Nonadditive probabilities in statistics

Frank Hampel

December 2007

Abstract:

The paper gives a short survey about the occurrence (sometimes hidden in the background) of nonadditive probabilities in statistics. It starts with the original meaning of "probability'' in statistics in the Ars conjectandi by Jakob (James) Bernoulli, and the ensuing misunderstanding which gave the term its present meaning. One chapter is about robustness theory, its use of (nonadditive) Choquet-capacities, and an attempt to clarify some widespread misunderstandings about it, which have consequences for the use of upper and lower probabilities. Also the uncertainty about model choice (including the conflict between purely mathematical reasoning and good statistical practice) and treatment of outliers is briefly discussed. The partial arbitrariness of additivity both in Bayes' famous Scholium and in modern Bayes theory is outlined. The infamous and almost forgotten fiducial probabilities can actually be corrected and find their place in a more general paradigm using upper and lower probabilities. Finally, a new (?) qualitative theory of inference is mentioned which (hopefully) contains some essentials of inductive reasoning in real life.

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