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On Ratio Measures of Confirmation

Critical Remarks on Zalabardo’s Argument for the Likelihood-Ratio Measure

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Abstract

There are different Bayesian measures to calculate the degree of confirmation of a hypothesis H in respect of a particular piece of evidence E. Zalabardo (Analysis 69:630–635, 2009) is a recent attempt to defend the likelihood-ratio measure (LR) against the probability-ratio measure (PR). The main disagreement between LR and PR concerns their sensitivity to prior probabilities. Zalabardo invokes intuitive plausibility as the appropriate criterion for choosing between them. Furthermore, he claims that it favours the ordering of pairs evidence/hypothesis generated by LR. We will argue, however, that the intuitive non-numerical example provided by Zalabardo does not show that prior probabilities do not affect the degree of confirmation. On account of this, we conclude that there is no compelling reason to endorse LR qua measure of degree of confirmation. On the other side, we should not forget some technicalities which still benefit PR.

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Notes

  1. Horwich (1982); Milne (1996) and Kuipers (2000) defend PR, while Good (1983); Jeffrey (2004); Roush (2005) and Fitelson (2006) prefer LR. Zalabardo (2009) is a new attempt to exploit the alleged intuitive advantages of LR.

  2. If p (H|E) > p(H|E*), then \( \frac{{p(E\left| {H)p(H)} \right.}}{{p(E\left| H \right.)p(H) + p(E\left| {\neg H)p(\neg H)} \right.}} > \frac{{p(E*\left| {H)p(H)} \right.}}{{p(E*\left| H \right.)p(H) + p(E*\left| {\neg H)p(\neg H)} \right.}} \) (by Bayes’s Theorem). This inequality will be satisfied when one at least of these conditions is also fulfilled: (i) p (E|H) p (H) > p (E*|H) (p (H); (ii) p (E|H) p (H) < p (E*|H) p (H). Obviously, C1 is the particular case where only (ii) is satisfied.

  3. Ellery Eells and Branden Fitelson, for instance, point out that many contemporary Bayesian resolutions of both the ravens paradox and the problem of evidential variety take this general principle for granted. See Eells and Fitelson (2000, 670).

  4. For the distinction between absolute and incremental confirmation, see the preface of Carnap (1962), where the respective expressions are firmness and degree of firmness.

  5. Roush (2005, 164) claims that LR is preferable to PR because LR is more adequate to discriminate between H and H. We do not want to make a point of their respective discriminative merits. We question what she takes for granted, that is, that discrimination is what a measure of confirmation should measure. Again, the aim is not so much discriminating as calculating an increment. Incidentally, if discrimination is what really is at concern, perhaps a pure “likelihoodist” approach should be considered a more promising alternative to PR than LR. Likelihoodism as developed in Royall (1997) is committed to an intrinsically comparative account of evidential support. According to it judgments of evidential support involve a three-place predicate ‘E supports/favours H over H*’. On the differences between Bayesianism and Likelihoodism, see Fitelson (2007).

  6. Intuitions about conditional probability are not only unclear, but very often misguiding, as experimental work on the base-rate fallacy shows (Kahneman et al. 1982, part II). The fallacy occurs precisely when we overlook the role played by prior probabilities. This spontaneous tendency to ignore priors perhaps could also explain why, at first sight at least, we do not pay attention to them in Zalabardo’s example. For a more recent account of empirical research on this fallacy see the papers included in Behavioral and Brain Sciences, 30 (2007), issue 3.

References

  • Carnap, R. (1962). Logical foundations of probability (2nd ed.). Chicago: University of Chicago Press.

    Google Scholar 

  • Crupi, V., Tentori, K., & González, M. (2007). On Bayesian measures of confirmational support: Theoretical and empirical issues. Philosophy of Science, 74, 229–252.

    Article  Google Scholar 

  • Eells, E., & Fitelson, B. (2000). Measuring confirmation and evidence. Journal of Philosophy, 97, 663–672.

    Article  Google Scholar 

  • Festa, R. (2012) “For unto every one that hath shall be given”: Matthew properties for incremental confirmation. Synthese, 184(1), 89–100.

    Google Scholar 

  • Fitelson, B. (2006). Logical foundations of evidential support. Philosophy of Science, 73, 500–512.

    Article  Google Scholar 

  • Fitelson, B. (2007). Likelihoodism, bayesianism and relational confirmation. Synthese, 156, 473–489.

    Article  Google Scholar 

  • Gillies, D. (1998). Confirmation theory. In D. M. Gabbay & Ph. Smets (Eds.), Handbook of defeasible reasoning and uncertainty management systems (Vol. I, pp. 135–167). Dordrecht: Kluwer.

    Google Scholar 

  • Good, I. J. (1983). Good thinking: The foundations of probability and its applications. Minneapolis: University of Minneapolis Press.

    Google Scholar 

  • Horwich, P. (1982). Probability and evidence. Cambridge: Cambridge University Press.

    Google Scholar 

  • Jeffrey, R. (2004). Subjective probability: The real thing. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Kahneman, D., Slovic, P., & Tversky, A. (Eds.). (1982). Judgment under uncertainty: Heuristics and biases. Cambridge: Cambridge University Press.

    Google Scholar 

  • Kuipers, Th. (2000). From instrumentalism to constructive realism. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Milne, P. (1996). log [p (h/eb)/p (h/b)] is the one true measure of confirmation. Philosophy of Science, 63, 21–26.

    Article  Google Scholar 

  • Roush, S. (2005). Tracking truth. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Royall, R. (1997). Statistical evidence: A likelihood paradigm. London: Chapman and Hall.

    Google Scholar 

  • Zalabardo, J. (2009). An argument for the likelihood-ratio measure of confirmation. Analysis, 69, 630–635.

    Article  Google Scholar 

Download references

Acknowledgments

This work was funded by the Spanish Ministry of Science and Innovation (research project FFI2008-01169). We should thank José Zalabardo, Vincenzo Crupi and an anonymous referee for their helpful comments.

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Authors and Affiliations

  1. Departamento de Lógica y Filosofía de la Ciencia, Universidad de Valencia, Valencia, Spain

    Valeriano Iranzo

  2. Departamento de Economía Aplicada, Universidad de Valencia, Valencia, Spain

    Ignacio Martínez de Lejarza

Authors
  1. Valeriano Iranzo
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  2. Ignacio Martínez de Lejarza
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Correspondence to Valeriano Iranzo.

Appendix: Proofs of Theorems

Appendix: Proofs of Theorems

Theorem 1

If likelihoods for H and H* are even and evidence (E and E*, respectively) is confirmatory, then PR favours the hypothesis with the lowest prior.

Proof

Consider that the following conditions are true:

  • (i) p(E|H) = p(E*|H*),

  • (ii) p(E|¬H) = p(E*|¬H*)

  • (iii) p(H) > p(H*)

  • (iv) p(E|H) > p(E|¬H)

  • By (iv):

  • p(E|H) > p(E|¬H).

  • By (iii), since p(H) − p(H*) is positive, we have:

  • p(E|H) [p(H) − p(H*)] > p(E|¬H) [p(H) − p(H*)];

  • p(E|H) [p(H) − p(H*)] > p(E|¬H) [p(¬H*) − p(¬H)];

  • p(E|H) p(H) − p(E|H) p(H*) > p(E|¬H) p(¬H*) − p(E|¬H) p(¬H);

  • p(E|H) p(H) + p(E|¬H) p(¬H) > p(E|H) p(H*) + p(E|¬H) p(¬H*).

  • By (i) and (ii):

  • p(E|H) p(H) + p(E|¬H) p(¬H) > p(E*|H*) p(H*) + p(E*|¬H*) p(¬H*);

  • p(E) > p(E*);

  • Then, relying on (i):

  • p(E|H)/p(E) < p(E*|H*)/p(E*);

  • p(H|E)/p(H) < p(H*|E*)/p(H*);

Therefore, PR (H, E) < PR (H*, E*).

Theorem 2

If likelihoods for H and H* are even and evidence (E and E*, respectively) is disconfirmatory, then PR favours the hypothesis with the highest prior.

Proof

Consider that the following conditions are true:

  • (i) p(E|H) = p(E*|H*),

  • (ii) p(E|¬H) = p(E*|¬H*)

  • (iii) p(H) < p(H*)

  • (iv) p(E|H) < p(E|¬H)

  • By (iv):

  • p(E|H) < p(E|¬H).

  • By (iii), since p(H*) − p(H) is positive, we have:

  • p(E|¬H) [p(H*) − p(H)] > p(E|H) [p(H*) − p(H)];

  • p(E|¬H) [p(¬H) – p(¬H*)] > p(E|H) [p(H*) − p(H)];

  • p(E|¬H) p(¬H) − p(E|¬H) p(¬H*) > p(E|H) p(H*) − p(E|H) p (H);

  • p(E|¬H) p(¬H) + p(E|H) p(H) > p(E|H) p(H*) + p(E|¬H) p(¬H*);

  • By (i) and (ii):

  • p(E|¬H) p(¬H) + p(E|H) p(H) > p(E*|H*) p(H*) + p(E*|¬H*) p(¬H*);

  • p(E) > p(E*);

  • Then, relying on (i):

  • p(E|H)/p(E) < p(E*|H*)/p(E*);

  • p(H|E)/p(H) < p(H*|E*)/p(H*);

Therefore, PR (H, E) < PR (H*, E*).

Theorem 3

Cases where likelihoods for H and H* are even and conditions (iii) and (iv) go in opposite directions, are reducible either to Theorem 1 or Theorem 2.

Proof

Consider that the following conditions are true:

  • (i) p(E|H) = p(E*|H*),

  • (ii) p(E|¬H) = p(E*|¬H*)

  • (iii) p(H) < p(H*)

  • (iv) p(E|H) > p(E|¬H)

  • By (i) and (ii):

  • p(E|H*) > p(E|¬H*)

By replacing this inequality for (iv), we get those conditions stated for Theorem 1 with H and H* switched.

Cases where (iii) p(H) > p(H*) and (iv) p(E|H) < p(E|¬H) are reducible to conditions stated for Theorem 2 by an analogous proof.

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Iranzo, V., Martínez de Lejarza, I. On Ratio Measures of Confirmation. J Gen Philos Sci 44, 193–200 (2013). https://doi.org/10.1007/s10838-012-9175-3

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