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Published online by Cambridge University Press: 28 February 2022
In the development of the theory of subjective probability and in discussions about it an important role has been played by de Finetti's representation theorem and by a number of related results presented in de Finetti's classic monograph. This theorem, together with the notion of exchangeability (among its aliases there are the terms equivalence, permutability, and symmetry), was originally put forward by Bruno de Finetti as a solution to a problem to which his subjectivistic interpretation of probability had led him. Although this theorem is mentioned in many expositions of the theory of subjective probability, there are not many satisfactory discussions of its significance available to philosophers. (The best one, and almost the only one, is Braithwaite's paper ‘On Unknown Probabilities’.)
1 Finetti, Bruno de, ‘La prévision: ses lois logiques, ses sources subjectives’, in Annales de l'institut Henri Poincaré, Vol. 7 (1937)Google Scholar. Translated by Kyburg, Henry E. under the title ‘Foresight: Its Logical Laws, Its Subjective Sources’, in Studies in Subjective Probability (ed. by Kyburg, H. E. and Smokier, H. E.), John Wiley & Sons, New York, 1964, pp. 93–158Google Scholar.
2 Braithwaite, R. B., ‘On Unknown Probabilities’, in Observation and Interpretation: A Symposium of Philosophers and Physicists (ed. by Körner, S.), Butterworth, London, 1957, pp. 3–11Google Scholar.
3 As some philosophers have urged - see e.g. Davidson, Donald, ‘The Logical Form of Action Sentences’, in The Logic of Action and Preference (ed. by Rescher, Nicholas), Pittsburgh University Press, Pittsburgh, 1967Google Scholar.
4 In other words, for any e of the form
where aj, ak are different from all the al's.
5 For instance, he speaks of “the nebulous and unsatisfactory definition of ‘independent events with fixed but unknown probability’” (Kyburg and Smokler, p. 142).
6 In his article, ‘Probability: Philosophy and Interpretation’, in the International Encyclopedia of Social Sciences, de Finetti says that for a subjectivist “’independent’ means ‘devoid of influence on my opinion’”. Is this not a fairly clear and not at all nebulous notion?
7 In de Finetti's own exposition (see Kyburg and Smolder, pp. 141-142), we find a contrast between biased coin and random drawing of balls from an urn of an unknown composition. What makes the difference between the two cases seems to be that the frequency of different kinds of balls in a (finite?) urn is “an objective fact which can be directly verified”.
8 As de Finetti himself expresses the point (Kyburg and Smokler, p. 142): “A rich enough experience leads us always to consider as probable future frequencies or distributions close to those which have been observed”.
9 Some caution is needed here, however. Although the notion of exchangeability together with the limit theorems that can be established in terms of it, throw some extremely interesting lights on the idea of learning from experience, the precise limits of the set of those probability distributions which can be said to represent (rational) learning from experience remains somewhat unclear.
10 Notice especially how the switch from independence to exchangeability makes a Bayesian treatment possible here. This point is not always appreciated. In his paper, ‘A New Approach to a Classical Statistical Decision Problem’, in Induction: Some Current Issues (ed. by Kyburg, H. E. and Nagel, E.), Wesleyan University Press, Middletown, 1961, pp. 101-10Google Scholar; Herbert Robbins has discussed situations closely similar to de Finetti's examples. He compares his own recommendations with two others which he labels the Bayesian decision functions (p. 108) and the minimax decision functions (p. 107). I shall not discuss the first or the third here. As to a Bayesian treatment, Robbins restricts it to a case in which the random variables analogous to our properties Ct are independent. This, however, represents precisely the kind of refusal to learn from experience which de Finetti has rightly rejected and which to me seems antithetical to the spirit of Bayesianism.
11 In his interesting book, Logic of Statistical Inference, Cambridge University Press, Cambridge 1965, pp. 215-16, Ian Hacking denies that it makes sense to bet on hypotheses like general laws. He appeals in so many words to de Finetti for support. However, it seems fairer to say that de Finetti's very results show precisely what it means for a subjectivist to bet on generalizations. The admissibility of generalizations to one's probabilistic discussion is shown by de Finetti's result to be a problem independent of whether one uses “betting rates as the central tool in subjective statistics”.
12 Gaifman, H., ‘Concerning Measures on First-Order Calculi’, Israel Journal of Mathematics 2 (1964) 1-18CrossRefGoogle Scholar.
13 In working on this paper, I have greatly profited from discussions with Dr. William K. Goosens. Among many other things, footnote 10 above is essentially due to him.