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The Inference Fallacy From Bernoulli to Kolmogorov

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  • Xavier De Scheemaekere
  • Ariane Szafarz

Abstract

Bernoulli’s (1713) well-known Law of Large Numbers (LLN) establishes a legitimate one-way transition from mathematical probability to observed frequency. However, Bernoulli went one step further and abusively introduced the inverse proposition. Based on a careful analysis of Bernoulli’s original proof, this paper identifies this appealing, but illegitimate, inversion of LLN as a strong driver of confusion among probabilists. Indeed, during more than two centuries this “inference fallacy” hampered the emergence of rigorous mathematical foundations for the theory of probability. In particular, the confusion pertaining to the status of statistical inference was detrimental to both Laplace’s approach based on “equipossibility” and Mises’ approach based on “collectives”. Only Kolmogorov’s (1933) axiomatization made it possible to adequately frame statistical inference within probability theory. This paper argues that a key factor in Kolmogorov’s success has been his ability to overcome the inference fallacy.

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Bibliographic Info

Paper provided by ULB -- Universite Libre de Bruxelles in its series Working Papers CEB with number 11-006.

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Length: 25 p.
Date of creation: Feb 2011
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Publication status: Published by:
Handle: RePEc:sol:wpaper:2013/77259

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Keywords: Probability; Bernoulli; Kolmogorov; Statistics; Law of Large Numbers;

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  1. Xavier De Scheemaekere & Ariane Szafarz, 2009. "The Special Status of Mathematical Probability: A Historical Sketch," ULB Institutional Repository 2013/95543, ULB -- Universite Libre de Bruxelles.
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