Bernstein polynomials and learning theory

When learning processes depend on samples but not on the order of the information in the sample, then the Bernoulli distribution is relevant and Bernstein polynomials enter into the analysis. We derive estimates of the approximation of the entropy function xlogx that are sharper than the bounds from Voronovskaja's theorem. In this way we get the correct asymptotics for the Kullback-Leibler distance for an encoding problem.

[1]  Jürgen Forster.  A linear lower bound on the unbounded error probabilistic communication complexity , 2002, J. Comput. Syst. Sci..

[2]  Martin Aigner,et al.  Diskrete Mathematik , 1993, Vieweg Studium Aufbaukurs Mathematik = Advanced lectures in mathematics.

[3]  V. Totik.  APPROXIMATION BY BERNSTEIN POLYNOMIALS , 1994 .

[4]  Thomas Sauer,et al.  The Genuine Bernstein-Durrmeyer Operator on a Simplex , 1994 .

[5]  Thomas M. Cover.  Admissibility properties or Gilbert's encoding for unknown source probabilities (Corresp.) , 1972, IEEE Trans. Inf. Theory.

[6]  Xinlong Zhou,et al.  The Lower Estimate for Linear Positive Operators (II) , 1994 .

[7]  George G. Lorentz,et al.  Deferred Bernstein polynomials , 1951 .

[8]  T. Cover.  Admissibility Properties of Gilbert ’ s Encoding for Unknown Source Probabilities , 1998 .

[9]  Hans Ulrich Simon,et al.  How to Achieve Minimax Expected Kullback-Leibler Distance from an Unknown Finite Distribution , 2002, ALT.

[10]  George G. Lorentz,et al.  Inverse Theorems for Bernstein Polynomials , 1997 .

[11]  Rafail E. Krichevskiy,et al.  Laplace's Law of Succession and Universal Encoding , 1998, IEEE Trans. Inf. Theory.

[12]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[13]  Yuan Xu,et al.  K-moduli, moduli of smoothness, and Bernstein polynomials on a simplex , 1991 .

[14]  Manfred K. Warmuth,et al.  Relative Expected Instantaneous Loss Bounds , 2000, J. Comput. Syst. Sci..

[15]  Xinlong Zhou,et al.  The lower estimate for linear positive operators, I , 1995 .