An Essay on the Double Nature of the Probability

ABSTRACT

Classical statistics and Bayesian statistics refer to the frequentist and subjective theories of probability respectively. Von Mises and De Finetti, who authored those conceptualizations, provide interpretations of the probability that appear incompatible. This discrepancy raises ample debates and the foundations of the probability calculus emerge as a tricky, open issue so far. Instead of developing philosophical discussion, this research resorts to analytical and mathematical methods. We present two theorems that sustain the validity of both the frequentist and the subjective views on the probability. Secondly we show how the double facets of the probability turn out to be consistent within the present logical frame.

Classical statistics and Bayesian statistics refer to the frequentist and subjective theories of probability respectively. Von Mises and De Finetti, who authored those conceptualizations, provide interpretations of the probability that appear incompatible. This discrepancy raises ample debates and the foundations of the probability calculus emerge as a tricky, open issue so far. Instead of developing philosophical discussion, this research resorts to analytical and mathematical methods. We present two theorems that sustain the validity of both the frequentist and the subjective views on the probability. Secondly we show how the double facets of the probability turn out to be consistent within the present logical frame.

KEYWORDS

Theory of Probability; Subjectivism; Frequentism; Theorem of Large Numbers; Probability Dualism

Theory of Probability; Subjectivism; Frequentism; Theorem of Large Numbers; Probability Dualism

Cite this paper

P. Rocchi and L. Gianfagna, "An Essay on the Double Nature of the Probability,"*Advances in Pure Mathematics*, Vol. 2 No. 5, 2012, pp. 305-309. doi: 10.4236/apm.2012.25041.

P. Rocchi and L. Gianfagna, "An Essay on the Double Nature of the Probability,"

References

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[6] K. R. Popper, “The Logic of Scientific Discovery,” Routledge, New York, 2002.

[7] T. Tao, “The Strong Law of Large Numbers,” 2012.http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/

[8] P. Rocchi and L. Gianfagna, “Probabilistic Events and Physical Reality: A Complete Algebra of Probability,” Physics Essays, Vol. 15, No. 3, 2002, pp. 331-118. doi.org/10.4006/1.3025535

[9] P. Rocchi, “The Structural Theory of Probability,” Kluwer/Plenum, New York, 2003.

[10] B. Gower, “Scientific Method: A Historical and Philosophical Introduction,” Taylor & Francis, London, 2007.

[11] L. J. Savage, “The Foundations of Statistics,” Courier Dover Publications, New York, 1972.

[12] J. Earman, “Aspects of Determinism in Modern Physics,” In: J. Butterfield and J. Earman, Eds., Philosophy of Physics, Part B, North Holland, 2007, pp. 1369-1434. doi.org/10.1016/B978-044451560-5/50017-8

[13] B. De Finetti, “Theory of Probability: A Critical Introductory Treatment,” John Wiley & Sons, New York, 1975.

[1] D. Gillies, “Philosophical Theories of Probability,” Routledge, London, 2000.

[2] D. G. Mayo, “Error and the Growth of Experimental Knowledge,” University of Chicago Press, Chicago, 1996.

[3] M. R. Forster, “How Do Simple Rules ‘Fit to Reality’ in a Complex World?” Minds and Machines, Vol. 9, 1999, pp. 543-564. doi:10.1023/A:1008304819398

[4] M. Denker, W. A. Woyczyński and B. Ycar, “Introductory Statistics and Random Phenomena: Uncertainty, Complexity and Chaotic Behavior in Engineering and Science,” Springer-Verlag Gmbh, Boston, 1998.

[5] G. D’Agostini, “Role and Meaning of Subjective Probability; Some Comments on Common Misconceptions,” 20th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, AIP Conference Proceedings, Vol. 568, 2001, 23 p. doi.org/10.1063/1.1381867

[6] K. R. Popper, “The Logic of Scientific Discovery,” Routledge, New York, 2002.

[7] T. Tao, “The Strong Law of Large Numbers,” 2012.http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/

[8] P. Rocchi and L. Gianfagna, “Probabilistic Events and Physical Reality: A Complete Algebra of Probability,” Physics Essays, Vol. 15, No. 3, 2002, pp. 331-118. doi.org/10.4006/1.3025535

[9] P. Rocchi, “The Structural Theory of Probability,” Kluwer/Plenum, New York, 2003.

[10] B. Gower, “Scientific Method: A Historical and Philosophical Introduction,” Taylor & Francis, London, 2007.

[11] L. J. Savage, “The Foundations of Statistics,” Courier Dover Publications, New York, 1972.

[12] J. Earman, “Aspects of Determinism in Modern Physics,” In: J. Butterfield and J. Earman, Eds., Philosophy of Physics, Part B, North Holland, 2007, pp. 1369-1434. doi.org/10.1016/B978-044451560-5/50017-8

[13] B. De Finetti, “Theory of Probability: A Critical Introductory Treatment,” John Wiley & Sons, New York, 1975.