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Qualitative Properties of Randomized Maximum Entropy Estimates of Probability Density Functions

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  • Yuri S. Popkov

    (Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, 119333 Moscow, Russia
    Institute of Control Sciences of Russian Academy of Sciences, 117997 Moscow, Russia
    Department of Software Engineering, ORT Braude College, 2161002 Karmiel, Israel)

Abstract

The problem of randomized maximum entropy estimation for the probability density function of random model parameters with real data and measurement noises was formulated. This estimation procedure maximizes an information entropy functional on a set of integral equalities depending on the real data set. The technique of the Gâteaux derivatives is developed to solve this problem in analytical form. The probability density function estimates depend on Lagrange multipliers, which are obtained by balancing the model’s output with real data. A global theorem for the implicit dependence of these Lagrange multipliers on the data sample’s length is established using the rotation of homotopic vector fields. A theorem for the asymptotic efficiency of randomized maximum entropy estimate in terms of stationary Lagrange multipliers is formulated and proved. The proposed method is illustrated on the problem of forecasting of the evolution of the thermokarst lake area in Western Siberia.

Suggested Citation

  • Yuri S. Popkov, 2021. "Qualitative Properties of Randomized Maximum Entropy Estimates of Probability Density Functions," Mathematics, MDPI, vol. 9(5), pages 1-13, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:548-:d:510995
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    References listed on IDEAS

    as
    1. Loubes, Jean-Michel & Rochet, Paul, 2012. "Approximate maximum entropy on the mean for instrumental variable regression," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 972-978.
    2. Golan, Amos, 2008. "Information and Entropy Econometrics — A Review and Synthesis," Foundations and Trends(R) in Econometrics, now publishers, vol. 2(1–2), pages 1-145, February.
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    Cited by:

    1. Mikhail Posypkin & Andrey Gorshenin & Vladimir Titarev, 2022. "Preface to the Special Issue on “Control, Optimization, and Mathematical Modeling of Complex Systems”," Mathematics, MDPI, vol. 10(13), pages 1-8, June.

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