Different Closed-Form Expressions for Generalized Entropy Rates of Markov Chains

Girardin, Valérie; Lhote, Loïck; Regnault, Philippe

doi:10.1007/s11009-018-9679-3

Cited by 7 publications

(3 citation statements)

References 31 publications

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“…Pardo [22], Vajda & Vasek [38], as well as e.g. Chen et al [265], Girardin & Lhote [266], Ren et al [267], Girardin et al [268] for exemplary applications). (D6) Burbea-Rao divergences [14] (see also e.g.…”

Section: A Further Divergences and Friendsmentioning

confidence: 99%

A Precise Bare Simulation Approach to the Minimization of Some Distances. I. Foundations

Broniatowski¹,

Stummer

2023

IEEE Trans. Inform. Theory

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The constrained minimization (respectively maximization) of directed distances and of related generalized entropies is a fundamental task in information theory as well as in the adjacent fields of statistics, machine learning, artificial intelligence, signal processing and pattern recognition. In our previous paper "A precise bare simulation approach to the minimization of some distances. I. Foundations", we obtained such kind of constrained optima by a new dimension-free precise bare (pure) simulation method, provided basically that (i) the underlying directed distance is of f −divergence type, and that (ii) this can be connected to a light-tailed probability distribution in a certain manner. In the present paper, we extend this approach such that constrained optimization problems of a very huge amount of directed distances and generalized entropies -and beyond -can be tackled by a newly developed dimension-free extended bare simulation method, for obtaining both optima as well as optimizers. Almost no assumptions (like convexity) on the set of constraints are needed, within our discrete setup of arbitrary dimension, and our method is precise (i.e., converges in the limit). For instance, we cover constrained optimizations of arbitrary f −divergences, Bregman distances, scaled Bregman distances and weighted ℓr−distances. The potential for wide-spread applicability is indicated, too; in particular, we deliver many recent references for uses of the involved distances/divergences in various different research fields (which may also serve as an interdisciplinary interface).

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Section: A Further Divergences and Friendsmentioning

confidence: 99%

A Precise Bare Simulation Approach to the Minimization of Some Distances. I. Foundations

Broniatowski¹,

Stummer

2023

IEEE Trans. Inform. Theory

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“…, where π = (π 1 , π 2 , .., π d ) is the initial stationary vector for P. The references [3], [18] and [19] consider statistical tests for Markov Chains. In the two by two case, we get that…”

Section: )mentioning

confidence: 99%

Decision Theory and Large Deviations for Dynamical Hypotheses Tests: Neyman-Pearson, Min-Max and Bayesian Tests

Ferreira¹,

Lopes²,

Lopes³

2021

Preprint

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We analyze hypotheses tests via classical results on large deviations for the case of two different Hölder Gibbs probabilities. The main difference for the the classical hypotheses tests in Decision Theory is that here the two considered measures are singular with respect to each other. We analyze the classical Neyman-Pearson test showing its optimality. This test becomes exponentially better when compared to other alternative tests, with the sample size going to infinity. We also consider both, the Min-Max and a certain type of Bayesian hypotheses tests. We shall consider these tests in the log likelihood framework by using several tools of Thermodynamic Formalism. Versions of the Stein's Lemma and the Chernoff's information are also presented.

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“…The Jacobian is the natural extension of the concept of stochastic matrix (see example 1, in [25] or page 27 in [31]). The references [2], [21] and [20] consider statistical tests for Markov Chains but using different methodologies.…”

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confidence: 99%

Decision Theory and large deviations for dynamical hypotheses tests: The Neyman-Pearson Lemma, Min-Max and Bayesian tests

Ferreira

Lopes

2022

JDG

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<p style='text-indent:20px;'>We analyze hypotheses tests using classical results on large deviations to compare two models, each one described by a different Hölder Gibbs probability measure. One main difference to the classical hypothesis tests in Decision Theory is that here the two measures are singular with respect to each other. Among other objectives, we are interested in the decay rate of the wrong decisions probability, when the sample size <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> goes to infinity. We show a dynamical version of the Neyman-Pearson Lemma displaying the ideal test within a certain class of similar tests. This test becomes exponentially better, compared to other alternative tests, when the sample size goes to infinity. We are able to present the explicit exponential decay rate. We also consider both, the Min-Max and a certain type of Bayesian hypotheses tests. We shall consider these tests in the log likelihood framework by using several tools of Thermodynamic Formalism. Versions of the Stein's Lemma and Chernoff's information are also presented.</p>

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Different Closed-Form Expressions for Generalized Entropy Rates of Markov Chains

Cited by 7 publications

References 31 publications

A Precise Bare Simulation Approach to the Minimization of Some Distances. I. Foundations

A Precise Bare Simulation Approach to the Minimization of Some Distances. I. Foundations

Decision Theory and Large Deviations for Dynamical Hypotheses Tests: Neyman-Pearson, Min-Max and Bayesian Tests

Decision Theory and large deviations for dynamical hypotheses tests: The Neyman-Pearson Lemma, Min-Max and Bayesian tests

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