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arXiv:physics/0211109

Statistical Inverse Problem

Yu. I. Bogdanov

Published Nov 27, 2002• 6 citations
arXivHTMLPDF

Abstract

A fundamental problem of statistical data analysis, distribution density estimation by experimental data, is considered. A new method with optimal asymptotic behavior, the root density estimator, is developed. The method proposed may be applied to its full extent to solve the statistical inverse problem of quantum mechanics, namely, estimating the psi function on the basis of the results of mutually complementing experiments.

DOI
10.48550/arxiv.physics/0211109

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A fundamental problem of statistical data analysis, distribution density estimation by experimental data, is considered. A new method with optimal asymptotic behavior, the root density estimator, is developed. The method proposed may be applied to its full extent to solve the statistical inverse problem of quantum mechanics, namely, estimating the psi function on the basis of the results of mutually complementing experiments.
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Multiparametric statistical model providing stable reconstruction of parameters by observations is considered. The only general method of this kind is the root model based on the representation of the probability density as a squared absolute value of a certain function, which is referred to as a psi-function in analogy with quantum mechanics. The psi-function is represented by an expansion in terms of an orthonormal set of functions. It is shown that the introduction of the psi-function allows one to represent the Fisher information matrix as well as statistical properties of the estimator of the state vector (state estimator) in simple analytical forms. A new statistical characteristic, a confidence cone, is introduced instead of a standard confidence interval. The chi-square test is considered to test the hypotheses that the estimated vector converges to the state vector of a general population and that both samples are homogeneous. The expansion coefficients are estimated by the maximum likelihood method. The method proposed may be applied to its full extent to solve the statistical inverse problem of quantum mechanics (root estimator of quantum states). In order to provide statistical completeness of the analysis, it is necessary to perform measurements in mutually complementing experiments (according to the Bohr terminology). The maximum likelihood technique and likelihood equation are generalized in order to analyze quantum mechanical experiments. It is shown that the requirement for the expansion to be of a root kind can be considered as a quantization condition making it possible to choose systems described by quantum mechanics from all statistical models consistent, on average, with the laws of classical mechanics.
A new method of solving classical and quantum problems of statistical data analysis based on the symbiosis of notions of quantum theory and mathematical statistics is considered. Particular attention is given to the specificity of quantum problems, determined by mutually complementary measurements (according to the Bohr complementarity principle), when, for example, a spatial-temporal picture is complemented by a momentum-energy one. The possibility of construction of multiparametric statistical models admitting a stable reconstruction of the parameters from observations (the inverse statistical problem) is studied. In this case, the only universal model of such a kind is the root model, based on the representation of the probability density as the square of the modulus of some function (called the psi function by analogy with quantum mechanics). The psi function is represented as an expansion in terms of an orthonormal basis, with the expansion coefficients being estimated by the maximum likelihood technique. The root approach makes it possible to represent the Fisher information matrix, covariance matrix, and statistical properties of the estimates of the reconstructed states in the simplest and a universal form. Being asymptotically efficient, the method allows one to reconstruct the states with an accuracy close to the theoretically attainable accuracy. It is shown that the requirement for the expansion to be of a root kind can be considered as a quantization condition, which makes it possible to choose, from among all the statistical models, which, on the average, are consistent with the laws of classical mechanics, those systems that are described by quantum mechanics.
The root estimator of quantum states based on the expansion of the psi function in terms of system eigenfunctions followed by estimating the expansion coefficients by the maximum likelihood method is considered. In order to provide statistical completeness of the analysis, it is necessary to perform measurements in mutually complementing experiments (according to the Bohr terminology). Estimation of quantum states by the results of coordinate, momentum, and polarization (spin) measurements is considered.