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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.

Suggested Citation

  • Xavier De Scheemaekere & Ariane Szafarz, 2011. "The Inference Fallacy From Bernoulli to Kolmogorov," Working Papers CEB 11-006, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:sol:wpaper:2013/77259
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    1. Xavier De Scheemaekere & Ariane Szafarz, 2008. "The special status of mathematical probability: a historical sketch," Working Papers CEB 08-017.RS, ULB -- Universite Libre de Bruxelles.
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    Keywords

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

    JEL classification:

    • N01 - Economic History - - General - - - Development of the Discipline: Historiographical; Sources and Methods
    • B31 - Schools of Economic Thought and Methodology - - History of Economic Thought: Individuals - - - Individuals
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools

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