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A Humean Guide to Spielraum Probabilities

  • Special Section Article: Kries and Objective Probability
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Abstract

The most promising accounts of ontic probability include the Spielraum conception of probabilities, which can be traced back to J. von Kries and H. Poincaré, and the best system account by D. Lewis. This paper aims at comparing both accounts and at combining them to obtain the best of both worlds. The extensions of both Spielraum and best system probabilities do not coincide because the former only apply to systems with a special dynamics. Conversely, Spielraum probabilities may not be part of the best system, e.g. if a broad class of random devices is not often used. Spielraum probabilities have the potential to provide illuminating explanations of frequencies with which outcomes of trials on gambling devices arise. They ultimately fail to account for such frequencies though because they are compatible with frequencies that grossly differ from the values of the respective probabilities. I thus follow recent proposals by M. Strevens and M. Abrams and strengthen the definition of Spielraum probabilities by adding a further condition that restricts some actual frequencies. The resulting account remains limited in scope, but stands objections raised against it in the recent literature: it is neither circular nor based upon arbitrary choices of a measure or of physical variables. Nor does it lack any explanatory power.

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Notes

  1. For more details on von Kries’s view of probabilities see Kamlah (1983) and Heidelberger (2001).

  2. According to von Plato (1983, 38), it is not clear whether Poincaré knew von Kries’s work when writing Poincaré (1896). See von Plato (1983) for the method of arbitrary functions. For the difference between Spielraum probabilities and the method of arbitrary functions see fn. 13 below.

  3. There are other accounts of ontic probabilities, notably frequentism and various propensity views. These accounts are not very popular though these days. See Gillies (2000) for an introduction to the philosophy of probability and Briggs (2010) for a recent discussion of ontic probabilities.

  4. See Meacham (2010a, b) for interpretations of probabilities in statistical physics.

  5. For a nuanced view about method in the philosophy of probability see Hájek (1997, 210–211) and Hájek (2010). See also Hájek (2012, 346).

  6. von Kries (1886), Ch. III considered a “Stoss-Spiel” as defined on page 49f.

  7. See Keller (1986), Diaconis (1998) and Diaconis et al. (2007) for the physics of coin flips.

  8. This result is called Liouville’s Theorem. See e.g. Arnol’d (1989, Sect. 3.5).

  9. Here we assume realism about phase space. We suppose that each coin flip is properly described using a phase space and that assertions made about this phase space literally pertain to the system itself. This may seem worrisome because most known phase-space descriptions are not literally true due to idealizations. This is not the place though to discuss this worry.

  10. We do not require that the patches are maximally connected, as does Rosenthal.

  11. My definition of SR probabilities is close to the one given by Strevens (2003, 128). However, I refer to a specific measure, viz. the Lebesgue measure in my definition, while Strevens does not. This does not much matter for the mathematical elaboration. Rosenthal (2012, 226) draws on the Lebesgue measure too to define SR probabilities, but his definition differs from mine because he is more restrictive about the geometry of the test regions. I assume that such differences about technical details do not matter for the purposes of my discussion.

  12. Properly speaking, the values of \(T_{\min }\) and \(T_{\max }\) are specific to a particular coin flip.

  13. This result is at the basis of the method of arbitrary functions. Unlike the definition of SR probabilities, the method assumes a probability distribution over initial conditions (von Plato 1983, 38).

  14. See e.g. Frigg and Hoefer (2010, 357).

  15. It is not clear how to combine these pro tanto desiderata in case they conflict; more problems of this kind will follow below.

  16. If SR referred to an initial probability distribution, matters would be more complicated and, maybe, different.

  17. Of course, formally, a system may describe a deterministic dynamics using probabilities that only take the values 0 or 1. But a best system cannot do so because such probabilities make the description of a deterministic dynamics more complicated than necessary.

  18. As Hoefer (2007, 558–560) points out, Lewis may have inferred the incompatibility between determinism and the existence of chances from a version of the Principal Principle, but Hoefer suggests that this very version is problematic. See Frigg and Hoefer (2010) and (2015) for more discussion.

  19. It may be objected that one and the same mosaic is compatible with different SR probabilities because the same coin flip may be described using different phase spaces and because SR probabilities depend on the phase space chosen. However, in this paper, we assume that there is a literally true phase space description of the system (see fn. 9). If there are several true phase space descriptions, we expect them to agree upon SR probabilities.

  20. See Rosenthal (2012, 223–224, and 226–227) for the explanatory power of SR probabilities and ibid., 219 and Abrams (2012, 245) for the way in which BS probabilities explain.

  21. This point is raised by Rosenthal (2012, 219). In this passage, Rosenthal refers to frequentist views, and his point is particularly convincing for actual frequentism as defined by Hájek (1997). But for Rosenthal, Lewis’s best system account of chances is a sort of frequentism (ibid., 218), so the criticism should carry to Lewisian chances.

  22. Note though that Hoefer (2007) and Frigg and Hoefer (2010) consider a slightly modified account of best system probabilities. But their modifications are largely irrelevant for the purposes of this paper.

  23. In a similar way, Frigg and Hoefer (2010, Sects. 3–4) argue that H follows from what they take to be a best system just because the macroscopic patterns of related outcomes call for a simple probabilistic description in terms of H independently of the microphysics.

  24. The type of problem described in this paragraph is also pointed out by Frisch (2011, 1004–1005) and by Beisbart (2014).

  25. Frigg and Hoefer (2010) argue that the best system contains a statement specifying the chances that certain symmetric solids with n faces land on one of their faces (359). They do not consider biased coins, dices etc. If biased coins count as regular solids, then my argument casts doubts on the idea that the rule given by Frigg and Hoefer (2010) is part of the best system. If biased coins are not part of the regular solids covered by the rule, the question arises whether they have Humean chances.

  26. Some of these problems discussed here may be avoided if we allow for unsharp Humean chances (Dardashti et al. 2014).

  27. Frigg and Hoefer (2010, Sect. 4) suggest that, what is the best system according to their criteria, entails a probability rule of the outcomes of coin flips and, independently, a probability model for the initial conditions. Both probability models have to be roughly consistent with each other.

  28. By considering a probability density in phase space, we move to a continuous probability model. The definition of fit for such models has additional difficulties, see Elga (2004), particularly App. B. The argument in this paragraph relies on intuitive assessments of fit. It is thus presupposed that we can define a precise notion of fit that accords well with intuitive assessments of fit.

  29. The precise formulation of the Principal Principle is a matter of debate (see e.g. Lewis 1994; Hoefer 2007, 552–553; Frigg and Hoefer 2010, 356–357) and need not concern us here.

  30. Maybe, it is easier to obtain a robustly best system if we restrict ourselves to chances at a fundamental level as does Lewis. But the question then arises how we obtain chances of macroevents (cf. Hoefer 2007, 557, Frigg and Hoefer 2010, Sect. 4, and Frigg and Hoefer 2015, Sect. 4).

  31. Why is it a problem that BS has not much explanatory power? There are two possible reasons to require that an account of ontic probability provide illuminating explanations. First, if chances have considerable explanatory power, an account of them should capture this. Second, it is a reasonable demand that an account of ontic probabilities itself display explanatory power, even if chances do not explain much. If we are not just aiming at an analysis of meaning, it is sensible to require that the account explains well why probabilities are applied to some types of physical systems and not to others.

  32. See also Strevens (2003). For the purposes of this paper, I prefer Strevens (2011) to Strevens (2003) because it is more succinct regarding the problems discussed here.

  33. Cf. Strevens (2003, Ch. 2), particularly Def. 2.5 on 129 for a more precise definition of macroperiodicity. Note though that my definition applies to empirical distributions directly, whereas the one by Strevens applies to probability densities, strictly speaking. There is another respect in which I slightly deviate from Strevens (2011). In his condition on empirical distributions (352), Strevens also considers infinite series of trials and demands that, in this case, the limiting frequencies converge to a macroperiodic distribution. This is problematic because the limit of the frequencies depends on the order in which the points in phase space are considered. Cf. Hájek (2009, 218–220). We need not consider infinite series because many of their finite sub-series are covered by the definition.

  34. I’m grateful to an anonymous referee for drawing my attention to this point.

  35. The notion of macroperiodicity is here defined on the basis of binning data. It may be suggested that the data in long, but finite series of coin flips rather be fitted using simple probability models. We may then demand that the best fitting model varies only slowly. This would well accord with ideas on which BS is built. It is doubtful though whether the alternative suggestion works. Fitting a simple probability model to the data leads to a model that is smooth and varies slowly independently of whether the empirical distribution of initial conditions is macroperiodic.

    Something like SR+ is already anticipated by von Kries (1886, e.g. 50–51).

  36. The distinction between empirical distributions and probability models is often neglected for two reasons. First, the latter are sometimes called probability distributions. Second, actual data are often conveniently described in terms of a probability model that fits the data well. It is then said that the data are normally distributed, for instance. Nevertheless, there is a distinction between empirical distributions and probability models, and it is not ad hoc. When statisticians say that they fit a probability model to the actual data, they implicitly refer to the distinction.

  37. In the following, I do not distinguish between physical characteristics and coordinates. Even though there is a distinction, a switch between different coordinates may be seen as a switch between two physical characteristics. I also sometimes call the characteristics variables.

  38. This is Liouville’s Theorem again. It does not exclude though other measures that are preserved by the dynamics. Nevertheless, generally, other measures will not be preserved by the dynamics. We will presently argue for the choice of certain variables, and once we have chosen these variables, we are only interested in measures that are simple to express in these variables, which should be sufficient to fix the measure.

  39. This is another way in which the Lebesgue measure is distinguished.

  40. Note also that Lewis uses considerations of simplicity to analyze chances. There are of course problems if simplicity is too vague or if people disagree on what is simple. I do not think that we run into this problem here.

  41. Strevens (2011, 351) contains a different reply to the objection that the notion of SR probabilities turns on an arbitrary choice of a measure: He only requires that there be a measure with respect to which microconstancy and macroperiodicity are fulfilled. I do not think that this will do. Suppose that microconstancy and macroperiodicity hold with respect to some very unnatural measure, but not with respect to other, more natural variables. Suppose further that, in the natural variables, the initial conditions leading to the \(o_i\) could be described very easily. That is, the patches do not display the geometry that we expect in a well-mixed dough. I doubt whether we would allow for chances in this case.

  42. See e.g. Hempel and Oppenheim (1948, 135–140).

  43. See Abrams (2012, 366) for the point about the value of the strike ratio.

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Acknowledgments

I’m very grateful for detailed and valuable criticism by two anonymous referees and for comments by the participants of the workshop at Bonn. Thanks also to Jacob Rosenthal and Carsten Seck for the invitation to contribute this paper and for comments on the paper. I’m also grateful to Joannes Campell, Martina Jakob and Lukas Lüscher for proofreading.

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    Claus Beisbart

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Beisbart, C. A Humean Guide to Spielraum Probabilities. J Gen Philos Sci 47, 189–216 (2016). https://doi.org/10.1007/s10838-015-9316-6

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