Abstract
Jeffrey conditioning tells an agent how to update her priors so as to grant a given probability to a particular event. Weighted averaging tells an agent how to update her priors on the basis of testimonial evidence, by changing to a weighted arithmetic mean of her priors and another agent’s priors. We show that, in their respective settings, these two seemingly so different updating rules are axiomatized by essentially the same invariance condition. As a by-product, this sheds new light on the question how weighted averaging should be extended to deal with cases when the other agent reveals only parts of her probability distribution. The combination of weighted averaging (for the events whose probability the other agent reveals) and Jeffrey conditioning (for the events whose probability the other agent does not reveal) is a comprehensive updating rule to deal with such cases, which is again axiomatized by invariance under embedding. We conclude that, even though one may dislike Jeffrey conditioning or weighted averaging, the two make a natural pair when a policy for partial testimonial evidence is needed.
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Notes
We are simplifying a bit: \(A_2\) may well disclose her subjective probabilities for other events, and maybe for all events in the algebra under consideration. But the point is that \(A_2\) may not do that and may only reveal parts of her subjective probabilities.
We leave aside the motivations for such a wish and the question whether perceptual evidence provides us with that kind of information.
The results could be extended to infinite \(\sigma \)-algebras, but we find it interesting that they already hold working just with finite ones.
In the following, varying S will prove to be crucial. This is the reason why we make it explicit, instead of merely working with probability distributions on a fixed algebra, as is commonly done in the literature on opinion pooling.
Continuity applies only to one of the arguments of an updating rule, namely the probability function. It means that, given a sequence of belief states of the form \(\langle S,p_i \rangle \) and a belief state \(\langle S,p \rangle \) such that \(\langle S,p \rangle = \underset{i\rightarrow \infty }{\lim }\langle S,p_i \rangle \), \(F_U\) commutes with limits, that is \(F_U(\langle S,p \rangle ) = \underset{i\rightarrow \infty }{\lim }F_U(\langle S,p_i \rangle )\). \(\langle S,p \rangle = \underset{i\rightarrow \infty }{\lim }\langle S,p_i \rangle \) is short for \(p(A) = \underset{i\rightarrow \infty }{\lim }p_i(A)\), for all \(A \in S\).
We require that \(0< p(A) < 1\) to guarantee that Jeffrey conditioning is well-defined on the whole domain.
This condition is reminiscent of the so-called Success Postulate in AGM-type belief revision.
Invariance under embedding could be unpacked into two distinct requirements of invariance under isomorphism and of invariance under substates (we are thankful to the editor in charge for this remark). This will also show up in Sect. 4, in connection with standard axioms for the aggregation of probability distributions, which similarly distinguish the formality and the locality constraints. In the present paper, we wish to push the intuition, familiar in model theory, of invariance under embedding as a sui generis idea: updatings should be insensitive to moving around the belief state in which they operate.
van Fraassen (1990) appeals to invariance under embedding in the wider context of a vindication of basic laws by symmetry principles. He does not provide a detailed vindication in that particular case, but he points out that, when a rule is invariant under embedding, “we are allowed to switch our attention to a more tractable ’equivalent’ probability space” (p. 334).
Dietrich et al. (2016) have recently proposed a characterization of Jeffrey conditioning in terms of a property called “conservativeness”, that generalizes rigidity. In a nutshell, conservativeness says that the updating process should leave unchanged the probabilities on which the update instruction is “silent”, in that case conditional probabilities.
We are grateful to an anonymous referee for pointing us to Teller and van Fraassen’s work, which we were unaware of when we first up with Theorem 1.
We thank the editor of this issue for having drawn our attention to this interpretive point.
We have formulated (IE) in a fully general form which applies to unrestricted rules. This is not needed for the present Theorem, which deals with restricted rules (dealing with total inputs, that is cases where \(S = S'\) and \(T = T'\)).
Actually, (ZU) is not implied by (N) in Wagner’s framework. The implication holds, however, in McConway’s and ours. This is because the empty set is among the considered events. Thus, as soon as there is 0-unanimity towards some arbitrary event A, the updated degree of belief must be zero in virtue of (N).
p is said to be regular if it assigns non-zero probabilities to all non-empty events.
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Acknowledgements
Earlier versions of this work were presented at seminars and workshops in Paris, Munich and Stockholm, as well as at a symposium at PSA 2014 in Chicago. The present paper greatly benefited from the audiences questions and comments, and special thanks are due to Richard Bradley, Jan-Willem Romeijn and Olivier Roy who were our partners in crime on several of those occasions. We are also particularly thankful to David Etlin, for pointing us to Teller’s work, to the reviewers and the editor of the present journal for their careful reading which made for significant improvements in both content and form, and to the members of the Décision, Rationalité et Interaction team in Paris. We both acknowledge support from the ANR-10-LABX-0087 IEC and ANR-10-IDEX-0001-02 PSL Grants. Mikael Cozic was also supported by the Institut Universitaire de France (Junior Fellowship).
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Bonnay, D., Cozic, M. Weighted averaging, Jeffrey conditioning and invariance. Theory Decis 85, 21–39 (2018). https://doi.org/10.1007/s11238-017-9639-3
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DOI: https://doi.org/10.1007/s11238-017-9639-3