Abstract
De Finetti is one of the founding fathers of the subjective school of probability. He held that probabilities are subjective, coherent degrees of expectation, and he argued that none of the objective interpretations of probability make sense. While his theory has been influential in science and philosophy, it has encountered various objections. I argue that these objections overlook central aspects of de Finetti’s philosophy of probability and are largely unfounded. I propose a new interpretation of de Finetti’s theory that highlights these aspects and explains how they are an integral part of de Finetti’s instrumentalist philosophy of probability. I conclude by drawing an analogy between misconceptions about de Finetti’s philosophy of probability and common misconceptions about instrumentalism.
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Notes
A few preliminary comments are in place: 1. As we shall see in Section 4.6, in de Finetti’s theory the notion of expectation is taken to be a ‘primitive’. A degree of expectation about an event (proposition) is the expectation of the binary random-variable that represents the occurrence of that event (truth-value of that proposition), which is not defined as a probability-weighted sum. 2. The distinction between the terms ‘degree of belief’ and ‘degree of expectation’ is interesting and important, but its consideration goes beyond the scope of this paper. In any case, this distinction will not play an essential role in most of the discussion below. 3. In what follows, I shall talk about degrees of expectation about both events and propositions. The analysis of degrees of expectation about events can easily be translated to the analysis of degrees of expectation about propositions and vice versa. In fact, de Finetti frequently used the term ‘event’ to refer to propositions. 4. For ease of presentation, in discussing the literature on de Finetti’s theory, I shall sometimes follow the common terminology and refer to subjective probabilities as coherent degrees of belief or coherent credences.
For a similar criticism, see for example Kyburg (1970, Chap. 6), Psillos (2007, p. 45), Bunge (2012, Chap. 11), Talbot (2016, Section 4.2F), and Williamson (2017, p. 78). Some authors agree with Hájek that Bayesianism of the style of de Finetti only imposes the constraints (i) and (ii), but do not conceive de Finetti’s epistemology as permissive. Rather, in analogy with deductive logic, they take subjective probability a la de Finetti to provide a logic of partial belief (Gillies 1973, Chap. 1; Howson 2000) or a logic of inductive inference (Howson and Urbach 2006, Chap. 9). Although some of de Finetti’s writings may suggest such an interpretation, I believe that it is inadequate. I will propose below that de Finetti conceived his theory of probability as much broader.
A bet is an option to receive (or give) a specified positive amount S (‘the stake’) if an event E occurs, sold (bought) by a bookie for a price Q. The ratio Q/(S − Q) is ‘the odds’, and an agent will accept to buy a bet at these odds only if her degree of belief in E is equal or larger than the betting quotient Q/S. Betting quotients are odds normalized, so that their values lie within the half-open unit interval [0, 1). This interval could be extended to the closed-unit interval [0, 1] by allowing the odds to take the ‘value’ ∞ (Howson 2000, pp. 125–126).
Mura (2009) is a notable exception.
The main focus in de Finetti’s later writings is on the Theory of Probability (Vols. 1 and 2), Probability, Induction and Statistics, and Philosophical Lectures on Probabilities, which constitute his comprehensive attempts to present the main ideas of his theory and philosophy of probability.
For Vailati’s philosophy, see for example Arrighi et al. (2010).
De Finetti (1937/1980, p. 61) takes the idea that betting reveals an agent’s subjective probability to be “trivial and obvious.” Yet, as we shall see below, in his later philosophy he considered this idea less obvious and opted for a different method of measuring probability.
De Finetti was not always careful in articulating his concept of probability and some of his writings could naturally be interpreted as advocating an operational definition of probability in the philosophical sense. For example, in his discussion in the Theory of Probability of two decision-theoretic frameworks for measuring probabilities and explicating coherence, he said that each of these frameworks will consist of “a scheme of decisions to which an individual (it could be You) can subject himself in order to reveal – in an operational manner – that value which, by definition, will be called his prevision of [a random quantity] X, or in particular his probability of [an event] E” (de Finetti 1974a, p. 85).
Quantum mechanics seems to provide a counterexample. For instance, in a measurement of position, a system that is in a superposition of various positions does not have any definite position before the measurement. But calling the operation that transforms a superposition of positions to a definite position a measurement of position is a misnomer. More generally, the so-called ‘measurement’ in quantum mechanics is a misnomer. The term ‘measurement’ in this theory is more akin to a preparation or transformation of a state of a system, or perhaps an operational definition of a quantity.
Gillies (2000, pp. 200–203) seems to advocate such a view in the context of the social sciences.
The argument here is that de Finetti was instrumentalist about probability and probabilistic theories. The question whether he was an instrumentalist about science in general, as the above quotation may suggest, is interesting but goes beyond the scope of this study.
Obviously, such inductive reasoning may be dependent on probabilities. On the proposed interpretation, this is not a problem for de Finetti as he did not attempt to define probability, let alone provide a reductive definition.
The question arises as to how such background knowledge could be integrated into the framework of the mathematical probability theory.
De Finetti also appealed to a weaker symmetry condition, ‘partial exchangeability’, where a series of events is divided into k classes and the probability distribution remains invariant only under permutations within each of these classes. For an analysis of various conceptions of exchangeability and de Finetti’s ‘representation theorem’ for exchangeable random-variables, see for example Diaconis (1977, 1988).
De Finetti’s (1974a, p. 134) presentation of conditional probability is both ambiguous and potentially misleading. “In precise terms, we shall write P(E/H) for the probability ‘of the event E conditional on the event H’ (or even the probability ‘of the conditional event E ∣ H’), which is the probability that You attribute to E if you think that in addition to your present information, i.e. the H0 which we understand implicitly, it will become known to You that H is true (and nothing else)”; where the ‘conditional event’ E ∣ H is a conditional proposition that is true when H and E are true, false when H is true and E is false, and has no truth-value when H is false. (For the sake of simplicity, I assume that H includes the background knowledge H0; nothing essential in what follows will depend on this assumption.) The statement in the first brackets of the above quotation suggests that the conditional probability of E given H is to be understood as the unconditional probability of the conditional event E ∣ H, whereas the last part of the sentence suggests my interpretation above. There are at least two reasons why the interpretation of de Finetti’s concept of conditional probability as unconditional probability of a conditional event cannot be correct. First, as noted above, in de Finetti’s theory all probabilities are conditional on background knowledge and unconditional probabilities do not make sense. But if conditional probability were defined as an unconditional probability of a conditional event, it would fundamentally be an unconditional probability of such event, and accordingly will not be related to background knowledge. Second, the interpretation of conditional probability as unconditional probability of a conditional event is incompatible with de Finetti’s characterization of the measurement of conditional probability in terms of conditional penalty in the Brier scoring-rule decision-theoretic framework. Consider the Brier scoring-rule decision-theoretic framework (see Section 4.1), and suppose an agent with coherent degrees of expectation. In this framework, the decision of such an agent is comprised of the probabilities that she post for various propositions (events) E1, ..., En, and the penalty she is subject to is proportional to the distance (i.e. the square root difference) between the posted probabilities and the values of the binary random-variables that represent whether these propositions (events) are true (occur). It is assumed that penalties for the various probabilities are additive and that the agent is trying to minimize her expected loss. Given these assumptions, the agent’s decision reflects her probabilities. The penalty for the conditional probability of E given H is conditional on H being true (occurring): H(E − P(E))2; where H and E are binary random-variables that represent respectively the truth-values (occurrences) of the propositions (events) E and H. This conditional penalty is compatible with interpreting conditional probability as a conditional with a probabilistic consequent but not with interpreting it as an unconditional probability of a conditional event. If the conditional probability of E given H were defined as the unconditional probability of the conditional event E ∣ H, the penalty would have to be (E| H − P(E|H))2; where E∣H is the random variable that is supposed to represent the truth-value of E∣H. But this penalty is ill defined, as E ∣ H has no truth-value when H is false.
De Finetti (1972, pp. 15–16, 1974a, Section 4.3) demonstrates that a coherence condition on the degrees of expectation about H conditional on H0, E&H conditional on H0 and E conditional on H&H0 implies that, when the degree of expectation about H conditional on H0 is non-zero, the probability of E conditional on H & H0 is equal to the ratio of the probability of E & H conditional on H0 and the probability of H conditional on H0. Thus, if we label the probability of E conditional on H & H0 as the ‘conditional probability of E given H’, the probability of E & H conditional on H0 as the ‘unconditional probability of E & H’ and the probability of H conditional on H0 as the ‘unconditional probability of H’, we obtain the equality (KCP).
Two comments: 1. Although de Finetti didn’t use this notation, it follows naturally from his representation of conditional probability (de Finetti 1974a, pp. 134–139) and his verificationism (de Finetti 1974a, p. 34, 1974b, pp. 266–267 and 302–313). For a detailed discussion of de Finetti’s verificationism and its implications for the logical structure of subjective probability, see Berkovitz (2012). 2. It is important to distinguish between zero loss in the case of degrees of expectation about unverifiable events and zero loss in the case of degrees of expectation about verifiable events. In the first case, the zero loss is independent of the values of degrees of expectation and it reflects the fact that degrees of expectation about unverifiable events are excluded from the accumulated loss because they have no instrumental value. In the second case, zero loss is only possible for extreme degrees of expectation about verifiable events and it represents maximum instrumental value.
Consider, for example, an agent who assigns degrees of expectation α and β to a verifiable event E and its absence ¬E, respectively. Suppose that α and β constitute an incoherent system of degrees of expectation: α + β ≠ 1. Then, the posted degree of expectation about E that minimizes the agent’s expected loss is α/α + β. For her subjective expected loss is Exp(LD) = (α(1 − d)2 + βd2)/k2, where d is the degree of expectation that she posts for E, and Exp(LD) is minimized when d = α/α + β. A similar reasoning demonstrates that the degree of expectation that the agent should post for ¬E is β/α + β. As is not difficult to see, these degrees of expectation are coherent.
The differences between the Brier scoring-rule and de Finetti’s variant of it are immaterial for the discussion in this section.
The distinction between pragmatic and epistemic justifications of the calculus of probability has been made by a number of authors. For some examples, see Skyrms (1984), Armendt (1992, 1993), Christensen (1996, 2004), Joyce (1998, 2009), Huber (2007), Gibbard (2008a, b), Leitgeb and Pettigrew (2010a, b), Weisberg (2015), and Pettigrew (2016).
In de Finetti’s theory, zero probability does not imply impossibility. Accordingly, a degree of expectation zero (one) about a contingent proposition E is compatible with E being true (false).
Joyce (2009, Section 5) proposes that different weights in the average sum that constitutes the accumulated loss could reflect the extent to which the epistemic utilities of degrees of belief contribute to the overall epistemic utility of a system of degrees of belief. While this proposal has some similarities to our interpretation of de Finetti’s variant of the Brier scoring-rule decision-theoretic framework, it is also quite different because it is situated in a different epistemic framework. In both cases, the scoring-rule can be thought of as measuring epistemic utility. Yet, Joyce (2009) bases his notion of epistemic utility on the idea that degrees of belief are estimates of truth-values of propositions the accuracy of which could be measured by the Brier scoring-rule, whereas in our interpretation of de Finetti the epistemic utility of degrees of expectation should be evaluated according to their instrumental value.
To assume that such a scale exists is of course an idealization.
Christensen holds that the classic formulations of the Dutch Book argument, due to Ramsey and de Finetti, demonstrate that agents who are Dutch Book vulnerable are ‘pragmatically’ inconsistent, and accordingly these arguments provide only pragmatic justifications for the probability calculus. His aim is to offer a non-pragmatic reading of the Dutch Book argument that will furnish epistemic reasons to conform to this calculus. Christensen (2004) presents his “depragmatized Dutch Book argument” in two stages. First, he formulates it for a ‘simple agent’, i.e. an agent “who values money positively, in a linear way” and “does not value anything else at all, positively or negatively” and whose “degrees of belief sanction as fair monetary bets at odds matching his degrees of belief” (ibid., p. 117). The argument is based on two principles for simple agents (ibid. pp. 118 and 119):
Bet Defectiveness. For a simple agent, a set of bets that is logically guaranteed to leave him
monetarily worse off is rationally defective.
Belief Defectiveness. If a simple agent’s beliefs sanction as fair each of a set of bets, and that
set of bets is rationally defective, then the agent’s beliefs are rationally defective.
The argument proceeds as follows. If a simple agent has incoherent degrees of belief, then there is a set of monetary bets at odds matching her degrees of belief that logically guarantees her monetary loss. By Bet Defectiveness, this set of bets is rationally defective. Thus, since each member of this set of bets is sanctioned by the agent’s degrees of belief, it follows from Belief Defectiveness that her degrees of beliefs are rationally defective (ibid., p. 121). Having established that in Dutch bookable scenarios the degrees of belief of a simple agent are rationally defective, Christensen then argues that the point of the Dutch Book argument is not dependent on the fact that it is formulated for such an agent. That is, he argues that since “the basic defect diagnosed in the simple agent is not a preference-defect”, the problem has to lie with the incoherent degrees of belief, and “the simple agent’s problematic preferences function in the [Dutch Book argument] merely as a diagnostic device” that “discloses a purely epistemic defect” (ibid., p. 123).
As Vineberg (2016, Section 2.2) notes, in Christensen’s reasoning the so-called pragmatic dimension of the Dutch Book argument seems to have been merely submerged. While Christensen’s claim that the model of simple agent functions only as a diagnostic device for disclosing the defect in incoherent degrees of belief, it is not clear from his argument why incoherence is irrational unless it has bearings for the instrumental value of beliefs. Indeed, Christensen claims that “[i]n severing the definitional or metaphysical ties between belief and preferences,” his depragmatized Dutch Book argument “frees us from seeing the basic problem with incoherent beliefs as a pragmatic one” (ibid., p. 123). But his argument for the belief defectiveness of incoherent degrees of belief relies heavily on what he takes to be a pragmatic consideration, i.e. the fact that incoherent degrees of belief could, in certain circumstances, leave agents worse off come what may.
For a similar proposal see Leitgeb and Pettigrew (2010a, b). Leitgeb and Pettigrew define inaccuracy relative to a world, and they propose to minimize the expectation of the inaccuracy of an agent’s degrees of belief (or more exactly, credence function) over all and only the worlds that are epistemically possible for the agent (2010a, pp. 205–206).
Two comments: 1. Joyce uses here the term ‘credence’. But since he seems to employ the terms ‘credence’ and ‘degree of belief’ interchangeably and for continuity with the terminology of previous sections, I use the term ‘degree of belief’. Nothing essential in what follows will hinge on this terminological choice. 2. (i) and (ii) correspond to claims (3) and (4) in Joyce (2009).
Following Jeffrey (1992, p. 44), Joyce (1998, pp. 575–576) characterizes probabilism as the doctrine that “any adequate epistemology must recognize that opinions come in varying gradations of strength and must make conformity to the axioms of probability a fundamental requirement of rationality for these graded or partial beliefs.”
Two comments: 1. For principles that relate subjective probabilities to objective probabilities, see, for example, Hacking’s (1965, Chap. 9) ‘frequency principle’, Lewis’ (1986, Chapter 19) ‘principal principle’, and Mellor’s (1995, Chap. 4) ‘evidence condition’. 2. In the example above I assumed for the sake of simplicity that the agent’s degree of belief is constrained by objective probability. Yet, this reasoning could be reformulated so as to apply to agents who follow de Finetti and reject the idea of objective probability or those who reject the idea that objective probabilities should constrain degrees of belief.
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Acknowledgments
For helpful comments on previous drafts of this paper and discussions of de Finetti’s philosophy, I am very grateful to the journal’s editors-in-chief, Phyllis Illari and Federica Russo, and Holger Andreas, Colin Elliott, Franz Huber, Joel Katzav, Noah Stemeroff, and in particular Donald Gillies, Aaron Kenna and Alberto Mura. Parts of this paper were presented at the CSHPS 2010, Montreal; GRECC Colloquium, Philosophy, Universitat Autònoma de Barcelona; IHPST Workshop, University of Toronto; Israel Foundations of Physics Discussion Group, Edelstein Center, Hebrew University of Jerusalem; Serious Metaphysics Group, Philosophy, University of Cambridge; Munich Center for Mathematical Philosophy, Ludwig Maximilian University of Munich; 42nd Dubrovnik Philosophy of Science conference; IHPST Colloquium, Paris; LSE Choice Group, CPNSS, London School of Economics; Theory Workshop, Sociology, University of Toronto; Probability and Models: Instrumentalism and Pragmatism in De Finetti’s Subjectivism workshop, HPS, University of Sassari; Philosophy, University of Rome III; CSHPS 2017, Toronto; Philosophy of Science Group, CONICET, University of Buenos Aires; Colloquialism, IHPST, University of Toronto; and the Philosophy Colloquium, Department of Economics, Philosophy and Political Science, University of British Columbia, Okanagan. I would like to thank the audiences in these forums for their helpful comments and suggestions. The research for this paper was supported by SSHRC Insight and SSHRC SIG grants.
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Berkovitz, J. On de Finetti’s instrumentalist philosophy of probability. Euro Jnl Phil Sci 9, 25 (2019). https://doi.org/10.1007/s13194-018-0226-4
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DOI: https://doi.org/10.1007/s13194-018-0226-4