Higher-Order Beliefs and the Undermining Problem for Bayesianism

Abstract

Jonathan Weisberg has argued that Bayesianism’s rigid updating rules make Bayesian updating incompatible with undermining defeat. In this paper, I argue that when we attend to the higher-order beliefs we must ascribe to agents in the kinds of cases Weisberg considers, the problem he raises disappears. Once we acknowledge the importance of higher-order beliefs to the undermining story, we are led to a different understanding of how these cases arise. And on this different understanding of things, the rigid nature of Bayesianism’s updating rules is no obstacle to its accommodating undermining defeat.

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Notes

  1. 1.

    Note that what goes by the name “undermining defeat” in Weisberg’s discussion is more commonly referred to as “undercutting defeat”. For the canonical account of undercutting defeat, see Pollock (1986, 2008).

  2. 2.

    The worry he raises is an extension of the argument from Weisberg (2009). The bulk of my discussion will focus on Weisberg (2015), where Weisberg attempts to sharpen this worry. However, we will return to consider some of the more general issues raised in Weisberg (2009) in Section 5.

  3. 3.

    And where we assume that p(B) > 0.

  4. 4.

    The correct philosophical interpretation of Jeffrey conditionalization is a complicated matter, made all the more complicated by the difference in how Jeffrey understood his own updating rule, and how others have appropriated it (and made more complicated still by the fact that Jeffrey’s own views about his updating rule were only clarified over time (cf. Jeffrey (1965) and Jeffrey(1983)). I follow Weisberg (and others) in understanding Jeffrey conditionalization as a substantive diachronic updating norm. See Weisberg (2015), 124.

  5. 5.

    It was Jeffrey himself who first coined this term. See Jeffrey (1965).

  6. 6.

    To avoid this cumbersome expression, in what follows, I will simply use the term “conditional probabilities” to denote those conditional probabilities that are conditional on the elements of the evidence partition. Where I do not have this more restricted usage in mind, context should make that clear.

  7. 7.

    Weisberg (2015), 125.

  8. 8.

    Weisberg (2015), 126. For the proof of this, see fn. 4 in the same.

  9. 9.

    Weisberg (2015), 141. Note that I have replaced E′ with AB in this passage, as well as in the passages that follow.

  10. 10.

    Weisberg (2015), 140–141.

  11. 11.

    Weisberg (2015), 141.

  12. 12.

    In other words, F∧HO slots into Y in the schema from Section 3:

    $$ p\left(\mathrm{X}\ |\ Y\right)=p\left(\mathrm{X}\right) $$
    $$ {p}^{\prime}\left(\mathrm{X}\ |\ Y\right)<{p}^{\prime}\left(\mathrm{X}\right) $$
  13. 13.

    Thanks to an anonymous referee for raising this objection.

  14. 14.

    Note that the above inequalities are not sufficient to establish that rigidity holds in the case at hand. In order to establish that rigidity holds, it must be shown that an agent who maintains these inequalities at t and t′ maintains them in virtue of the fact that they govern the same conditional probabilities at t and t′. In other words, in order to establish that rigidity holds, it must be shown that: p(E | F∧RAB) = p′(E | F∧RAB) and p(E | RAB) = p′(E | RAB).

    The intuitions I have appealed to in the argument above do indeed establish these equalities. I have argued that since the agent’s higher-order beliefs screen off (i.e., make probabilistically irrelevant) the impact of the experience in question in these cases, there is nothing that differentiates the agent’s conditional probabilities at t from her conditional probabilities at t′. Thanks to an anonymous referee for pointing out to me the need to establish this point.

  15. 15.

    Thanks to an anonymous referee for pressing me to address the following issues.

  16. 16.

    See Jeffrey (1965, 1988, 2002). See also Wagner (2013) for how this observation bears on Weisberg’s discussion.

  17. 17.

    For discussion of information minimization rules, see, for instance, Williams (1980), van Frasssen (1981) and Diaconis and Zabell (1982, p. 827–828). Some have resisted the idea that Jeffrey conditionalization should be associated with rigidity and contrasted with information minimization techniques. Diaconis and Zabell (1982, p. 827–828) argue that rigidity and information minimization measures are actually two ways of justifying Jeffrey Conditionalization. Along similar lines, Wagner (2013, n.5) insists that different strategies for requantifying a probability distribution all formally qualify as Jeffrey conditionalization. Wagner takes the difference between rigidity and other means of updating to be in the amount of ‘labor’ they employ.

  18. 18.

    There are rules, other than information minimization rules, that also clearly classify as forms of Bayesian updating (see, for instance, Bradley (2005)). The general response offered in this section to the use of information minimization rules applies to these other rules as well.

  19. 19.

    There is not much discussion in the literature on the connection between the choice of updating rule and the complexity of an agent’s evidence. Nevertheless, it seems reasonable to assume, as many have, that different updating rules (or justifications) will be appropriate for dealing with different types of evidence. For instance, Diaconis and Zabell (1982, p.827) take the labor employed by rigidity to be “subjective” and to involve “introspection”, since updating in this way requires identifying one’s evidence before and after an update. One might hypothesize that this sort of labor is most appropriate for cases involving simple, two-membered partitions—partitions that the agent can easily introspect as her evidence—than for cases that involve more complicated evidence.

  20. 20.

    Weisberg (2015, p. 147) also makes this point. For discussion of the general strategy of updating on a more complicated evidence partition, see Wagner (2013).

  21. 21.

    Sturgeon (2014), 117.

  22. 22.

    Sturgeon (2014), 117.

  23. 23.

    For a description of the Judy Benjamin problem, see again van Frasssen (1981).

  24. 24.

    For a recent account of this, see Schwan and Stern (2017).

  25. 25.

    Weisberg (2009), 5.

  26. 26.

    Weisberg (2009), 1.

  27. 27.

    Since undermining defeat is simply an instance of Jeffrey conditionalization, and Jeffrey conditionalization entails Holism (general) and is compatible with Holism (Weisberg), undermining defeat entails Holism (general) and is compatible with Holism (Weisberg).

  28. 28.

    Weisberg (2009).

  29. 29.

    Weisberg (2015), 122.

  30. 30.

    Thanks to an anonymous referee for this comparison, and for prompting me to address the issues in this section.

  31. 31.

    Carnap wrote this in a correspondence with Jeffrey, which is reprinted in Jeffrey (1975).

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Acknowledgements

Thanks to several anonymous referees for very helpful comments and suggestions. I am especially grateful to Chris Meacham for extensive discussion and detailed written comments on multiple earlier drafts.

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Cassell, L. Higher-Order Beliefs and the Undermining Problem for Bayesianism. Acta Anal 34, 197–213 (2019). https://doi.org/10.1007/s12136-018-0366-3

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Keywords

  • Higher-order Beliefs
  • Bayesian Updating
  • Update Rule
  • Jeffrey Conditionalization
  • Evidential Partition