The Annals of Statistics

Belief Function Representations of Statistical Evidence

Peter Walley

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In Glenn Shafer's theory of parametric statistical inference, observational evidence and prior evidence are separately represented by belief or commonality functions $Q$ and $R$, which are then combined by Dempster's rule. We characterise, for finite parameter spaces, the functionals $Q$ and $R$ for which statistically independent observations may be combined by Dempster's rule, and those for which Dempster's rule is consistent with Bayes' rule. The functionals are determined up to an arbitrary partition of the parameter space and an arbitrary scale parameter, which might be chosen to reflect aspects of the evidence on which the statistical model is based. Our results suggest that Dempster's rule is not generally suitable for combining evidence from independent observations nor for combining prior beliefs with observational evidence.

Article information

Ann. Statist., Volume 15, Number 4 (1987), 1439-1465.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62A99: None of the above, but in this section
Secondary: 60A05: Axioms; other general questions

Belief functions Dempster's rule Bayesian inference likelihood likelihood principle prior probabilities Bayes' rule upper and lower probabilities


Walley, Peter. Belief Function Representations of Statistical Evidence. Ann. Statist. 15 (1987), no. 4, 1439--1465. doi:10.1214/aos/1176350603.

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