Abstract
Imprecise probabilism—which holds that rational belief/credence is permissibly represented by a set of probability functions—apparently suffers from a problem known as dilation. We explore whether this problem can be avoided or mitigated by one of the following strategies: (a) modifying the rule by which the credal state is updated, (b) restricting the domain of reasonable credal states to those that preclude dilation.
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Notes
The term “representor” is due to van Fraassen (1990).
This consequence also seems to be in conflict with the weight/balance motivation for imprecise probabilism mentioned earlier.
Throughout the paper it will normally be obvious what rule and representor we are discussing, and so we do not always make this explicit when describing a case of dilation.
Joyce (2011) attributes this term to Adam Elga.
The terminology is from Gilboa and Schmeidler (1993).
We won’t discuss more radical depatures from the standard Bayesian model, such as Kyburg’s Evidential Probabilities model (Kyburg and Teng 2001). Kyburg’s model doesn’t really have a concept of updating: there is simply the rationally permissible belief function given a certain body of evidence.
The updated belief for B is still [0, 1], whatever value \(\mathcal{P}(X)\) takes. For values of \(\mathcal{P}(X)\) not equal to {0.5}, the prior belief for B is imprecise.
We return to restrictions on the priors in the next section.
Recall that classical rules have credence match up, in some sense, with your conditional beliefs. Walley shows that, if these prior conditional attitudes are what govern your conditional betting behaviour, then generalised conditioning is the right sort of update rule to avoid the possibility of a Dutch book being made against you.
Note that \(\mathcal{P}(B|X)\) is a prior conditional belief, not an updated belief. It is the set of values assigned to B conditional on X by the set of prior probabilities \(\mathcal{P}\). That is, \(\mathcal{P}(B|X)=\{\hbox{Pr}(B|X) : \hbox{Pr}\in\mathcal{P}\}\). This is extensionally the same as the updated belief in B after updating on X (assuming generalised conditioning), but we vary the formalism to highlight the conceptual distinction between updated belief and conditional belief.
Pedersen and Wheeler (n.d.) explore a number of cases where the initial model is somehow misdescribed, and once the appropriate redescription is done, certain troubling instances of dilation disappear. These are the cases of improper dilation; where the initial model does not take into account certain kinds of irrelevance that should be built into the model. Our example is not such a case.
Earlier we implicitly assumed that the domain of the update rule was universal: Upd was a function on all of \({2^{\Uppi^{\mathcal{M}}}}\). Now we are considering restricting Upd to some subset of \({2^{\Uppi^{\mathcal{M}}}}\). The hope is that there is some plausible restriction that precludes Upd being vulnerable to dilation.
Note that Joyce does not offer a norm of belief, but rather a definition of epistemic (or “evidential” irrelevance) that is based on the pattern of likelihoods across some evidence partition. Joyce’s definition of irrelevance is different from those discussed by Pedersen and Wheeler (n.d.). Our norm amounts to roughly the following restriction: “If X is irrelevant to B (in Joyce’s sense) then X should be epistemically irrelevant to B (in the sense of Pedersen and Wheeler)”.
Of course, beliefs given X and beliefs given \(\neg X\) will differ with respect to, say, the truth of X. But what is required here is that they amount to the same beliefs about B.
This is not a mathematical truth, of course, but rather a substantial restriction on rational belief functions. This is why we describe it as a norm of belief.
This is a limited version of the reflection principle. It applies just to cases where you know what your beliefs will be in the future, presumably because, whatever the evidence, you will have the same beliefs. We do not deny that more general versions of reflection are interesting and may constrain rational belief, but in the context of our discussion, the limited version of the principle is more pertinent and is already controversial.
Topey (2012) offers some interesting reasons why conditional beliefs of [0, 1] should be treated as identical, and thus why the reflection principle does apply in cases like our urns example.
We owe this example of non-symmetric dilation to Teddy Seidenfeld (in correspondence).
The position we describe here seems to be the consensus view in statistics. Our aim is to present that view to philosophy.
Walley (1991) suggests something in this vein as an attempt to reconcile his readers to dilation.
Note that this deflationary suggestion about dilation does nothing to rehabilitate the “weight of evidence” motivation for imprecise probabilism that we mentioned earlier. This motivation for imprecise probabilism may well be misguided, but note that there are other motivations that imprecise probabilists can appeal to. See, for instance, Joyce (2011).
See Weatherson (n.d.), Joyce (2011) and Bradley and Steele (n.d.a), Bradley and Steele (n.d.b).
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Acknowledgments
Thanks to Patryk Dziurosz-Serafinowicz for helpful comments when serving as discussant of the paper for the Erasmus Institute EIPE Research Seminar. Thanks also to the anonymous referees of this journal for detailed and helpful comments. SB's research supported by the Alexander von Humboldt Foundation.
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Bradley, S., Steele, K. Uncertainty, Learning, and the “Problem” of Dilation. Erkenn 79, 1287–1303 (2014). https://doi.org/10.1007/s10670-013-9529-1
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DOI: https://doi.org/10.1007/s10670-013-9529-1