Abstract
In several papers, John Norton has argued that Bayesianism cannot handle ignorance adequately due to its inability to distinguish between neutral and disconfirming evidence. He argued that this inability sows confusion in, e.g., anthropic reasoning in cosmology or the Doomsday argument, by allowing one to draw unwarranted conclusions from a lack of knowledge. Norton has suggested criteria for a candidate for representation of neutral support. Imprecise credences (families of credal probability functions) constitute a Bayesian-friendly framework that allows us to avoid inadequate neutral priors and better handle ignorance. The imprecise model generally agrees with Norton’s representation of ignorance but requires that his criterion of self-duality be reformulated or abandoned.
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Notes
This assumption is contentious (see, e.g., (Aguirre 2001) for an alternative proposal).
See Weinberg, S. (2000). A priori probability distribution of the cosmological constant. arXiv preprint astro-ph/0002387.
The median value of the distribution obtained by such anthropic prediction is about 20 times the observed value \(\rho _V^{\text {obs}}\), whereas predictions based on existing theories are 120 orders of magnitude higher than the observed value (Pogosian et al. 2004).
(1) For a probability function \(p\), \(\forall \alpha , p(\alpha )\ge 0\); (2) if \(\alpha \) is logically true, then \(p(\alpha )=1\); (3) additivity: if \(\alpha ,\beta \) are incompatible \( (p(\alpha \& \beta )=0)\), then \(p(\alpha \vee \beta )=p(\alpha )+p(\beta ).\) It follows from these laws that \(\forall \alpha , p(\alpha )+p(\lnot \alpha )=1\).
The invariance under redescription only requires that the probability value that corresponds to neutral support for a same event must not depend on how this event is described. For instance, in the example given above in § 1, book length was given in terms of number of words and could be redescribed in terms of number of pages or lines.
Strictly speaking, it is not entirely appropriate to define this condition in terms of additivity. For a representation of credence to be ‘non-additive’ in the sense of interest to Norton here, it has to fulfill the following condition: \(\forall \alpha ,\beta \) incompatible propositions about which we are completely indifferent or ignorant, we can have neither \(p(\alpha \vee \beta )>p(\alpha )\) nor \(p(\alpha \vee \beta )>p(\beta )\).
\(\top \) is an unconditionally true statement, and \(\bot \) an unconditionally false one.
‘Imprecise credence’ is more appropriate than ‘imprecise probability’ since it does not necessarily obey the laws of probability. Here I nevertheless use both expressions interchangeably, as is done in the literature.
Here and later, what is implied is “expectation with respect to credence.”
As discussed in (Cozman 2012), this definition violates convexity.
This requirement corresponds in fact more to a state of indifference than to one of ignorance. Indeed, one may argue that a credal set that gives the set of values \(\{0.1,0.8\}\) is a better representation of ignorance—but not one of indifference—than one that gives the set of values \(\{0.49,0.51\}\). I am here overlooking distinctions between these two notions.
i.e., \(\forall c\in C, c(\top )=1=1-c(\bot ).\)
If the functions in this set are described as Dirichlet distributions, then this criterion will be satisfied (see, e.g., de Cooman et al. 2009).
This remark also applies to non-convex sets.
The following passage makes it clear that Norton thinks of the use of a set of probability functions as allowing the simultaneous representation of several states of belief: “the use of sets renders ignorance as a second order sort of belief. We allow that many different belief-disbelief states are possible. We represent ignorance by presenting them all, in effect saying that we dont know which is the pertinent one.” (Norton 2007a, § 6.2, p. 248).
Unless we are dealing with only two mutually exclusive propositions.
As Gott (1994) recalls, this choice of prior is fairly standard (albeit contentious) in statistical analysis. It is the Jeffreys prior for the unbounded parameter \(N\), such that \(p(N)\,\mathrm {d}N\propto \,\mathrm {d}\ln N\propto \dfrac{\,\mathrm {d}N}{N}\). This means that the probability for \(N\) to be in any logarithmic interval is the same. This prior is called improper because it is not normalizable, and it is usually argued that it is justified when it yields a normalizable posterior.
For reasons expressed earlier in footnotes 14 and 19, this should preferably be done by means of Dirichlet distributions.
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Acknowledgments
I am grateful to Wayne Myrvold for initial discussions, Jim Joyce and John Norton for stimulating exchanges. I am indebted to Chris Smeenk for many comments and suggestions. I also thank an anonymous reviewer for helpful comments.
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Benétreau-Dupin, Y. The Bayesian who knew too much. Synthese 192, 1527–1542 (2015). https://doi.org/10.1007/s11229-014-0647-3
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DOI: https://doi.org/10.1007/s11229-014-0647-3