Characterization of Coherent Conditional Probabilities as a Tool for Their Assessment and Extension

Abstract

A major purpose of this paper is to show the broad import and applicability of the theory of probability as proposed by de Finetti, which differs radically from the usual one (based on a measure-theoretic framework). In particular, with reference to a coherent conditional probability, we prove a characterization theorem, which provides also a useful algorithm for checking coherence of a given assessment. Moreover it allows to deepen and generalise in useful directions de Finetti’s extension theorem (dubbed as “the fundamental theorem of probability”), emphasising its operational aspects in many significant applications.

The role of coherence for handling probabilistic evaluations and independence

“…. ; S k (see, e.g., [5,6]). However, in this construction some atoms of C F can have never positive probability, so we denote by C kþ1 such subset of atoms, and, if it is not empty, we define an unconditional probability P kþ1 in a way that for every C i 2 C kþ1 we have P kþ1 ðC i Þ > 0 and P…”

“…Given a coherent conditional probability P on a family F , the class agreeing P of P is not necessarily unique. However, if F ¼ A Â A 0 , with A an algebra, then the agreeing class is uniquely determined by P (see, e.g., [6]). Definition 3.…”

“…In particular, supposed that P on F is coherent, if F 0 ¼ fEjH g [ F , the coherent values p ¼ P ðEjH Þ are all the values of a suitable closed interval ½p; p ½0; 1, with p p. In [6] a procedure to find the extreme values p; p has been proposed.…”

Nonmonotonic probabilistic reasoning under variable-strength inheritance with overriding

“…Another important approach to probabilistic reasoning with conditional constraints is based on the coherence principle of de Finetti and generalizations of it [7,11,12,13,14,27,28,29,54], or on similar principles that have been adopted for lower and upper probabilities [51,57]. The main tasks in this framework are checking the consistency of a probabilistic assessment, and the propagation of a given assessment to further conditional events.…”

Probabilistic logic under coherence, model-theoretic probabilistic logic, and default reasoning in SystemP

“…In such cases, a general approach is obtained by using (conditional and/or unconditional) probabilistic constraints, based on the coherence principle of de Finetti and suitable generalizations of it [5,8,9,10,11,20,21,22,37], or on similar principles that have been adopted for lower and upper probabilities [36,41]. Two important aspects in dealing with uncertainty are: (i) checking the consistency of a probabilistic assessment, and (ii) the propagation of a given assessment to further uncertain quantities.…”