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EPSA15 Selected Papers pp 285–295Cite as

Generalized Confirmation and Relevance Measures

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Part of the book series: European Studies in Philosophy of Science ((ESPS,volume 5))

Abstract

The main point of the paper is to show how popular probabilistic measures of incremental confirmation and statistical relevance with qualitatively different features can be embedded smoothly in generalized parametric families. In particular, I will show that the probability difference, log probability ratio, log likelihood ratio, odds difference, so-called improbability difference, and Gaifman’s measures of confirmation can all be subsumed within a convenient biparametric continuum. One intermediate step of this project may have interest on its own, as it provides a unified representation of graded belief of which both probabilities and odds are special cases.

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Notes

  1. 1.

    A regular probability function never assigns probability 0 to a statement unless it expresses a logical falsehood (i.e., for any α ∈ L C , P(α) > 0). Regularity can be motivated as a way to represent credences that are non-dogmatic as concerns L C (see Howson 2000, p. 70). It is known to be a convenient but not entirely innocent assumption (see Festa 1999; Kuipers 2000 for discussion; also see Pruss 2013).

  2. 2.

    Two ordinally equivalent measures C and C* are such that for any h,k,e,f ∈ L c and any P ∈ P, C(h,e) ⋛ C(k,f) if and only if C*(h,e) ⋛ C*(k,f).

  3. 3.

    A different way to connect and subsume probabilities and odds was already suggested by Festa (2008). Festa defined a parametric family of “belief functions” B α (x) = P(x)/[1 + αP(x)] with α ∈[−1,∞), so that B –1(x) = O(x) and B 0(x) = P(x).

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Authors and Affiliations

  1. Center for Logic, Language, and Cognition, Department of Philosophy and Education, University of Turin, Turin, Italy

    Vincenzo Crupi

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  1. Vincenzo Crupi
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Correspondence to Vincenzo Crupi .

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Editors and Affiliations

  1. University of Edinburgh, Edinburgh, United Kingdom

    Michela Massimi

  2. University of Groningen, Groningen, The Netherlands

    Jan-Willem Romeijn

  3. University of Düsseldorf, Düsseldorf, Nordrhein-Westfalen, Germany

    Gerhard Schurz

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Crupi, V. (2017). Generalized Confirmation and Relevance Measures. In: Massimi, M., Romeijn, JW., Schurz, G. (eds) EPSA15 Selected Papers. European Studies in Philosophy of Science, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-53730-6_23

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