Abstract
Single-case and long-run propensity theories are among the main objective interpretations of probability. There have been various objections to these theories, e.g. that it is difficult to explain why propensities should satisfy the probability axioms and, worse, that propensities are at odds with these axioms, that the explication of propensities is circular and accordingly not informative, and that single-case propensities are metaphysical and accordingly non-scientific. We consider various propensity theories of probability and their prospects in light of these objections. We argue that while propensity theories face challenges, these challenges do not undermine their validity as prospective interpretations of probability in science.
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Notes
Poincaré (1902/1952), p. 186.
That is not to say that the interpretation of probability cannot change the broadly accepted calculus of probability. In this paper, we shall focus on concepts of probability that satisfy the probability calculus.
Here and henceforth by ‘frequency’ we mean relative frequency.
Three comments: (1) Long-run propensity theories have also been attributed, retroactively, to Cramer (1946) and Braithwaite (1953) (Mellor 1971, p. 66, Levi 1973, p. 525, Kyburg 1974, p. 358). (2) Both Hacking’s and Levi’s theories rely on Braithwaite’s (1953) ideas about how to define frequency in the long-run, and accordingly they share various aspects. Levi (1967, p. 192) describes his view as a “qualified propensity interpretation.” But, later on, he (1977, endnotes 13 and 15, pp. 453–454) objects to the view that he and Hacking are propensity theorists. He (ibid., p. 446) emphasizes that in describing his view as a qualified propensity interpretation he “wished to treat chance predicates as primitives and to refuse to offer a semantics for chance predicates in terms of test behavior.” And although he “meant to suggest a certain analogy between chance predicates and disposition predicates in that both sorts of predicates should be taken as primitive, [he] meant to deny that reduction sentences could serve to illuminate the connections between chance and test behavior in the way they can illuminate the connections between disposition and test behavior.” (3) Gillies (2000a, p. 209) notes that he conceived his (1973) theory as an objective, non-frequency theory of probability that is not a propensity theory, partly because it differed from Popper’s theory. But in the meantime the term propensity “has taken on the broader meaning of an objective but non-frequency view of probability” and accordingly he would now classify his earlier position as “one particular example of a propensity theory.”
The distinction between tendency and dispositional propensity theories is inspired by Mellor’s (1971, pp. 68–70). Mellor maintains that propensities are dispositions, and he criticizes the tendency interpretation of propensities as either viciously circular or inadequate.
In Sect. 5.3.6, we shall discuss Fetzer’s response to Humphreys’ paradox.
For a critical review of propensity theories of probability up to the early 1970s see Kyburg (1974), and for a recent critical discussion of propensity theories see Eagle (2004).
Two comments: (1) For ease of presentation, we shall sometimes say that the conditioning events determine the propensity of the conditioned event, leaving open the nature of the determination. (2) In theories in which propensities are not probabilities, the above conditional probability will denote the probability distribution that is the display of the propensity, and the conditioning events will denote the conditions that display the propensity.
Hacking (1965, pp. 13, 25) notes that “von Mises’ probability is a property of a series, but it is intended as a model of a property of what [von Mises] calls an experimental set-up.” Hacking says that he “copied the very word ‘set-up’ from [von Mises’] English translators” and followed “Venn and von Mises, intending to explain frequency in the long run by means of chance set-ups.” And Gillies (2000b, p. 811) remarks that Popper’s suggestion that probabilities should be related to repeatable conditions rather than collectives had already been made by Kolmogorov (1933/1950, pp. 3–4).
Popper (1959a, p. 29).
For want of space, we cannot discuss the application of Hacking’s theory to statistical inference.
Hacking borrows the following theses from Koopman’s logic: (I) Implication: If \( h \) implies \( i \), then \( h/e\,\; \le \;\;i/e \); (II) Conjunction: If \( e \) implies \( i \), then \( h/e\,\; \le \;\;i/e \); (III) Transitivity: If \( h/e\;\; \le \;\;i/d \) and \( i/d\;\; \le \;\;j/c \), then \( h/e\;\; \le \;\;j/c \); (IV) Identity: \( h/e\,\le\, i/i \); where here ‘implies’ means logically implies and ‘\(h/e\,\le\, i/e \)’ means \( e \) supports \( i \) at least as well as it supports \( h \).
Hacking also proposes a generalization of the principle (under certain conditions) to cases in which the knowledge is that the chance of \( E \) is one of a set of possible chances.
SCIKT stands for ‘same-causal-influences kind of trial’ or ‘same-causal-influences kinds of trials’, and DCIKT stands for ‘different-causal-influences kind of trial’ or ‘different-causal-influences kinds of trials’. Context will determine whether each of these acronyms refers to a kind of trial or kinds of trials.
It is important to note that due to the uncertainty principle, experimenters cannot realize such a SCIKT at will.
Let \( Q_{t} (X_{t} ,Y_{t} ) \) be the actual position configuration of the universe at a certain time \( t \), where \( X \) is a subsystem of the universe (e.g. the particle and the measurement apparatus) and \( Y \) is its actual environment (i.e. the rest of the universe). Dürr, Goldstein and Zanghì call \( \psi_{t} (x) = \varPsi_{t} (x,Y_{t} ) \) the conditional wavefunction of \( X \); where \( \varPsi \) is the wavefunction of the universe, and \( x \) is a variable ranging over the possible position configurations of \( X .\)
Conditional wavefunctions do not generally evolve according to the Schrödinger equation. However, when certain conditions are met—when the subsystem and the environment do not interact or, more generally, when the universal wavefunction evolves into a wide separation of components in the configuration space of the entire system—a conditional wavefunction will evolve according to the Schrödinger equation. When this is the case, the conditional wavefunction becomes the effective wavefunction.
As Gillies (ibid., p. 92) notes, the title of this law was suggested by Keynes (1921, p. 336).
Here we followed von Mises’ terminology. Gillies calls this principle the Law of Excluded Gambling Systems.
The claim that von Mises’ philosophy of probability is operationalist is questionable, as the infinite frequencies that explicate probabilities in von Mises’ theory are not observable.
Recall that (INT) applies only for sufficiently long sequences, where a binomial distribution approximates a normal distribution.
To simplify notation, we suppress the spacing condition in \( S \).
\( H_{1} \) and \( H_{2} \) are not joint propositions, but they can easily be converted to such propositions by adding to each of them the proposition that the outcome of our trial is a 0.3 frequency of heads.
Obviously, such data will also falsify \( H_{2} \).
In analyzing Popper’s (1957, 1959a) propensity theory, Rosenthal (2006, pp. 106–107) argues against the idea embodied in the Frequency Postulate that “it presupposes a non-probabilistic connection between probabilities and relative frequencies, which definitely does not exist.” Rosenthal points out that if propensity is characterized as a disposition to produce certain frequencies in the long run, then it is a disposition that is only probabilistically connected to its manifestations, and accordingly we face a circularity: “the concept of probability that should be explicated via the idea of propensity is in fact presupposed by it”.
More precisely, to break even the monetary value of \( \delta S \) should be sufficiently small so as to stop any transfer of money.
Similarly, Rosenthal (2006, p. 107) argues that if we think about the single-case propensity of an event \( A \) as an indeterministic disposition of a set-up \( S \) to produce \( A \), as some of Popper’s early writings on propensity suggest, the analysis of single-case propensity will be circular. Dispositions are characterized through their manifestations: the propensity of \( S \) to produce \( A \) is displayed just in case \( A \) actually occurs upon the realization of \( S \). But since \( A \) occurs only with a certain probability \( p \), the disposition of \( S \) to produce \( A \) is displayed only with a certain probability. Thus, the very concept of probability that is supposed to be interpreted by the concept of single-case propensity is presupposed.
Miller actually does not support a single-case propensity theory, though he provides a sympathetic interpretation of what he considers as the “most interesting and fertile strain” of Popper’s “numerous expositions of the propensity interpretation” (Miller 1994, p. 175).
For such an account of theoretical terms, see Lewis (1970).
Mellor calls these possibilities metaphysical to distinguish them from epistemic possibilities that reflect knowledge and ignorance.
Two comments: (1) Humphreys (2004, p. 669) notes that “CI stands for conditional independence, not causal independence.” (2) Humphreys (ibid.) proposes two other variants of this argument. In Sect. 5.2, footnote 40, we argue that our proposed resolution also applies to the other versions of the argument.
In fact, things are a bit more complicated. First, some interpretations of QM postulate the existence of backward causation (for a discussion of such interpretations and the challenges they encounter, see Berkovitz 2008 and 2011 and references therein). If such causal influences existed, there might not be an initial state of the universe in the conventional sense of influencing but not being influenced by other states. Yet, even in universes in which backward causation exists, there might be events in the probability space that have no conditions to determine their propensities. Second, it may be suggested that if an initial state of the universe existed, some laws of nature could determine its propensity. It is noteworthy, however, that none of our current or previous scientific theories provides such a propensity, and there seems to be no reason to think that future theories will be different.
Humphreys (2004) proposes two other versions of his paradox. Each of these versions replaces CI by another principle. One version is based on the zero influence principle, which states that when \( t_{2} < t_{3} \),\( Pr_{{B_{{t_{1} }} }} (I_{{t_{2} }} /T_{{t_{3} }} ) = 0 \). The other version is based on the fixity principle, which states that when \( t_{2} < t_{3} \),\( Pr_{{B_{{t_{1} }} }} (I_{{t_{2} }} /T_{{t_{3} }} ) = 0\;\text{or}\;1 \). Like CI, both of these principles violate the symmetry between conditioned and conditioning events in standard conditional probability and accordingly they do not obtain in the above representation of propensity.
Humphreys (2004, pp. 671–672) characterizes the co-production interpretation for time-dependent propensities. “A co-production interpretation considers the conditional propensity to be located in structural conditions present at an initial time \( t \), with \( Pr_{t} ( \cdot / \cdot ) \) being a propensity at \( t \) to produce the events which serve as the two arguments of the conditional propensity…. The key feature of co-productions interpretations is that the representations of the conditions present at the initial time \( t \) are not included in the algebra or σ-algebra of events within the probability space.” But this characterization could be easily translated to time-independent propensities.
The suggestion here is that such conditionals may represent causal dispositions, not that they provide a reductive analysis of causation. In Lewis’s (1986, pp. 175–184) influential account of indeterministic causation, causation is explicated in terms of counterfactuals with probabilistic consequent. Although the above conditionals are substantially different, arguably they could express causal dispositions or tendencies.
McCurdy argues that CI fails because the conditioned and conditioning events in CI are effects of a common cause and accordingly they are not probabilistically independent of each other. Humphreys (2004, Sect. 6) replies that although a plausible case can be made for McCurdy’s co-production interpretation in the photon example discussed above, a formally identical example demonstrates that McCurdy’s attempt to resolve Humphreys’ paradox fails. Humphreys suggests the following example. A spherical radiation detector surrounds a radioactive source of alpha particles. The detector, which is shielded from other sources, is not perfectly reliable so that not all emitted particles are detected. The emission of an alpha particle in a specified time period is assumed to be an indeterministic process, as the case is in the orthodox and various other interpretations of QM. Let \( Pr_{{t_{0} }} (E_{{t_{1} }} /D_{{t_{2} }} \& B_{{t_{0} }} ) \) be the propensity at time \( t_{0} \) for an alpha particle to be emitted during a short time interval \( t_{I} \) conditional upon the alpha particle being detected at \( t_{2} \); where \( t_{0} < t_{1} < t_{2} \) and \( B_{{t_{0} }} \) are the ‘background conditions’ that include all the features that affect the propensity value at \( t_{0} \). Humphreys (ibid., p. 675) believes that in this example CI – \( Pr_{{t_{0} }} (E_{{t_{1} }} /D_{{t_{2} }} \& B_{{t_{0} }} ) = Pr_{{t_{0} }} (E_{{t_{1} }} /B_{{t_{0} }} ) \)—is evidently true and accordingly his challenge still stands. But in McCurdy’s account of propensity, CI fails; for it follows from the probability calculus that if \( Pr_{{t_{0} }} (D_{{t_{2} }} /E_{{t_{1} }} \& B_{{t_{0} }} ) \ne Pr_{{t_{0} }} (D_{{t_{2} }} /B_{{t_{0} }} ) \), then \( Pr_{{t_{0} }} (E_{{t_{1} }} /D_{{t_{2} }} \& B_{{t_{0} }} ) \ne Pr_{{t_{0} }} (E_{{t_{1} }} /B_{{t_{0} }} ) \). While Humphreys is right to object to McCurdy’s reasoning against CI on the basis of the existence of a common cause, the objection to CI need not rely on such reasoning: it could simply be based on the symmetry of standard conditional probability.
For continuity with the notation of previous sections, here and henceforth in this section we substitute ‘\( A \)’ for ‘\( a \)’ and ‘\( E \)’ for ‘\( b \)’ in the quotations.
For a similar criticism, see Drouet (2011), pp. 158–159.
De Finetti (1974a, Chap. 4, 1974b, pp. 307–310) characterizes subjective conditional probabilities as the probabilities of conditional events. But in his case, conditional events and their probabilities are not mysterious. One may plausibly interpret conditional events as three-valued statements and probabilities of conditional events as conditionals with probabilistic consequences (see Sect. 7.3 and Berkovitz 2012a).
The same reasoning may apply to other scientific disciplines that are highly mathematical. For example, in modern economics mathematics constitutes fundamental features of the economic realm.
There are different ways to define independent trials. A trial may be said to be independent of other trials if and only if the probability of its occurrence is independent of the occurrence of the other trials. On this definition, we need to add a qualification to the weak law of large numbers: if the propensity of E in a trial of kind T is p, then the frequency of E in a long series of unbiased independent trials of kind T will almost certainly be equal to \( p \). A series of independent trials of a certain kind may be biased because of backward causation, e.g. when the outcome of a trial of that kind is a partial cause of the trial’s occurrence. In such a series, trials could be causally and accordingly probabilistically independent. Yet, due to backward causation, the series will be biased, so that there will be a low probability that the long-run frequency of the outcomes of trials will reflect the probability of the outcome in any individual trial. Berkovitz (2008, 2011) argues that such scenarios are expected in retro-causal interpretations of quantum mechanics, where closed causal loops occur due to backward causation. Another way to define independent trials is in terms of the outcomes of trials: a series of trials is independent if and only if the probability of the joint outcome of the series factorizes into the probabilities of the individual outcomes. On this definition, the weak law of large numbers does not require any modification. For an example of such a concept of independence, see Hacking’s definition of independent trials in Sect. 3.2.2.
Lewis uses the term ‘credence’, but his concept of credence is akin to subjective probability.
This is a consequence of Lewis’ reformulation of the Principal Principle.
The mathematical expression of the weak law of large numbers is the following. Let \( P_{C} (E) \) be the probability of \( E \) in conditions \( C \) and \( f_{{C^{n} }} (E) \) be the frequency of \( E \) in a series of \( n \) independent repetitions of \( C \), where \( C^{n} \) denotes such a series. Then, for any \( \varepsilon > 0 \), \( \lim_{n \to \infty } (\left| {P(f_{{C^{n} }} (E)) - P_{C} (E)} \right| < \varepsilon ) = 1 \).
Similarly, in consistent infinite long-run propensity theories the relation between propensities and (infinite) frequencies is indirect: it is expressed in probabilistic terms. For example, let the infinite long-run propensity of an event \( E \) in conditions \( C \) be \( p \). The propensity that the frequency of \( E \) in an infinite series of independent repetitions of \( C \) will be \( p \), is 1. But there exist infinitely many possible series in which the frequency of \( E \) deviates substantially (extremely) from \( p \).
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Acknowledgments
Parts of this paper were presented at the Current Projects Seminar, Department of Philosophy, University of Sydney, the Probability in Biology and Physics Workshop, IHPST, Paris, and the 41st Dubrovnik Philosophy of Science Conference, and I thank audiences at these venues for their helpful comments. For discussions and comments on earlier drafts of this paper, I am grateful to Ian Hacking and Alan Hájek and especially Isabelle Drouet, Delia Gavrus, Carl Hoefer, Philippe Huneman, Joel Katzav, Duncan Maclean and Noah Stemeroff. This research was supported by a SSHRC Insight Grant, a SSHRC Institutional Grant, a Victoria College Travel Grant and by the Centre for Time, Department of Philosophy, University of Sydney.
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Berkovitz, J. The Propensity Interpretation of Probability: A Re-evaluation. Erkenn 80 (Suppl 3), 629–711 (2015). https://doi.org/10.1007/s10670-014-9716-8
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DOI: https://doi.org/10.1007/s10670-014-9716-8