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Dominating countably many forecasts. (English) Zbl 1295.62006

Coherence in preferences is associated to the notion that elementary rules on rational preferences can make personal probabilities to satisfy the axioms of mathematical probability. Since de Finetti two senses of coherence have been used: the first one, named coherence\(_1\), requires that probabilistic forecasts for random variables put together in a finite set of fair contracts cannot be uniformly dominated by abstaining. The second one, coherence\(_2\), requires that a finite set of probabilistic forecasts, under Brier scores, cannot be uniformly dominated by a rival set of forecasts. These two definitions are equivalent in the sense that each version of coherence uses expectations of random variables as its forecasts. These expectations are assumed to be based on finitely additive only probabilities. Later, the authors extended this equivalence to a third definition of coherence, named coherence\(_3\), where a large class of strictly proper scoring rules (not just Brier scores) is considered to compare forecasts for general random variables [Int. J. Approx. Reasoning 49, No. 1, 148–158 (2008; Zbl 1185.91087)]. This article studies differences/similarities between coherence\(_1\) and coherence\(_3\) when facing cases where personal probabilities are countably additive and cases where these probabilities are finitely but not countably additive. The first result in this study states that putting together countable many unconditional forecasts based on a finitely additive personal probability may result in a single option that can be dominated by abstaining. Of course, such result is not true when the forecasts are based on a countable additive probability. One second result states that, under certain conditions, strictly proper scoring rules perform a distinct behavior. Summing together countable many coherent\(_3\) unconditional forecasts based on finitely additive probability cannot be dominated by a rival set of forecasts. The other result deals with conditional forecasts based on conditional probabilities, given elements of a partition \(\pi\). Extensions of coherence\(_1\) to include countable sum of individually conditional forecasts and of coherence\(_3\) to include countable sum of strictly proper scores from conditional forecasts hold, if and only if, the involved coherent quantities are based on conditional expectations that are conglomerable in \(\pi\). The authors of this paper proved earlier that, under certain conditions, conglomerability is equivalent to the law of total previsions.

MSC:

62A01 Foundations and philosophical topics in statistics
62C05 General considerations in statistical decision theory

Citations:

Zbl 1185.91087
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References:

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