Probability interpretations

The word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly one believes it will occur, or does it draw on both these elements? In answering such questions, mathematicians interpret the probability values of probability theory.

There are two broad categories[1][2] of probability interpretations which can be called "physical" and "evidential" probabilities. Physical probabilities, which are also called objective or frequency probabilities, are associated with random physical systems such as roulette wheels, rolling dice and radioactive atoms. In such systems, a given type of event (such as a die yielding a six) tends to occur at a persistent rate, or "relative frequency", in a long run of trials. Physical probabilities either explain, or are invoked to explain, these stable frequencies. The two main kinds of theory of physical probability are frequentist accounts (such as those of Venn,[3] Reichenbach[4] and von Mises)[5] and propensity accounts (such as those of Popper, Miller, Giere and Fetzer).[6]

Evidential probability, also called Bayesian probability, can be assigned to any statement whatsoever, even when no random process is involved, as a way to represent its subjective plausibility, or the degree to which the statement is supported by the available evidence. On most accounts, evidential probabilities are considered to be degrees of belief, defined in terms of dispositions to gamble at certain odds. The four main evidential interpretations are the classical (e.g. Laplace's)[7] interpretation, the subjective interpretation (de Finetti[8] and Savage),[9] the epistemic or inductive interpretation (Ramsey,[10] Cox)[11] and the logical interpretation (Keynes[12] and Carnap).[13] There are also evidential interpretations of probability covering groups, which are often labelled as 'intersubjective' (proposed by Gillies[14] and Rowbottom).[6]

Some interpretations of probability are associated with approaches to statistical inference, including theories of estimation and hypothesis testing. The physical interpretation, for example, is taken by followers of "frequentist" statistical methods, such as Ronald Fisher[dubious – discuss], Jerzy Neyman and Egon Pearson. Statisticians of the opposing Bayesian school typically accept the frequency interpretation when it makes sense (although not as a definition), but there is less agreement regarding physical probabilities. Bayesians consider the calculation of evidential probabilities to be both valid and necessary in statistics. This article, however, focuses on the interpretations of probability rather than theories of statistical inference.

The terminology of this topic is rather confusing, in part because probabilities are studied within a variety of academic fields. The word "frequentist" is especially tricky. To philosophers it refers to a particular theory of physical probability, one that has more or less been abandoned. To scientists, on the other hand, "frequentist probability" is just another name for physical (or objective) probability. Those who promote Bayesian inference view "frequentist statistics" as an approach to statistical inference that is based on the frequency interpretation of probability, usually relying on the law of large numbers and characterized by what is called 'Null Hypothesis Significance Testing' (NHST). Also the word "objective", as applied to probability, sometimes means exactly what "physical" means here, but is also used of evidential probabilities that are fixed by rational constraints, such as logical and epistemic probabilities.

It is unanimously agreed that statistics depends somehow on probability. But, as to what probability is and how it is connected with statistics, there has seldom been such complete disagreement and breakdown of communication since the Tower of Babel. Doubtless, much of the disagreement is merely terminological and would disappear under sufficiently sharp analysis.

— Savage, 1954, p. 2[9]
  1. ^ Hájek, Alan (21 October 2002), Zalta, Edward N. (ed.), Interpretations of Probability, The Stanford Encyclopedia of Philosophy The taxonomy of probability interpretations given here is similar to that of the longer and more complete Interpretations of Probability article in the online Stanford Encyclopedia of Philosophy. References to that article include a parenthetic section number where appropriate. A partial outline of that article:
    • Section 2: Criteria of adequacy for the interpretations of probability
    • Section 3:
      • 3.1 Classical Probability
      • 3.2 Logical Probability
      • 3.3 Subjective Probability
      • 3.4 Frequency Interpretations
      • 3.5 Propensity Interpretations
  2. ^ de Elía, Ramón; Laprise, René (2005). "Diversity in interpretations of probability: implications for weather forecasting". Monthly Weather Review. 133 (5): 1129–1143. Bibcode:2005MWRv..133.1129D. doi:10.1175/mwr2913.1. S2CID 123135127. "There are several schools of thought regarding the interpretation of probabilities, none of them without flaws, internal contradictions, or paradoxes." (p 1129) "There are no standard classifications of probability interpretations, and even the more popular ones may suffer subtle variations from text to text." (p 1130) The classification in this article is representative, as are the authors and ideas claimed for each classification.
  3. ^ Venn, John (1876). The Logic of Chance. London: MacMillan.
  4. ^ Reichenbach, Hans (1948). The theory of probability, an inquiry into the logical and mathematical foundations of the calculus of probability. University of California Press. English translation of the original 1935 German. ASIN: B000R0D5MS
  5. ^ Mises, Richard (1981). Probability, statistics, and truth. New York: Dover Publications. ISBN 978-0-486-24214-9. English translation of the third German edition of 1951 which was published 30 years after the first German edition.
  6. ^ a b Rowbottom, Darrell (2015). Probability. Cambridge: Polity. ISBN 978-0745652573.
  7. ^ Cite error: The named reference LaPlace was invoked but never defined (see the help page).
  8. ^ de Finetti, Bruno (1964). "Foresight: its Logical laws, its Subjective Sources". In Kyburg, H. E. (ed.). Studies in Subjective Probability. H. E. Smokler. New York: Wiley. pp. 93–158. Translation of the 1937 French original with later notes added.
  9. ^ a b Savage, L.J. (1954). The foundations of statistics. New York: John Wiley & Sons, Inc. ISBN 978-0-486-62349-8.
  10. ^ Ramsey, F. P. (1931). "Chapter VII, Truth and Probability (1926)" (PDF). In Braithwaite, R. B. (ed.). Foundations of Mathematics and Other Logical Essays. London: Kegan, Paul, Trench, Trubner & Co. pp. 156–198. Retrieved 15 August 2013. Contains three chapters (essays) by Ramsey. The electronic version contains only those three.
  11. ^ Cox, Richard Threlkeld (1961). The algebra of probable inference. Baltimore: Johns Hopkins Press.
  12. ^ Keynes, John Maynard (1921). A Treatise on Probability. MacMillan. Retrieved 15 August 2013.
  13. ^ Carnap, Rudolph (1950). Logical Foundations of Probability. Chicago: University of Chicago Press. Carnap coined the notion "probability1" and "probability2" for evidential and physical probability, respectively.
  14. ^ Gillies, Donald (2000). Philosophical theories of probability. London New York: Routledge. ISBN 978-0415182768.