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A Graded Bayesian Coherence Notion

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Abstract

Coherence is a key concept in many accounts of epistemic justification within ‘traditional’ analytic epistemology. Within formal epistemology, too, there is a substantial body of research on coherence measures. However, there has been surprisingly little interaction between the two bodies of literature. The reason is that the existing formal literature on coherence measure operates with a notion of belief system that is very different from—what we argue is—a natural Bayesian formalisation of the concept of belief system from traditional epistemology. Therefore, formal epistemology has so far only been concerned with one particular—arguably not even very natural—way of formalising coherence of belief systems; it has by no means refuted the viability of coherentism. In contrast to the existing literature, we formalise belief systems as families of assignments of (conditional) degrees of belief (which may be compatible with several subjective probability measures). Within this framework, we propose a Bayesian formalisation of the thrust of BonJour’s coherence concept in The structure of empirical knowledge (Harvard University Press, Cambridge, 1985), using a combination of Bayesian confirmation theory and basic graph theory. In excursions, we introduce graded notions for both logical and probabilistic consistency of belief systems—the latter being based on certain geometrical structures induced by probabilistic belief systems. For illustration, we reconsider BonJour’s “ravens” challenge (op. cit., p. 95f.). Finally, potential objections to our proposed formal coherence notion are explored.

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Article 22 July 2014

Jeffrey Sanford Russell, John Hawthorne & Lara Buchak

Notes

  1. For some well-known papers, cf. e.g. Shogenji (1999), Akiba (2000) and Fitelson (2003)—as well as Dietrich and Moretti (2005) for a systematic comparison.

  2. Cf. e.g. Klein and Warfield (1994, 1996), Bovens and Hartmann (2003, 2005, 2006), Olsson (2002, 2005a, b); for a critical assessment of those “impossibility results”, cf. Meijs and Douven (2007) and Schupbach (2008). That coherence measures generally fail to satisfy a rather intuitive criterion has also been demonstrated by Siebel and Wolff (2008).

  3. Douven and Meijs (2006) represent an exception to this rule. They propose a quantitative theory of coherence in the framework of a quantitative bootstrap confirmation theory à la Glymour (1980). But in this setting, the theories to be confirmed consist of propositions given some prior. Therefore, even the theories in Douven and Meijs’ (2006) paper are not belief systems as Bayesians would normally understand them.

  4. One may wonder where such an understanding of sets of beliefs is coming from. However, unless one is a Bayesian, it is very natural to identify beliefs with propositions and thus belief sets with sets of propositions. Moreover, as Hansson and Olsson (1999) have pointed out, there is a long-standing association of the AGM theory of belief revision due to Alchourrón et al. (1985) with coherentism.

  5. Note that the expression “in proportion” in BonJour’s desiderata (II) and (V) need not be understood in a literal manner (as if it required linear dependence). Otherwise, BonJour would have presupposed that coherence-salient qualities of a doxastic system such as the “degree of probabilistic consistency” or the “presence of unexplained anomalies in the believed content of the system” admit a canonical quantification with values in a vector space.

  6. There is, of course, an ongoing discussion in formal epistemology about how to optimally defend probabilism, cf. e.g. Joyce (2009), Easwaran and Fitelson (2012), Fitelson and McCarthy (2012), Leitgeb and Pettigrew (2010a, b), Wedgwood (2012); reviewing this discussion is, however, not within the scope of this paper.

  7. We write “essentially” because even this framework will under some conditions entail the assignment of interval probabilities to certain propositions/events: The monotonicity of probability measures always implies P(B|C) ≥ α for all α, A, B, C with P(A|C) = α and \(A\subseteq B\). Therefore, whenever B belongs to the algebra generated by the propositions featuring in \(\mathcal{S}\) and both \(\langle A|C\|\alpha\rangle\in\mathcal{S}\) and \(A\subseteq B\) holds, then one will have P(B|C) ≥ α for all P compatible with the belief system \(\mathcal{S}\). In other words, any probability measure P compatible with \(\mathcal{S}\) will assign a probability ≥α to the conditional event B-given-C.

  8. In the sense of Walley (1991, 2000) or Weichselberger (2000).

  9. Another possible name for the compatible-with relation would be “support”, cf. Herzberg (2013). This term, however, is already defined in measure theory, whence we shall not use it in the sense of the compatible-with relation lest we create confusion.

  10. This is what de Finetti (1937, 1970, 1974) already calls coherence.

  11. Cf. e.g. the classical paper by Stone (1936, Theorem 12).

  12. Statements that can be interpreted as expressing or at least supporting this thesis occur repeatedly in Locke’s Essay (1979, Book IV, Chapters XIV–XVII).

  13. One might perhaps wonder whether \(\tilde\beta_1(\mathcal{S})\) equals the somewhat simpler expression

    $$ \bar \beta_1({\mathcal{S}}):=1/\inf\left\{c\in[0, 1] : \exists \omega\in\Upomega\quad \bigcap_{A\in{\mathcal{A}} \atop \forall P\in{\mathcal{P}}_{\mathcal{S}} P(A)>c}A=\{\omega\}\right\}. $$

    To be sure,

    $$ \left\{c\in[0, 1] : \exists \omega\in\Upomega \bigcap_{A\in{\mathcal{A}} \atop \forall P\in{\mathcal{P}}_{\mathcal{S}} P(A)>c}A=\{\omega\}\right\}\subseteq \left\{c\in[0, 1] : \bigcap_{\mathop{A\in{\mathcal{A}}}\limits_{\forall P\in{\mathcal{P}}_{\mathcal{S}} P(A)>c}}A \neq\varnothing\right\}, $$

    so the infimum of the former set is greater or equal than that of the latter. The converse estimate, however, need not hold: For, it may happen that \(\bigcap_{A\in\mathcal{A} \atop \forall P\in\mathcal{P}_\mathcal{S} P(A)>c}A=\left\{\omega_1,\ldots,\omega_m\right\}\) with pairwise distinct \(\omega_1,\ldots,\omega_m\) while P{ω1} = P{ω i } for all \(P\in\mathcal{P}_\mathcal{S}\) and i ≤ m. Then, for any \(c^{\prime}<c\) for which \(\omega_1\not\in\bigcap\left\{A\in\mathcal{A} : \forall P\in\mathcal{P}_\mathcal{S} \quad P(A)>c^{\prime}\right\}\), one also has \(\omega_i\not\in\bigcap\left\{A\in\mathcal{A} : \forall P\in\mathcal{P}_\mathcal{S} \quad P(A)>c^{\prime}\right\}\) for all other i ≤ m, all the while \(\bigcap\left\{A\in\mathcal{A} : \forall P\in\mathcal{P}_\mathcal{S} \quad P(A)>c^{\prime}\right\}\subseteq \left\{\omega_1,\ldots,\omega_m\right\}\) and thus \(\bigcap\left\{A\in\mathcal{A} : \forall P\in\mathcal{P}_\mathcal{S} \quad P(A)>c^{\prime}\right\}=\varnothing\). For example, if \(\mathcal{S}\) is the belief system of ignorance, i.e. \(\mathcal{P}_\mathcal{S}\) is the singleton consisting of the uniform distribution on a finite set \(\Upomega\) of states of the world, then \(0=\bar\beta_1(\mathcal{S})< \tilde\beta_1(\mathcal{S})=1\).

  14. In very liberal belief systems—viz. those that are compatible with several extreme probability measures that assign probability zero to different states of the world—it may happen that \(\eta\left(\mathcal{S},\{1\}\right)=\varnothing\) and therefore a fortiori \(\eta\left(\mathcal{S},(c,1]\right)=\varnothing\) for all c, which entails \(\beta_1(\mathcal{S})=0\).

  15. In other words, if the values of \(\tilde\beta_2\) are used to measure probabilistic consistency, they should to be read in lexicographic order, so that the first component of \(\tilde\beta_2\) always trumps the second one.

  16. The graph-theoretic terminology is taken from Diestel (2010).

  17. A collection of paths is independent if and only if none of the paths in that collection contains an inner vertex of another path from that collection.

  18. For example, the assignment of a probability of α1 to the conditional event R 3|R 2 requires that the probability of R 2 ∩ R 3 equals α1 times the probability of R 2. The former proposition corresponds to \(\left\{\left\langle \dot R_2,\dot R_3\right\rangle\right\}=\left\{\omega^{(1)}\right\}\), the latter proposition to \(\left\{\left\langle\dot R_2,\dot R_3\right\rangle, \left\langle \dot R_2,\dot\neg \dot R_3\right\rangle\right\}=\left\{\omega^{(1)},\omega^{(2)}\right\}\). Therefore, the conditional subjective probability assignment \(\left\langle R_3|R_2\|\alpha_1\right\rangle\) from \(\mathcal{S}\) corresponds to the equation \(x^{(1)}=\left(x^{(1)}+x^{(2)}\right)\alpha_1\) in the above formula for \(\iota[\mathcal{P}_\mathcal{S}]\). In a similar vein, the conditional subjective probability assignment \(\left\langle R_2|\Upomega\|\alpha_2\right\rangle\) from \(\mathcal{S}\) corresponds to the equation x (1) + x (2) = α2 in the above formula for \(\iota[\mathcal{P}_\mathcal{S}]\), and the degree of belief \(\left\langle R_3|\Upomega\|\alpha_3\right\rangle\) corresponds to the equation x (1) + x (3) = α3.

  19. By elementary linear algebra, one first gets

    $$ \iota[{\mathcal{P}}_{{\mathcal{S}}^{\prime}}]= \left\{x\in [0,1]^8 : \begin{array}{c} x^{(1)}+\cdots+x^{(8)}=1, \\ x^{(1)}+x^{(2)}+x^{(3)}+x^{(4)}=\alpha_1^{\prime}, \\ x^{(1)}+x^{(2)}=\alpha_1^{\prime}\alpha_2^{\prime}, \\ x^{(1)}+x^{(3)}=\alpha_1^{\prime}\alpha_3^{\prime}, \\ x^{(1)}+x^{(2)}+x^{(5)}+x^{(6)}=\alpha_2^{\prime}, \\ x^{(1)}+x^{(2)}=\alpha_1^{\prime}\alpha_2^{\prime}, \\ x^{(1)}+x^{(5)}=\alpha_2^{\prime}\alpha_3^{\prime}, \\ x^{(1)}+x^{(3)}+x^{(5)}+x^{(7)}=\alpha_3^{\prime}, \\ x^{(1)}+x^{(3)}=\alpha_1^{\prime}\alpha_3^{\prime}, \\ x^{(1)}+x^{(5)}=\alpha_2^{\prime}\alpha_3^{\prime}. \end{array}\right\} \\ $$

    which after dropping repeated equations reduces to

    $$ \iota[{\mathcal{P}}_{{\mathcal{S}}^{\prime}}]= \left\{x\in [0,1]^8 : \begin{array}{c} x^{(1)}+\cdots+x^{(8)}=1, \\ x^{(1)}+x^{(2)}+x^{(3)}+x^{(4)}=\alpha_1^{\prime}, \\ x^{(1)}+x^{(2)}=\alpha_1^{\prime}\alpha_2^{\prime}, \\ x^{(1)}+x^{(3)}=\alpha_1^{\prime}\alpha_3^{\prime}, \\ x^{(1)}+x^{(2)}+x^{(5)}+x^{(6)}=\alpha_2^{\prime}, \\ x^{(1)}+x^{(3)}+x^{(5)}+x^{(7)}=\alpha_3^{\prime}, \\ x^{(1)}+x^{(5)}=\alpha_2^{\prime}\alpha_3^{\prime}. \end{array}\right\} \\ $$

    and finally, solving for x (1) and inserting the solutions,

    $$ \iota[{\mathcal{P}}_{{\mathcal{S}}^{\prime}}]= \left\{x\in [0,1]^8 : \begin{array}{c} x^{(2)}=\alpha_1^{\prime}\alpha_2^{\prime}-x^{(1)}, \\ x^{(3)}=\alpha_1^{\prime}\alpha_3^{\prime}-x^{(1)}, \\ x^{(4)}=\alpha_1^{\prime}-\alpha_1^{\prime}\alpha_2^{\prime}-\alpha_1^{\prime} \alpha_3^{\prime}+x^{(1)}, \\ x^{(5)}=\alpha_2^{\prime}\alpha_3^{\prime}-x^{(1)}, \\ x^{(6)}=\alpha_2^{\prime}-\alpha_1^{\prime}\alpha_2^{\prime}-\alpha_2^{\prime} \alpha_3^{\prime}+x^{(1)}, \\ x^{(7)}=\alpha_3^{\prime} -\alpha_1^{\prime}\alpha_3^{\prime}-\alpha_2^{\prime} \alpha_3^{\prime}+x^{(1)}, \\ x^{(8)}=1-\alpha_1^{\prime}-\alpha_2^{\prime}+\alpha_1^{\prime}\alpha_2^{\prime} +\alpha_2^{\prime}\alpha_3^{\prime}- \alpha_3^{\prime} +\alpha_1^{\prime}\alpha_3^{\prime}-x^{(1)} \end{array}\right\} \\ $$

  20. Very recently, Hájek (2012) has formulated (i) a powerful challenge to a core tenet of Bayesianism, viz. regularity, and (ii) D.W. Miller has offered a vigorous critique of Bayesian confirmation theory on account of its use of the relevance measure of confirmation (partly unpublished, but summarised in Miller (2013) and building upon the joint paper by Popper and Miller (1987)). These are, of course, serious challenges to the overall project of a revived Bayesian coherentism, but a rebuttal would be way beyond its scope. We shall respond to both challenges in two forthcoming papers.

    Another complaint against our graded coherence notion might be (iii) that our proposed formalisation could assign a non-minimal degree of coherence even to belief systems \(\mathcal{S}\) that encode infinite regresses of probabilistic justification, provided \({\mathcal{P}_{\mathcal{S}}}\) is sufficiently large. In response, we point to a recent result by Herzberg (2010, 2013) that gives a criterion for the probabilistic consistency simpliciter of certain probabilistic regresses of epistemic justification which in turn can be used to refute influential arguments against regresses of epistemic justification.

    Moreover, a critic might argue (iv) that our use of Robinsonian nonstandard analysis should subject us to a methodological criticism, since nonstandard analysis is allegedly fundamentally non-constructive (not just in a technical intuitionist sense). This objection can relatively easily be countered by referring to the metamathematical justification of nonstandard analysis (in terms of definable nonstandard models of the reals and entire nonstandard enlargements) contained in Kanovei and Shelah (2004), Kanovei and Reeken (2004), and Herzberg (2008, 2008).

  21. Except perhaps for the requirement that a sufficiently high degree of partial epistemic justification should constitute a approximation—for pragmatic purposes—of full epistemic justification.

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Acknowledgments

This work was financially supported by the Alexander von Humboldt Foundation through a Visiting Fellowship of the Munich Center for Mathematical Philosophy at Ludwig Maximilian University of Munich. I am deeply grateful to Professor Hannes Leitgeb and Professor Stephan Hartmann for very helpful discussions and comments on an earlier version of this paper. Moreover, I would like to thank Professors David McCarthy, Julian Nida-Rümelin, Ted Poston, Günter Zöller and not least two anonymous referees for their suggestions and comments.

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  1. Chair of Logic and Philosophy of Language, Munich Center for Mathematical Philosophy, Faculty of Philosophy, Philosophy of Science and Study of Religion, Ludwig Maximilian University of Munich, Geschwister-Scholl-Platz 1, 80539, Munich, Germany

    Frederik Herzberg

  2. Center for Mathematical Economics, Bielefeld University, Universitätsstraße 25, 33615, Bielefeld, Germany

    Frederik Herzberg

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Herzberg, F. A Graded Bayesian Coherence Notion. Erkenn 79, 843–869 (2014). https://doi.org/10.1007/s10670-013-9569-6

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