Extended probabilities in Statistics
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We propose a new, more general definition of extended probability measures. We study their properties and provide a behavioral interpretation. We use them in an inference procedure, whose environment is canonically represented by the probability space (Ω,ℱ,P), when both P and the composition of Ω are unknown. We develop an ex ante analysis – taking place before the statistical analysis requiring knowledge of Ω – in which we progressively learn the true composition of Ω. We describe how to update extended probabilities in this setting, and introduce the concept of lower extended probabilities. We provide two examples in the fields of ecology and opinion dynamics.
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