Abstract
According to Williamson (In defense of objective Bayesianism, Oxford University Press, Oxford, 2010), the difference between empirical subjective Bayesians and objective Bayesians is that, while both hold reasonable credence to be calibrated to evidence, the objectivist also takes such credence to be as equivocal as such calibration allows. However, Williamson’s prescription for equivocation generates constraints on reasonable credence that are objectionable. Herein Williamson’s calibration norm is explicated in a novel way that permits an alternative equivocation norm. On this alternative account, evidence calibrated probability functions are recognised as implications of evidence calibrated density functions defined over chance hypotheses. The objective Bayesian equivocates between these calibrated density functions rather than between the calibrated probability functions themselves. The result is an objective Bayesianism that avoids the main problem afflicting Williamson’s original proposal.
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Notes
The indefinite/definite terminology is preferred by Pollock (1984).
Obviously, in those atypical circumstances where there is no mismatch—either because the credence to be calibrated is generic, or because the objective probability is single case—the problem does not arise.
The texts cited here are just those of particular note in this area, and should not be taken to be exhaustive of either the authors contributions or of the authors who have contributed.
Because this paper’s subject is calibration and equivocation norms and not the development or critique of evidential probability theory variants, the examples given here are deliberately terse. The interested reader is directed to the works of Kyburg (2006) and Wheeler and Williamson (2011) for further detail.
Among other things, this doubly protects Williamson from Levi’s (1981) complaint that direct inference is incompatible with synchronic, and diachronic, conditionalisation. He is also protected on this score as his protocol endorses the minor chance premises Levi argues are required for compatibility of direct inference with conditionalisation.
Both Williamson and I freely allow that there may be other types of evidence and that these will place their own constraints on reasonable credence. Hence we allow that the calibration norm as explicated herein may be incomplete. This does not make what is presented here unsound, it merely means that it may not be the full story on calibration.
Wheeler (2012) has a superficially similar argument that Williamson’s prescription is not only insensitive to certain changes in evidence but that this insensitivity leaves that agent vulnerable to patently unreasonable betting behaviour. We need not go into the details of this argument because it is built upon a false premise; namely, that Williamson’s agent learns in some dynamic fashion. However, Wheeler’s argument does point to the genuine potential problem raised here: Williamson’s Straight Rule does imply a certain amount of insensitivity to evidence.
As noted earlier, though Miller’s Principle is commonly accepted it is not accepted by Williamson due to his commitment to the chance as ultimate belief thesis.
Pettigrew (2013) shows that a variant of Miller’s Principle can be motivated in much the same fashion as Williamson motivates his Straight Rule.
Williamson’s motivation of his equivocation norm is that in the absence of any information to the contrary we may assume that loss is a logarithmic. This assumption—originally from, and argued for in, Grünwald and Dawid (2004)—can be formalised as follows: In absence of any information to the contrary, where \(w\) is a proposition describing the true state of the world, \(C\) is a reasonable credence function and \(L(C,w)\) is the loss function, then \(L(C,w)=-log(C(w)).\) On this assumption, the highest entropy distributions—those that are most equivocal—minimise worst case expected loss Williamson(2010, pp.64–5).
As \(\theta \)’s role in the argument of the chance function—that of being a proper name of a sentence of the language and nothing more—is purely referential, so we can avail ourselves of Kaplan’s (1969) approach to quantifying into referentially opaque contexts to ensure the sense of this specification of the evidence.
These density distributions are defined as a mapping from the reals to the positive reals in order to allow for dirac distributions. Were evidence of chances never to include sharp valued chances—e.g., \(ch(\theta )=0.1\)—then these densities could be defined as \(\rho _{\mathcal {E}}^{\ \cdot }(x):\mathcal {SL}\times [0,1]\mapsto [0,\infty ]\).
By a theorem by Skyrms (1988); if \(P_{\mathcal {E}}(\theta |ch(\theta )=x)=x\), then \(P_{\mathcal {E}}(\theta |ch(\theta |\psi )=x,\psi )=x\). A further principle of conditionalizing out is often adhered to so that \(P_{\mathcal {E}}(ch(\theta |\psi )=y|\psi )=P_{\mathcal {E}}(ch(\theta |\psi )=y)\). Finally, we can also define a conditional evidence calibrated density function.
$$\begin{aligned} \rho _{\mathcal {E}}^{\theta |\psi }(y)&:\ \mathcal {SL}\times \mathcal {SL}\times [-\infty ,\infty ]\mapsto [0,\infty ],\\ \rho _{\mathcal {E}}^{\theta |\psi }(y)&=0,\ \text{ where } y\not \in [0,1],\\ \rho _{\mathcal {E}}^{\theta |\psi }(y)&=0,\ \text{ where } y\not \in \mathbb {T}_{\mathcal {E}}^{\theta |\psi }, \text{ and } \text{ is } \text{ such } \text{ that } P_{\mathcal {E}}^{\mathcal {L}^+}\in \langle \mathbb {P}^{*}\cap \mathbb {S}\rangle \\ P_{\mathcal {E}}(ch(\theta |\psi )\in [a,b])&:=\int \limits _{a}^{b}\rho _{\mathcal {E}}^{\theta |\psi }(y)dy,\\ 1&=\int \limits _{a}^{b}\rho _{\mathcal {E}}^{\theta |\psi }(y)dy. \end{aligned}$$Putting all this together we could have the following more general specification of the set of evidence calibrated density functions:
$$\begin{aligned} \mathbb {C}_{\mathcal {E}}^{\mathcal {L}}=\left\{ \rho _{\mathcal {E}}^{\mathcal {L}}:P_{\mathcal {E}}^{\mathcal {L}^+}(\cdot |\cdot )=\int \limits _{-\infty }^{\infty }x\rho _{\mathcal {E}}^{\ \cdot |\cdot }(x)dx,\ \int \limits _{\mathbb {T}_{\mathcal {E}}^{\ \cdot |\cdot }\in \mathbb {T}_{\mathcal {E}}^{\mathcal {L}}}\rho _{\mathcal {E}}^{\ \cdot |\cdot }(x)dx = 1,\ P_{\mathcal {E}}^{\mathcal {L}^+}\in \langle \mathbb {P}^{*}\cap \mathbb {S}\rangle \ne \varnothing \right\} . \end{aligned}$$This suggests that one could generalise the specification to conditional probabilities; however, as these are defined in terms of marginal probabilities the original specification is sufficient.
As \(x\) ranges over chances of \(\theta \), \(\int \limits _{-\infty }^{\infty }x\rho _{\mathcal {E}}^{\theta }(x)dx\) is the expectation of the chance of \(\theta \) relative to \(\rho _{\mathcal {E}}^{\theta }(x)\).
All such proofs are based on a theorem by Boltzmann that implies, when applied to densities over chances, that—as there is an \(a\in [0,1]\) that is the expectation of the chance in question, which is a consequence of Miller’s Principle as demonstrated in Sect. 3—if there is a maximum entropy density, then it has the form \(\rho (x)=c\cdot e^{\lambda x}\) where \(\lambda \) and \(c\) are given by constraints \(\int _{-\infty }^{\infty }x\rho (x)dx=a\) and \(1=\int _{-\infty }^{\infty }\rho (x)dx\) (Park and Bera 2009). This implies that the continuous distribution over chances with maximum entropy when no constraints are in place is always a uniform density.
I have been unable to find any proof of this in the literature; however, as it is certainly the case that the entropy of a discrete distribution with all its probability mass located at a point is zero, so I shall assume that the same is true of the continuous delta distributions. If I err in this, then at least I am not alone in doing so.
For instance, this protocol would treat \(\mathcal {E}=\{ch^{\mathcal {L}}:ch(\theta )\in [a,b]\}\) and \(\mathcal {E'}=\{ch^{\mathcal {L}}:ch(\theta )\in [a,b)\cup \{b+\epsilon \}, 0\approx \epsilon >0\}\) very differently, giving a reasonable credence in \(\theta \) of \(\frac{a+b}{2}\) in the first case and \(\frac{1}{2}\left( \frac{a+b}{2}+b+\epsilon \right) \) in the second. Intuitively, these reasonable credences should be almost identical, so doubt is cast on the protocol.
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Acknowledgments
With thanks to the Swedish Research council for funding this paper as part of the postdoc project Objective Degrees of Belief. I also wish to thank the Erkenntnis reviewers for all their help in bringing this paper to fruition; in particular, the 4th reviewer who deserves a special acknowledgement for all their efforts on my behalf.
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Masterton, G. Equivocation for the Objective Bayesian. Erkenn 80, 403–432 (2015). https://doi.org/10.1007/s10670-014-9649-2
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DOI: https://doi.org/10.1007/s10670-014-9649-2