On the converse theorem in statistical hypothesis testing for Markov chains

Abstract

Discussion on the converse theorem in statistical hypothesis testing. Hypothesis testing for two Markov chains is considered. Under the constraint that the first-kind error probability is less than or equal to exp(-T R), the second-kind error probability is minimized. The geodesic that connects the two Markov chains is defined. By analyzing the geodesic, the power exponents are calculated and then represent in terms of Kullback-Leibler divergence.

Information geometry approach to parameter estimation in Markov chains

“…For this purpose, we notice another type of exponential family of transition matrices by Nakagawa and Kanaya [2] and Nagaoka [5]. They defined the Fisher information matrix in their sense.…”

“…That is, when the unknown distribution belongs to a curved exponential family, the asymptotic efficient estimator can be treated in the informationgeometrical framework. Therefore, to deal with these problems in a wider class of families of transition matrices, we introduce a curved exponential family of transition matrices as a subset of an exponential family of transition matrices in the sense of [2,5]. Since any exponential family of transition matrices is a curved exponential family, the class of curved exponential families is a larger class of families of transition matrices than the class of exponential families.…”

“…In fact, the authors applied it to finite-length evaluations of the tail probability, the error probability in simple hypothesis testing, source coding, channel coding, and random number generation in Markov chain as well as the estimation error of parametric family of transition matrices [17,18]. Thus, we revisit the exponential family of transition matrices [2,5] in a manner consistent with the above purpose by using Bregmann divergence [21,20]. In particular, the relative Rényi entropy for transition matrices plays an important role in the finite-length analysis; we define the relative entropy for transition matrices so that it is a special case of the relative Rényi entropy, which is different from the definitions in the literatures [2,5].…”

“…Since any smooth parametric subfamily of transition matrices on a finite-size system forms a curved exponential family, our treatment for curved exponential families has a wide applicability for the estimation of Markovian process. This is reason why we adopted the definition of an exponential family by [2,5].…”

“…Firstly, we show that, for an exponential family of transition matrices in the sense of [2,5], an estimator of a simple form asymptotically attains the Cramér-Rao bound, which is given as the inverse of Fisher information matrix. That is, the estimator for the expectation parameter is asymptotically efficient and is written as the sample mean of n + 1-observations.…”