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A New Argument for the Likelihood Ratio Measure of Confirmation

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Abstract

This paper presents a new argument for the likelihood ratio measure of confirmation by showing that one of the adequacy criteria used in another argument (Zalabardo Analysis 69: 630–635, 2009) can be replaced by a more plausible and better supported criterion which is a special case of the weak likelihood principle. This new argument is also used to show that the likelihood ratio measure is to be preferred to a measure that has recently received support in the literature.

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Notes

  1. Two measures c 1 and c 2 are said to be ordinally equivalent if and only if for any evidence-hypothesis pairs (E 1,H 1) and (E 2,H 2), c 1(H 1,E 1) > c 1(H 2,E 2) if and only if c 2(H 1,E 1) > c 2(H 2,E 2) and similarly for ‘<’ and ‘=’.

  2. See Zalabardo (2009), Fitelson (1999) and references therein for further details on these measures. The likelihood ratio, Pr(E|H)/Pr(E| ∼H) is often referred to as the Bayes factor due to its role in Bayesian inference where it is the ratio of the posterior to prior odds.

  3. Actually, Zalabardo’s criterion is stronger than (C1). It reads ‘E 1 confirms H to a higher degree than E 2 does just in case Pr(H|E 1) > Pr(H|E 2)’. However, the weaker condition expressed in (C1) also suffices to rule out ld and s.

  4. l satisfies (C4) provided division by zero is equated with infinity. To avoid this, the ordinally equivalent measure proposed by Kemeny & Oppenheim (1952) can be used instead. It is given by k(H,E) = [Pr(E|H) − Pr(E| ∼H)]/[Pr(E|H) + Pr(E|∼H)].

  5. There may well be various ways to think about confirmation. For example, in the field of data mining, it has been argued that measures to quantify the strength of association rules should be confirmation measures but that they should satisfy some criteria differing from those generally accepted in the philosophy literature (Glass 2013).

  6. Joyce also states a slightly different version which does not allow for equality between Pr(E|H 1) and Pr(E|H 2). Clearly, (C5) is not a special case if it is stated in that way, but it would at most be a very modest extension of it.

  7. More precisely it is the log-likelihood ratio that satisfies the symmetry requirements of Eells and Fitelson. The measure of Kemeny and Oppenheim (see footnote 4) also satisfies these requirements.

  8. Crupi et al. (2007) state that cf satisfies the weak likelihood principle; so how can it fail to satisfy (C5) which is a special case of it? This is due to slightly different formulations of the principle (see footnote 6).

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Acknowledgments

This publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this paper are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

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  1. School of Computing and Mathematics, University of Ulster, Shore Road, Newtownabbey, Co. Antrim, BT37 0QB, UK

    David H. Glass & Mark McCartney

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  1. David H. Glass
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  2. Mark McCartney
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Correspondence to David H. Glass.

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Glass, D.H., McCartney, M. A New Argument for the Likelihood Ratio Measure of Confirmation. Acta Anal 30, 59–65 (2015). https://doi.org/10.1007/s12136-014-0228-6

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