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Belief function representations of statistical evidence. (English) Zbl 0645.62003

In the approach advocated by G. Shafer [see, for example, his book, A mathematical theory of evidence. (1976; Zbl 0359.62002)], the assessment of the evidence concerning \(\theta\) (parametrising a finite set of probability models) provided by a statistical observation and represented by a commonality function Q, is separated from the assessment of prior evidence, represented by a commonality function R. The combination of Q and R then yields a posterior assessment of the total evidence.
In this paper three basic problems for a belief-function model of parametric inference are posed. In the light of these it is concluded that Dempster’s rule is unsuitable for combining evidence from statistically independent observations.
Extension of Q to all subsets of \(\theta\) includes formalised versions of fiducial and likelihood inference as extreme cases, and the class of extensions suggests an alternative to Dempster’s rule. Both of these rules, however, are inadequate in certain cases, and the author concludes that “there are serious objections to any theory of statistical inference which is based on Dempster’s rule of combination” (p. 1441).
Reviewer: A.Dale

MSC:

62A01 Foundations and philosophical topics in statistics
60A05 Axioms; other general questions in probability

Citations:

Zbl 0359.62002
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