Why Probability?

Bernardo and Smith (2000), page 86

"Why is probability a reasonable way of quantifying uncertainty? The following reasons are often advanced.

  1. By analogy: physical randomness induces uncertainty, so it seems reasonable to describe uncertainty in the language of random events. Common speech uses many terms such as ‘probably’ and ‘unlikely,’ and it appears consistent with such usage to extend a more formal probability calculus to problems of scientific inference.
  2. Axiomatic or normative approach: related to decision theory, this approach places all statistical inference in the context of decision-making with gains and losses. Then reasonable axioms (ordering, transitivity, and so on) imply that uncertainty must be represented in terms of probability. We view this normative rationale as suggestive but not compelling.
  3. Coherence of bets. [...]

Gelman, et al. (2004), page 13

Howson and Urbach (1993), chapter 5 in second edition

Abstract: "Arguments are adduced to support the claim that the only satisfactory description of uncertainty is probability. Probability is described both mathematically and interpretatively as a degree of belief. The axiomatic basis and the use of scoring rules in developing coherence are discussed. A challenge is made that anything that can be done by alternative methods for handling uncertainty can be done better by probability. This is demonstrated by some examples using fuzzy logic and belief functions. The paper concludes with a forensic example illustrating the power of probability ideas."
page 17: "Our thesis is simply stated: the only satisfactory description of uncertainty is probability. By this is meant that every uncertainty statement must be in the form of a probability; that several uncertainties must be combined using the rules of probability; and that the calculus of probabilities is adequate to handle all situations involving uncertainty. In particular, alternative descriptions of uncertainty are unnecessary. These include the procedures of classical statistics; rules of combination such as Jeffrey’s (1965); possibility statements in fuzzy logic, Zadeh (1983); use of upper and lower probabilities, Smith (1961), Fine (1973); and belief functions, Shafer (1976). We speak of “the inevitability of probability.”"
Conclusion: "Our argument may be summarized by saying that probability is the only sensible description of uncertainty and is adequate for all problems involving uncertainty. All other methods are inadequate. The justification for the position rests on the formal, axiomatic argument that leads to the inevitability of probability as a theorem and also on the pragmatic verification that probability does work. My challenge that anything that can be done with fuzzy logic, belief functions, upper and lower probabilities, or any other alternative to probability, can better be done with probability, remains."
Lindley (1987)

"The “inevitability of probability” is strengthened by the fact that three distinct axiom systems lead to that result. The first is firmly based on decision theory and is due to Ramsey (1931) and independently to Savage (1954). The second uses scoring rules, originating with de Finetti (1974/75) who also, in common with others, used a method based on betting. A third follows the usual mensuration procedure of comparison with a standard (Pratt, Raiffa and Schlaifer, 1964). There is an admirable survey by Shafer (1986; with discussion) that concentrates on Savage's method."
Lindley (1990), page 51

Paris (1994)

Summary and Conclusion

Various distinct axiom systems developed to deal with uncertainty (the axiomatic approach, Dutch book argument, scoring rules and using a standard) all lead to probability.