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2020 | OriginalPaper | Buchkapitel

Distorted Probabilities and Bayesian Confirmation Measures

verfasst von : Andrea Ellero, Paola Ferretti

Erschienen in: Modeling Decisions for Artificial Intelligence

Verlag: Springer International Publishing

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Abstract

Bayesian Confirmation Measures (BCMs) assess the impact of the occurrence of one event on the credibility of another. Many measures of this kind have been defined in literature. We want to analyze how these measures change when the probabilities involved in their computation are distorted. Composing distortions and BCMs we define a set of Distorted Bayesian Confirmation Measures (DBCMs); we study the properties that DBCMs may inherit from BCMs, and propose a way to measure the degree of distortion of a DBCM with respect to a corresponding BCM.

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Vorheriges Kapitel Probabilistic Measures and Integrals: How to Aggregate Imprecise Data

Nächstes Kapitel Constructive k-Additive Measure and Decreasing Convergence Theorems

Throughout the paper, the formulas are assumed to be well defined, e.g. denominators do not vanish, as implicitly done in the literature.

Feasibility of (x, y, z) requires that probabilities x, y and z satisfy in particular the Total Probability Theorem.

A BCM that satisfies all the above considered symmetries is Carnap’s b [1] which is defined as \( b(H,E)=P(E \cap H)-P(E) \, P(H). \)

Most of the literature reports in fact the measure as \(R(H,E)=1-[P(\lnot H |E)/P(\lnot H)]\), see, e.g., [5].

Computations in this paper were performed with Wolfram’s software Mathematica (version 11.0.1.0).

Moreover, depending on the chosen DBCM, some of the above recalled inequalities may be required to be strict.

Note that in the case of \(R_g\), confirmation and disconfirmation cases are considered separately. We decided to split the computation of the \(L_2\) norm in the latter case, due to the fact that in some cases (distortions \(g_3\) and \(g_4\)) the value of \({\mu }^p(C,g)\) diverges on its whole domain, but converges in the subset where \(P(H|E)>P(H)\), i.e., the confirmation case.

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Titel: Distorted Probabilities and Bayesian Confirmation Measures
verfasst von: Andrea Ellero
Paola Ferretti
Verlag: Springer International Publishing
Buch: Modeling Decisions for Artificial Intelligence
Print ISBN: 978-3-030-57523-6

Electronic ISBN: 978-3-030-57524-3

Copyright-Jahr: 2020
DOI: https://doi.org/10.1007/978-3-030-57524-3_8

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