# On the preference for more specific reference classes

## Abstract

In attempting to form rational personal probabilities by direct inference, it is usually assumed that one should prefer frequency information concerning more specific reference classes. While the preceding assumption is intuitively plausible, little energy has been expended in explaining why it should be accepted. In the present article, I address this omission by showing that, among the principled policies that may be used in setting one’s personal probabilities, the policy of making direct inferences with a preference for frequency information for more specific reference classes yields personal probabilities whose accuracy is optimal, according to all proper scoring rules, in situations where all of the relevant frequency information is *point-valued*. Assuming that frequency information for narrower reference classes is preferred, when the relevant frequency statements are point-valued, a dilemma arises when choosing whether to make a direct inference based upon (i) relatively *precise-valued* frequency information for a broad reference class, R, or upon (ii) relatively *imprecise-valued* frequency information for a more specific reference class, \(\hbox {R}^{\prime }\) (\(\hbox {R}^{\prime }\subset \hbox {R}\)). I address such cases, by showing that it is often possible to make a precise-valued frequency judgment regarding \(\hbox {R}^{\prime }\) based on precise-valued frequency information for R, using standard principles of direct inference. Having made such a frequency judgment, the dilemma of choosing between (i) and (ii) is removed, and one may proceed by using the precise-valued frequency estimate for the more specific reference class as a premise for direct inference.

## Keywords

Direct inference Statistical syllogism Specificity Scoring rules The reference class problem Imprecise probabilities The principle of indifference## Notes

### Acknowledgments

Work on this paper was supported by DFG Grant SCHU1566/9-1 as part of the priority program “New Frameworks of Rationality” (SPP 1516). For helpful comments on a presentation of this paper, I am thankful for an audience at EPSA 2015. For helpful discussions, I am thankful to Ludwig Fahrbach, Gerhard Schurz, and Ioannis Votsis. Finally, I am especially thankful two anonymous referees for Synthese who provided excellent comments and suggestions concerning an earlier draft of the paper.

## References

- Bacchus, F. (1990).
*Representing and reasoning with probabilistic knowledge*. Cambridge, MA: MIT Press.Google Scholar - Bradley, S., & Steele, K. (2014). Uncertainty, learning, and the “Problem” of dilation.
*Erkenntnis*,*79*(6), 1287–1303.CrossRefGoogle Scholar - Brier, G. (1950). Verification of forecasts expressed in terms of probability.
*Monthly Weather Review*,*78*, 1–3.CrossRefGoogle Scholar - Carnap, R. (1962).
*Logical foundations of probability*. Chicago: University of Chicago Press.Google Scholar - de Finetti, B. (1974).
*Theory of probability (vol. 1)*. New York: Wiley.Google Scholar - Easwaran, K. (2013). Expected accuracy supports conditionalization and conglomerability and reflection.
*Philosophy of Science*,*80*(1), 119–142.CrossRefGoogle Scholar - Gould, H. (2010). Combinatorial identities: Table I: Intermediate techniques for summing finite series. In J. Quaintance (Ed.), http://www.math.wvu.edu/~gould/Vol.4.PDF. Accessed 3 Feb 2016.
- Greaves, H., & Wallace, D. (2006). Justifying conditionalization: Conditionalization maximizes expected epistemic utility.
*Mind*,*114*, 607–632.CrossRefGoogle Scholar - Joyce, J. (1998). A nonpragmatic vindication of probabilism.
*Philosophy of Science*,*65*(4), 575–603.CrossRefGoogle Scholar - Kyburg, H. (1974).
*The logical foundations of statistical inference*. Dordrecht: Reidel Publishing Company.CrossRefGoogle Scholar - Kyburg, H., & Teng, C. (2001).
*Uncertain inference*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Leitgeb, H., & Pettigrew, R. (2010a). An objective justification of Bayesianism I: Measuring inaccuracy.
*Philosophy of Science*,*77*(2), 201–235.CrossRefGoogle Scholar - Leitgeb, H., & Pettigrew, R. (2010b). An Objective Justification of Bayesianism II: The Consequences of Minimizing Inaccuracy.
*Philosophy of Science*,*77*(2), 236–272.CrossRefGoogle Scholar - Levinstein, B. (2012). Leitgeb and Pettigrew on accuracy and updating.
*Philosophy of Science*,*79*(3), 413–424.CrossRefGoogle Scholar - Pollock, J. (1990).
*Nomic probability and the foundations of induction*. Oxford: Oxford University Press.Google Scholar - Reichenbach, H. (1949).
*A theory of probability*. Berkeley: Berkeley University Press.Google Scholar - Selten, R. (1998). Axiomatic characterization of the quadratic scoring rule.
*Experimental Economics*,*1*, 43–62.CrossRefGoogle Scholar - Stone, M. (1987). Kyburg, Levi, and Petersen.
*Philosophy of Science*,*54*(2), 244–255.CrossRefGoogle Scholar - Thorn, P. (2014). Defeasible conditionalization.
*Journal of Philosophical Logic*,*43*, 283–302.CrossRefGoogle Scholar - Venn, J. (1866).
*The logic of chance*. New York: Chelsea Publishing Company.Google Scholar - White, R. (2009). Evidential symmetry and mushy credence. In T. Szabo Gendler & J. Hawthorne (Eds.),
*Oxford studies in epistemology*(Vol. 3, pp. 161–186). Oxford: Oxford University Press.Google Scholar