockmarket portfolio selection.
Statistics
"... The relative gentropy of two finite, discrete probability distributions x = (Xi,., x,) and y = (yi, , y,,) is defined as H,(x, y) = Ckxkg(yk/xk I), where g: ( 1, M)a ‘8 is convex and g(O) = 0. When g(t) =1ogfl + t), then H,(x, y) = Ckxk log(xk/yk), the usual relative entropy. Let P, = {x,5 ..."
Abstract

Cited by 1 (0 self)
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The relative gentropy of two finite, discrete probability distributions x = (Xi,., x,) and y = (yi, , y,,) is defined as H,(x, y) = Ckxkg(yk/xk I), where g: ( 1, M)a ‘8 is convex and g(O) = 0. When g(t) =1ogfl + t), then H,(x, y) = Ckxk log(xk/yk), the usual relative entropy. Let P, = {x,5!R ” : Cixi = 1, xi> 0 Vi}. Our major result is that, for any m X n columnstochastic ‘matrix A, the contraction coefficient defined as ng ( A) = sup(H,, ( Ax, Ay)/H (x, y): x, y E P,,, x f y} satisfies ng ( A) = $ 1 cr ( A), where a ( A) = min,, k X3, min?aij, a,,) is Dobrushin’s coefficient of ergodicity. Consequently, ng ( A) < 1 if and only if A is scrambling. Upper and lower bounds onng ( A) are established. Analogous results hold for Markov chains in continuous time. 1.