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Bayesian coherentism

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Abstract

This paper considers a problem for Bayesian epistemology and proposes a solution to it. On the traditional Bayesian framework, an agent updates her beliefs by Bayesian conditioning, a rule that tells her how to revise her beliefs whenever she gets evidence that she holds with certainty. In order to extend the framework to a wider range of cases, Jeffrey (1965) proposed a more liberal version of this rule that has Bayesian conditioning as a special case. Jeffrey conditioning is a rule that tells the agent how to revise her beliefs whenever she gets evidence that she holds with any degree of confidence. The problem? While Bayesian conditioning has a foundationalist structure, this foundationalism disappears once we move to Jeffrey conditioning. If Bayesian conditioning is a special case of Jeffrey conditioning, then they should have the same normative structure. The solution? To reinterpret Bayesian updating as a form of diachronic coherentism.

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Notes

  1. Thanks to Jonathan Weisberg for this last example.

  2. The arrow analogy is borrowed from Weisberg (2015). This guiding feature of our conditional probabilities is often referred to as ‘rigidity’ (see Jeffrey 1965).

  3. A couple of notes about terminology. First, I will for the most part use the expressions ‘belief transitions’ and ‘updates’ interchangeably. Notably both expressions will sometimes be used in contexts where their standing as diachronically coherent is what is at issue. Relatedly, I will use the term ‘evidence’ a little ambiguously throughout. For instance, I will talk about ‘uncertain evidence’ in contexts where whether uncertain information meets the relevant normative constraint on evidence is, again, the question we are trying to decide. While the ambiguity here isn’t ideal, context should make clear enough what is being intended.

  4. For the proof of this, see Blackwell and Girshick (1979, p. 218) and Diaconis and Zabell (1982, p. 824). As Diaconis and Zabell note, there will be cases where our conditional probabilities are undefined for some partition—namely, where we assign a member of our partition a credence of zero. However, their result still holds for all updates if we take a sufficient partition to be a partition that is sufficient to represent a probabilistic belief transition as an update that is conditional on every proposition in this partition for which a conditional probability is defined (Cf. Blackwell and Girshick (1979, §8.4.3)).

  5. For some canonical examples of the Orthodox Account, see Skyrms (1987), Howson and Urbach (1989) and Jeffrey (1992), among many others.

  6. Perhaps an easier way of understanding what makes knowledge and access and reliability substantive, rather than formal, requirements is that they can be made sense of out of context. They can be defined independently of the other epistemic commitments one happens to hold. Formal norms are different in this regard. Take, for instance, the formal norm of consistency. The way that a Bayesian defines consistency will differ from the way that a defender of Dempster-Shafer theory does this. While the Bayesian will spell out her notion of consistency by means of probability functions (e.g., “X is consistent if it treats these probability functions in this way”), a Dempster-Shafer theorist, who trades in belief functions, will define her notion of consistency in terms of those. Unlike knowledge or access or reliability, the norm of consistency is so thin that it isn’t complete without a framework already in place to fill it in.

  7. A consequence of this is that the Normative Approach is more perspicuously formulated in terms of a set of evidence partitions, instead of just one. How this can be done will become clearer in Sect. 5.

  8. Despite this, I will continue to speak more loosely in terms of ‘norms’ and ‘oughts’. I do this mainly for ease of exposition, but also because it seems plausible that this language can be appropriately applied, in an evaluative way, to states of affairs. (For this point, see, for instance, Chrisman (2008)).

  9. I continue to take ‘Bayesian conditioning’ to refer, very generally as in Sect. 1, to Bayesian updating on certain evidence. Those who have explicitly taken Bayesian conditioning to instantiate a foundationalist structure include Christensen (1992), Bradley (2005), Weisberg (2009) and Titelbaum (forthcoming), among others, though these discussions do not necessarily take this foundationalism to provide the sort of formal account of evidence I have in mind.

  10. By contrast, many non-classical foundationalist accounts, like Goldman (1988)’s reliabilism, Plantinga (1991)’s proper basicality, Pryor (2000)’s dogmatism and Huemer (2007)’s phenomenal conservativism defend some form of fallible foundationalism. Each of these accounts maintains that the property that makes beliefs basic is something other than their infallibility. We’ll return to consider the relevance of these accounts in the next section.

  11. One might object that to identify a certain belief with an infallible one is to make precisely the sort of substantive assumption that a formal account of evidence is supposed to avoid. However, this assumption seems less divisive than the ones that we appealed to earlier as paradigmatically substantive assumptions, e.g., that internalism or externalism is the correct theory of justification, etc. Moreover, if one is uncomfortable with this assumption, one can do without it since, as we’ve already said, (3) is not a necessary component of a foundationalist account. As I will go on to argue in the next section, it doesn’t matter much how we think about the property that makes a state foundational. In assuming that the property that makes some state a foundation is certainty, I am simply following other Bayesians who have made this assumption (see fn. 9).

  12. I take this to be consistent with the idea that before a belief becomes evidence—when it is merely a proposition that the agent has some credence in—it is able to receive some inferential support.

  13. Thanks to an anonymous referee for raising this objection.

  14. The following line of argument has a steady, if diffused, presence in the literature on Jeffrey conditioning. There are references to it as early as Carnap (1957) (reprinted in Jeffrey 1975) and as late as Weisberg (2009).

  15. Thanks to an anonymous referee for drawing my attention to this proposal.

  16. For two early proponents of this view, see Domotor (1980) and Doring (1999).

  17. Of course, there are many coherentist accounts, and not all of them endorse all three of these constraints. Ewing (1934), for instance, takes coherence to be a matter of logical coherence alone, while Lewis (1946) takes coherence to be a matter of evidential coherence. Notably, BonJour (1985) takes coherence to be a matter of logical and probabilistic coherence, as well as a number of other requirements that might be held to fall under the heading of evidential coherence (see pp. 97-99 for the details).

  18. Many contemporary coherentist accounts spell this out probabilistically [for one notable account, see Fitelson (2003)]. For instance, a very simple view might hold that, in the previous case, what accounts for the increased coherence provided by \(\textit{P}_{3}\) is that \(\textit{p}(\textit{P}_{1}|\textit{P}_{2})<\textit{p}(\textit{P}_{1}|\textit{P}_{2}\wedge \textit{P}_{3}\)) and \(\textit{p}(\textit{P}_{2}|\textit{P}_{1})<\textit{p}(\textit{P}_{2}|\textit{P}_{1}\wedge \textit{P}_{3}\)).

  19. Thanks to an anonymous referee for raising this objection.

  20. Thanks to an anonymous referee for raising this objection.

  21. For a discussion of the distinction between wide- and narrow-scope requirements, see, for instance, Broome (1999) and Kolodny (2007).

  22. This follows from standard deontic logic, which says that a norm can’t require X if X is logically impossible.

  23. For a more detailed discussion of this point, see Cassell (2020).

  24. Can we reject one of these two claims? I’ve already suggested that (1) seems unimpeachable. What about (2)? Perhaps one might want to argue that the kind of defect the non-commutativity of Jeffrey conditioning represents is not a defect of particular updates, but is a sort of defect that inheres in the framework in general. However, it’s difficult to imagine what it could mean for a framework to be defective in a way that doesn’t directly derive from the particular updates that it produces.

  25. See Schervish et al. (2000, 2002, 2003). For a more recent account, see Staffel (2015).

  26. Thanks to an anonymous referee for raising this concern.

  27. The necessity claim is easy to see. Since Jeffrey updating fixes the probabilities on the partition (i.e., \(\textit{P}_\mathcal {EF}(F_{j})=\textit{q}_{j}\) and \(\textit{P}_\mathcal {FE}(E_{i})=\textit{p}_{i}\)), an update will be commutative only if \(\textit{P}_\mathcal {EF}(E_{i})=\textit{p}_{i}\) and \(\textit{P}_\mathcal {FE}(F_{j})=\textit{q}_{j}\), for all i and j (p.825). Diaconis and Zabell don’t give a proof of the sufficiency claim, but note that it follows from Csiszar (1975, theorem 3.2).

  28. Of course, Jeffrey himself did not defend this account of evidence. However, our discussion assumes that there are different ways of interpreting the core commitments of Jeffrey conditioning, and of the Bayesian updating framework more generally, that have different consequences for how we should think about evidence. This assumption seems especially plausible in light of the fact that Jeffrey’s own views about his updating rule seem to have evolved over time.

  29. See Diaconis and Zabell (1982, §4) for a similar proposal.

  30. See de Finetti (1937, 1972) for a justification of Probabilism, Lewis (1999) and Teller (1973) for a justification of Bayesian updating, and Skyrms (1987) for a justification of Jeffrey conditioning.

  31. See Joyce (1998, 2009) for justifications of Probabilism, Greaves and Wallace (2006) for a justification of Bayesian updating, and Leitgeb and Pettrigrew (2010a); Leitgeb and Pettigrew (2010b) and Pettigrew (2016) for justifications of both Probabilism and Bayesian updating.

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Acknowledgements

Thanks to Eric Campbell, Pete Graham, Sophie Horowitz, Thomas Icard, Hilary Kornblith, Mike Nance, Luis Oliveira, Alejandro Pérez Carballo, Jessica Pfeifer, Joel Pust, Jonah Schupbach, Whitney Schwab, Julia Staffel, Mike Titelbaum, Steve Yalowitz and audiences at the University of Delaware, the Pacific APA, the Formal Epistemology Workshop, the Society for Exact Philosophy, the Canadian Philosophical Association annual meeting and the Northern New England Philosophical Association annual meeting for very helpful comments and suggestions. Special thanks to Chris Meacham, Willam Talbott, Jonathan Weisberg and two anonymous referees for the journal for detailed written feedback on earlier versions of this paper that helped improve it considerably.

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  1. Department of Philosophy, University of Maryland, Baltimore County, Performing Arts and Humanities Building, Room 461, Baltimore, MD, 21250, USA

    Lisa Cassell

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Cassell, L. Bayesian coherentism. Synthese 198, 9563–9590 (2021). https://doi.org/10.1007/s11229-020-02656-6

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