Information geometry approach to parameter estimation in Markov chains

Abstract

Abstract:We consider the parameter estimation of Markov chain when the unknown transition matrix belongs to an exponential family of transition matrices. Then, we show that the sample mean of the generator of the exponential family is an asymptotically efficient estimator. Further, we also define a curved exponential family of transition matrices. Using a transition matrix version of the Pythagorean theorem, we give an asymptotically efficient estimator for a curved exponential family. Primary 62M05.

Uniform Random Number Generation From Markov Chains: Non-Asymptotic and Asymptotic Analyses

“…Under more restrictive assumption, i.e., Assumption 2, we also introduce the upper conditional Rényi entropy for a transition matrix (cf. (34)). Then, we evaluate the upper Rényi entropy for the Markov chain in terms of its transition matrix counterpart.…”

“…We also derive converse bounds for every problem by using the change of measure argument developed by the authors in the accompanying paper on information geometry [34], [35]. When there is no information leakage, the converse bounds are described by the Rényi entropy for transition matrices.…”

“…The purpose of this section is to introduce transition matrix counterparts of those measures in Section II-A. For this purpose, we first need to introduce some assumptions on transition matrices: Assumption 1 (Non-Hidden [30], [34], [35]) We say that a transition matrix W is non-hidden (with respect to Y) if…”

“…In converse proofs, we use some techniques to bound tail probabilities in [34], [35]. For this purpose, we need to translate some terminologies in statistics into terminologies in information theory.…”

“…For this purpose, we need to translate some terminologies in statistics into terminologies in information theory. In this appendix, we introduce some terminologies and bounds from [34], [35]. For proofs, see [34], [35].…”