Non-Algorithmic Theory of Randomness

This paper proposes an alternative language for expressing results of the algorithmic theory of randomness. The language is more precise in that it does not involve unspecified additive or multiplicative constants, making mathematical results, in principle, applicable in practice. Our main testing ground for the proposed language is the problem of defining Bernoulli sequences, which was of great interest to Andrei Kolmogorov and his students.

[1]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[2]  J. Lukasiewicz.  Logical foundations of probability theory , 1970 .

[3]  Andrei N. Kolmogorov,et al.  Logical basis for information theory and probability theory , 1968, IEEE Trans. Inf. Theory.

[4]  Vladimir Vovk,et al.  Learning about the Parameter of the Bernoulli Model , 1997, J. Comput. Syst. Sci..

[5]  Michel Souslin.  Sur une definition des ensembles mesurables B sans nombres transfinis , 1917 .

[6]  V. Vovk,et al.  On the Empirical Validity of the Bayesian Method , 1993 .

[7]  HENRY STEINITZ,et al.  KOLMOGOROV COMPLEXITY AND ALGORITHMIC RANDOMNESS , 2013 .

[8]  Glenn Shafer,et al.  On the Nineteenth-Century Origins of Significance Testing and P-Hacking , 2019, SSRN Electronic Journal.

[9]  Péter Gács,et al.  Uniform test of algorithmic randomness over a general space , 2003, Theor. Comput. Sci..

[10]  A. N. Kolmogorov.  Combinatorial foundations of information theory and the calculus of probabilities , 1983 .

[11]  F. Y. Edgeworth,et al.  The theory of statistics , 1996 .

[12]  Vladimir Vovk,et al.  Game‐Theoretic Foundations for Probability and Finance , 2019, Wiley Series in Probability and Statistics.

[13]  G. Shafer,et al.  Test Martingales, Bayes Factors and p-Values , 2009, 0912.4269.

[14]  V. Vovk.  On the concept of the Bernoulli property , 1986 .

[15]  Ilia Nouretdinov,et al.  Transductive Confidence Machine Is Universal , 2003, ALT.

[16]  Vladimir Vovk,et al.  On the concept of Bernoulliness , 2016, 1612.08859.

[17]  A. Kolmogorov.  On Logical Foundations of Probability Theory , 1983 .

[18]  Nicolas Bourbaki,et al.  Elements of mathematics , 2004 .

[19]  Wouter M. Koolen,et al.  Safe Testing , 2019, 2020 Information Theory and Applications Workshop (ITA).

[20]  M. Schervish.  Theory of Statistics , 1995 .

[21]  Vladimir Vovk,et al.  Test statistics and p-values , 2017, COPA.

[22]  G. Shafer.  The Language of Betting as a Strategy for Statistical and Scientific Communication , 2019, 1903.06991.

[23]  Alexander Gammerman,et al.  Machine-Learning Applications of Algorithmic Randomness , 1999, ICML.

[24]  Sander Greenland,et al.  Valid P-Values Behave Exactly as They Should: Some Misleading Criticisms of P-Values and Their Resolution With S-Values , 2019, The American Statistician.

[25]  Vladimir Vovk,et al.  Kolmogorov's Contributions to the Foundations of Probability , 2003, Probl. Inf. Transm..

[26]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 1997, Texts in Computer Science.

[27]  Vladimir V. V'yugin,et al.  Algorithmic Complexity and Stochastic Properties of Finite Binary Sequences , 1999, Comput. J..

[28]  S. Lauritzen.  Extremal Families and Systems of Sufficient Statistics , 1988 .