STATISTIC STRUCTURE GENERATED BY RANDOMIZE DENSITY

Differential geometry methods of have applications in the information files study (families of probability distributions of spaces of quantum states, neural networks, etc.). Research on geometry information back to the S. Rao that based by Fisher information matrix defined the Riemannian metric of probability distributions manifold. Further investigation led to the concept of statistical manifold. Statistical manifold is a smooth finite-dimensional manifold on which a metrically-affine structure, ie, metric and torsion-free linear connection that is compatible with a given metric; while the condition Codazzi. Geometric manifold and the manifold is given statistical structure tensor. In the present study examines the statistical structure of the generated randomized density of the normal distribution and the Cauchy distribution. The study put the allegation that a randomized probability density of the normal distribution can be regarded as the solution of the Cauchy problem for the heat equation, and randomized probability density of the Cauchy distribution can be considered as a solution to the Dirichlet problem for the Laplace equation. Conversely, the solution of the Cauchy problem for the heat equation can be regarded as a randomized probability density of the normal distribution, and the solution of the Dirichlet problem for the Laplace equation as randomized probability density of the Cauchy distribution. The main objective of the study was the fact that for each of these two cases to find the Fisher information matrix components and structural tensor. We found nonlinear differential equations of the first, second and third order for the density of the normal distribution and Cauchy density computational difficulties to overcome. The components of the metric tensor (the Fisher information matrix) and the components of the strain tensor are calculated according to formulas in which there is the log-likelihood function, ie, logarithm of the density distribution. Because of the positive definiteness of the Fisher information matrix obtained inequality, which obviously satisfy the Cauchy problem solution with nonnegative initial conditions in the case of the Laplace equation and the heat equation.

1. Rao C. R. 2002, "Linear Statistical Inference and its Applications" , (2nd ed. ), Wiley. 656 p.

2. Prokopenko, M., Lizier, Joseph T., Lizier, J. T., Obst, O. & Wang, X. R. 2011, "Relating Fisher information to order parameters" , Physical Review E. vol. 84(4), pp. 1–11.

3. Norden A. P. 1976, Prosttranstva affinnoj svjaznosti [Spaces with affine connection], (2nd ed. ). M, Nauka. 432 p.

4. Lang S. 2002, "Introduction to differentiable manifolds" . (2nd. edition) Springer-Verlag. 250 p.

5. Dodson, C. T. J. & Poston, T. 1991, "Tensor geometry, Graduate Texts in Mathematics"(2nd ed. ), Berlin, New York: Springer-Verlag. 447 p.

6. Dubrovin B. A., Fomenko A. T. & Novikov S. P. 1984, "Modern Geometry. Methods and Applications" , Springer-Verlag.

7. Hazewinkel, Michiel 2001, "Levi-Civita connection" , Encyclopedia of Mathematics, Springer.

8. Sternberg, Shlomo 1983, "Lectures on differential geometry" , New York. Chelsea. 137 p.

9. Rashevskii P. K. 1967, Rimanova geometrija i tenzornyj analiz, [Riemannian geometry and tensor analysis], (3rd Edition). M, Nauka. 664 p.

10. Feller W. 1966, "An introduction to probability theory and its applications" , Wiley., vol. 2, 528 p.

11. Evans, L. C. 1998, "Partial Differential Equations" , American Mathematical Society. 749 p.

12. Tichonov A. N. and Samarskii A. A. 1963, "Equations of mathematical physics" , Pergamon Press. 480 p.

13. Johnson N. L., S. Kotz, and N. Balakrishnan 1994. "Continuous Univariate Distributions" , New York: Wiley., vol. 1, 784 p.

14. Ventsel A. 1996, "Teorija sluchajnyh processov" . [Theory of random processes], М. : Nauka. 400 p.

15. Carslaw, H. S. & Jaeger, J. C. 1959, "Conduction of Heat in Solids"(2nd edition), Oxford University Press. 510 p.

16. Vladimirov V. S. 1971, "Equations of Mathematical Physics" , M. Dekker. 640 p.

17. Polyanin A. D. and Zaitsev V. F. 2003, "Handbook of Exact Solutions for Ordinary Differential Equations" , (2nd edition), Chapman Hall/CRC Press, Boca Raton. 1543 p.

18. Korolyuk V. S. , Portenko N. I. , Skorokhod A. V. & Turbin, A. F. 1985, "Spravochnik po teorii verojatnostej i matematicheskoj statistike" . [Handbook on probability theory and mathematical statistics]. М. : Nauka.

19. Chentsov N. N. 1972, "Statisticheskie reshajushhie pravila i optimal’nye vyvody" . [ Statistical decisive rules and optimum conclusions], M.: Nauka.

20. Amari S. I. 1985, "Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics" , Springer-Verlag, vol. 28, 127 p.

21. McCullagh P. 1987, "Tensor Methods in Statistics" , Chapman Hall/CRC Monographs on Statistics Applied Probability. 280 p.

22. Stepanov S. E., Shandra I. G. & Stepanova E. S. 2007, "Conjugate connectivities on statistical manifolds" , Izv. Mathematics, no. 11, pp. 90–98.

23. Pratt, J. W. 1976, "F. Y. Edgeworth and R. A. Fisher on the Efficiency of Maximum Likelihood Estimation" , The Annals of Statistics vol. 4(3), pp. 501– 514.