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- Carl Wagner, Jeffrey Conditioning and External Bayesianity.Abstract. Suppose that several individuals who have separately assessed prior probability distributions over a set of possible states of the world wish to pool their individual distributions into a single group distribution, while taking into account jointly perceived new evidence. They have the option of (i) first updating their individual priors and then pooling the resulting posteriors or (ii) first pooling their priors and then updating the resulting group prior. If the pooling method that they employ is such that they arrive at the same final distribution in both cases, the method is said to be externally Bayesian, a property first studied by Madansky (1964). We show that a pooling method for discrete distributions is externally Bayesian if and only if it commutes with Jeffrey conditioning, parameterized in terms of certain ratios of new to old odds, as in Wagner (2002), rather than in terms of the posterior probabilities of members of the disjoint family of events on which such conditioning originates.
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Since Ramsey, much discussion of the relation between probability and belief has taken for granted that there are degrees of belief, i.e., that there is a real-valued function, B, that characterizes the degree of belief that an agent has in each statement of his language. It is then supposed that B is a probability. It is then often supposed that as the agent accumulates evidence, this function should be updated by conditioning: BE(·) should be B(·E)/B(E). Probability is also important in classical statistics, where it is generally supposed that probabilities are frequencies, and that inference proceeds by controlling error and not by conditioning. I will focus on the tension between these two approaches to probability, and in the main part of the paper show where and when Bayesian conditioning conflicts with error based statistics and how to resolve these conflicts.
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| Export citation | Other links: journals.uchicago.edu dx.doi.org | Scholar | At my library | More options ...
| | Other links: journals.uchicago.edu dx.doi.org | Scholar | At my library | More options ... A simple rule of probability revision ensures that the final result of a sequence of probability revisions is undisturbed by an alteration in the temporal order of the learning prompting those revisions. This Uniformity Rule dictates that identical learning be reflected in identical ratios of certain new-to-old odds, and is grounded in the old Bayesian idea that such ratios represent what is learned from new experience alone, with prior probabilities factored out. The main theorem of this paper includes as special cases (i) Field's theorem on commuting probability-kinematical revisions and (ii) the equivalence of two strategies for generalizing Jeffrey's solution to the old evidence problem to the case of uncertain old evidence and probabilistic new explanation.
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| | Scholar | At my library | More options ... Jeffrey updating is a natural extension of Bayesian updating to cases where the evidence is uncertain. But, the resulting degrees of belief appear to be sensitive to the order in which the uncertain evidence is acquired, a rather un-Bayesian looking effect. This order dependence results from the way in which basic Jeffrey updating is usually extended to sequences of updates. The usual extension seems very natural, but there are other plausible ways to extend Bayesian updating that maintain order-independence. I will explore three models of sequential updating, the usual extension and two alternatives. I will show that the alternative updating schemes derive from extensions of the usual rigidity requirement, which is at the heart of Jeffrey updating. Finally, I will establish necessary and sufficient conditions for order-independent updating, and show that extended rigidity is closely related to these conditions.
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| Discuss | Edit | Categorize | My bibliography | Export citation | Other links: dx.doi.org faculty-staff.ou.edu jstor.org | Scholar | At my library | More options ...
| | Other links: dx.doi.org faculty-staff.ou.edu jstor.org | Scholar | At my library | More options ... Jeffrey updating is a natural extension of Bayesian updating to cases where the evidence is uncertain. But, the resulting degrees of belief appear to be sensitive to the order in which the uncertain evidence is acquired, a rather un-Bayesian looking effect. This order dependence results from the way in which basic Jeffrey updating is usually extended to sequences of updates. The usual extension seems very natural, but there are other plausible ways to extend Bayesian updating that maintain order-independence. I will explore three models of sequential updating, the usual extension and two alternatives. I will show that the alternative updating schemes derive from extensions of the usual rigidity requirement, which is at the heart of Jeffrey updating. Finally, I will establish necessary and sufficient conditions for order-independent updating, and show that extended rigidity is closely related to these conditions.
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| Discuss | Edit | Categorize | My bibliography | Export citation | Other links: springerlink.com dx.doi.org jstor.org | Scholar | At my library | More options ...
| | Other links: springerlink.com dx.doi.org jstor.org | Scholar | At my library | More options ... If a group as a whole is modelled as a single Bayesian agent, what should its beliefs be? I propose an axiomatic model that connects group beliefs to beliefs of group members, who are themselves modelled as Bayesian agents, possibly with di¤erent priors and di¤erent information. Group beliefs are shown to take a simple multiplicative form if people’s information is independent, and a more complex form if information can overlap arbitrarily. This shows that group beliefs can incorporate all information spread over the individuals without the individuals having to communicate their (possibly complex and hard-to-describe) private information; communicating prior and posterior beliefs su¢ ces.
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| | Scholar | More options ... Weighted averaging is a method for aggregating the totality of information, both regimented and unregimented, possessed by an individual or group of individuals. The application of such a method may be warranted by a theorem of the calculus of probability, simple conditionalization, or Jeffrey's formula for probability kinematics, all of which average in terms of the prior probability of evidence statements. Weighted averaging may, however, be applied as a method of rational aggregation of the probabilities of diverse perspectives or persons in cases in which the weights cannot be articulated as the prior probabilities of statements of evidence. The method is justified by Wagner's Theorem exhibiting that any method satisfying the conditions of the Irrelevance of Alternatives and Zero Unanimity must, when applied to three or more alternatives, be weighted averaging.
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| | Other links: jstor.org | Scholar | At my library | More options ... Richard Jeffrey's generalization of Bayes' rule of conditioning follows, within the theory of belief functions, from Dempster's rule of combination and the rule of minimal extension. Both Jeffrey's rule and the theory of belief functions can and should be construed constructively, rather than normatively or descriptively. The theory of belief functions gives a more thorough analysis of how beliefs might be constructed than Jeffrey's rule does. The inadequacy of Bayesian conditioning is much more general than Jeffrey's examples of uncertain perception might suggest. The ``parameter α '' that Hartry Field has introduced into Jeffrey's rule corresponds to the "weight of evidence" of the theory of belief functions.
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| | Other links: jstor.org journals.uchicago.edu dx.doi.org | Scholar | At my library | More options ... Jeffrey (1983) proposed a generalization of conditioning as a means of updating probability distributions when new evidence drives no event to certainty. His rule requires the stability of certain conditional probabilities through time. We tested this assumption (“invariance”) from the psychological point of view. In Experiment 1 participants offered probability estimates for events in Jeffrey’s candlelight example. Two further scenarios were investigated in Experiment 2, one in which invariance seems justified, the other in which it does not. Results were in rough conformity to Jeffrey (1983)’s principle.
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| | Scholar | More options ... It has often been recommended that the differing probability distributions of a group of experts should be reconciled in such a way as to preserve each instance of independence common to all of their distributions. When probability pooling is subject to a universal domain condition, along with state-wise aggregation, there are severe limitations on implementing this recommendation. In particular, when the individuals are epistemic peers whose probability assessments are to be accorded equal weight, universal preservation of independence is, with a few exceptions, impossible. Under more reasonable restrictions on pooling, however, there is a natural method of preserving the independence of any fixed finite family of countable partitions, and hence of any fixed finite family of discrete random variables.
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| | Other links: springerlink.com dx.doi.org | Scholar | At my library | More options ... Jeffrey conditioning allows updating in Bayesian style when the evidence is uncertain. A weighted average, essentially, over classically updating on the alternatives. Unlike classical Bayesian conditioning, this allows learning to be unlearned.
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| | Other links: stevepetersen.net | Scholar | More options ... Discussion of Carl Wagner, Jeffrey conditioning and external bayesianity
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