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Reviving frequentism

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Abstract

Philosophers now seem to agree that frequentism is an untenable strategy to explain the meaning of probabilities. Nevertheless, I want to revive frequentism, and I will do so by grounding probabilities on typicality in the same way as the thermodynamic arrow of time can be grounded on typicality within statistical mechanics. This account, which I will call typicality frequentism, will evade the major criticisms raised against previous forms of frequentism. In this theory, probabilities arise within a physical theory from statistical behavior of almost all initial conditions. The main advantage of typicality frequentism is that it shows which kinds of probabilities (that also have empirical relevance) can be derived from physics. Although one cannot recover all probability talk in this account, this is rather a virtue than a vice, because it shows which types of probabilities can in fact arise from physics and which types need to be explained in different ways, thereby opening the path for a pluralistic account of probabilities.

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Notes

  1. Eberhard and Glymour call them “sets of ur-events” because they are the irreducible basis for von Kries’s probabilities (see Reichenbach 2008, Introduction, section 4.2).

  2. For simplicity’s sake and to be as close to Boltzmann’s reasoning as possible, I assume Newtonian mechanics as the microscopic theory.

  3. A measure \(\mu \) is absolutely continuous with respect to a measure \(\lambda \) (symbolically \(\mu \ll \lambda \)), if all the null sets of \(\lambda \) are null sets of \(\mu \), that is \(\forall X \left( \lambda (X)=0 \Rightarrow \mu (X)=0\right) \).

  4. There has been a long debate to make Boltzmann’s argument more mathematically and conceptually precise. For our purpose, we do not need to dive into these details (see, e.g., Volchan 2007; Frigg 2009; Werndl 2013; Lazarovici and Reichert 2015; Myrvold 2019.)

  5. There are some subtleties when one generalizes this scheme to infinite sample spaces, like, forming a \(\sigma \)-algebra. These are treated in standard textbooks on probability theory and are not the focus of this paper.

  6. If one were to embed this discussion in quantum theory, one would need to replace phase space with configuration space.

  7. Whenever I refer to the law of large numbers, I always mean the weak law of large numbers.

  8. More precisely, there are three parameters in the law of large numbers that are fixed successively. First, one chooses an \(\epsilon \), then a \(\delta \), and then sufficiently large \(N>N_0\) such that:

    $$\begin{aligned} \lambda \left( \left|\frac{1}{N}\sum _{k=1}^{N}X_k(x)-\frac{1}{2}\right|<\epsilon \right) >1-\delta . \end{aligned}$$

    Such an \(N_0\) exists, because according to the Chebychev inequality

    $$\begin{aligned} \lambda \left( \left|\frac{1}{N}\sum _{k=1}^{N}X_k(x)-\frac{1}{2}\right|<\epsilon \right) >1-\frac{1}{\epsilon ^2N}. \end{aligned}$$

  9. e.g., \(\frac{1}{1,000,000}=0.000001\), \(\frac{1}{\ln (1,000,000)}\approx 0.072\), and \(\frac{1}{\ln (\ln (1,000,000))}\approx 0.38\).

  10. I thank an anonymous referee for raising many of the following points.

  11. There is one caveat: even if all the assumptions of the law of large numbers were fulfilled it is still possible for a sequence to have a different limit or no limit at all; the initial conditions leading to these sequences have measure zero though.

  12. I also thank here an anonymous referee for raising these issues.

  13. Hoefer (2007, 2011, 2019) developed a more pragmatic account of Humean probabilities, which is closely linked to the Albert–Loewer account.

  14. The concept of fit leads to the zero-fit problem; Elga (2004) proposes a solution by invoking a certain notion of typicality.

  15. Thanks to an anonymous referee for raising this point.

  16. Something similar has been proposed in certain versions of the propensity interpretation. Miller (1994, p. 56) says that propensities are determined by “the complete situation of the universe (or the light-cone) at the time, and, for Fetzer (1982, p. 195), they are determined by “a complete set of (nomically and/or causally) relevant conditions [...] which happens to be instantiated in that world at that time.” These solutions, however, are not satisfactory for Hajek (2007, p. 576) because the propensities, such defined, are not accessible to an agent to assign probabilities in practice. Therefore, he subsumes these proposals under no-theory theories of probabilities.

  17. Thanks to an anonymous referee for raising this issue, who I quote almost verbatim.

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Acknowledgements

I wish to thank Frederick Eberhardt, Christopher Hitchcock, and Charles Sebens for their helpful and detailed comments on previous drafts of this paper. I also wish to thank David Albert, Jeffrey Barrett, Detlef Dürr, Sheldon Goldstein, Dustin Lazarovici, Barry Loewer, Tim Maudlin, Isaac Wilhelm, and Nino Zanghì for many invaluable hours of discussions. I also thank the members of the Caltech Philosophy of Physics Reading Group, in particular Joshua Eisentahl and James Woodward. I want to thank two anonymous reviewers for their helpful comments, which significantly improved the paper. Especially one of the anonymous reviewers spent considerable time and effort in the review process; I particularly thank this reviewer.

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    Mario Hubert

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Hubert, M. Reviving frequentism. Synthese 199, 5255–5284 (2021). https://doi.org/10.1007/s11229-021-03024-8

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