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.
Authors are listed in alphabetical order. All authors contributed equally to this work, they discussed the results and implications and commented on the manuscript at all stages.
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- 1.
Throughout the paper, the formulas are assumed to be well defined, e.g. denominators do not vanish, as implicitly done in the literature.
- 2.
Feasibility of (x, y, z) requires that probabilities x, y and z satisfy in particular the Total Probability Theorem.
- 3.
A BCM that satisfies all the above considered symmetries is Carnap’s b [1] which is defined as \( b(H,E)=P(E \cap H)-P(E) \, P(H). \)
- 4.
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].
- 5.
Computations in this paper were performed with Wolfram’s software Mathematica (version 11.0.1.0).
- 6.
Moreover, depending on the chosen DBCM, some of the above recalled inequalities may be required to be strict.
- 7.
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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Ellero, A., Ferretti, P. (2020). Distorted Probabilities and Bayesian Confirmation Measures. In: Torra, V., Narukawa, Y., Nin, J., Agell, N. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2020. Lecture Notes in Computer Science(), vol 12256. Springer, Cham. https://doi.org/10.1007/978-3-030-57524-3_8
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