Qualitative Properties of Randomized Maximum Entropy Estimates of Probability Density Functions

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.

Preface to the Special Issue on “Control, Optimization, and Mathematical Modeling of Complex Systems”

“…Popkov [15] has formulated the problem of randomized maximum entropy estimation for the probability density function of random model parameters with real data and measurement noises. This estimation procedure maximizes an information entropy functional on a set of integral equalities depending on the real dataset.…”

“…Numerical simulation in physical, social, and life sciences [1][2][3][4]; • Modeling and analysis of complex systems based on mathematical methods and AI/ML approaches [5,6]; • Control problems in robotics [3,[7][8][9][10][11][12]]; • Design optimization of complex systems [13]; • Modeling in economics and social sciences [4,14]; • Stochastic models in physics and engineering [1,[15][16][17][18]; • Mathematical models in material science [19]; • High-performance computing for mathematical modeling [20].…”

Nearest neighbor estimates of Kaniadakis entropy

“…Wang and Gui [28] used the maximum likelihood and Bayesian methods to obtain the estimators of the entropy for a two-parameter Burr type XII distribution under progressive type-II censored data. Popkov [15] approached the problem of randomized maximum entropy estimation for the probability density function of random model parameters with real data and measurement noises.This estimation procedure maximizes an information entropy functional, taken from a set of integral equalities. The technique of the Gteaux derivatives was developed to solve the problem under an analytical form.…”