## Abstract

Dilation occurs when an interval probability estimate of some event *E* is properly included in the interval probability estimate of *E* conditional on every event *F* of some partition, which means that one’s initial estimate of *E* becomes less precise no matter how an experiment turns out. Critics maintain that dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. However, two points are often overlooked: (1) knowing that *E* is stochastically independent of *F* (for all *F* in a partition of the underlying state space) is sufficient to avoid dilation, but (2) stochastic independence is not the only independence concept at play within imprecise probability models. In this paper we give a simple characterization of dilation formulated in terms of deviation from stochastic independence, propose a measure of dilation, and distinguish between proper and improper dilation. Through this we revisit the most sensational examples of dilation, which play up independence between *dilator* and *dilatee*, and find the sensationalism undermined by either fallacious reasoning with imprecise probabilities or improperly constructed imprecise probability models.

## Notes

### Acknowledgments

Thanks to Horacio Arló-Costa, Jim Joyce, Isaac Levi, Teddy Seidenfeld, and Jon Williamson for their comments on early drafts, and to Clark Glymour and Choh Man Teng for a long discussion on dilation that began one sunny afternoon in a Lisbon café. This research was supported in part by award (LogICCC/0001/2007) from the European Science Foundation.

## References

- Couso, I., Moral, S., & Walley, P. (1999). Examples of independence for imprecise probabilities. In G. de Cooman (Ed.),
*Proceedings of the first symposium on imprecise probabilities and their applications (ISIPTA)*, Ghent, Belgium.Google Scholar - Cozman, F. (2000). Credal networks.
*Artificial Intelligence, 120*(2), 199–233.CrossRefGoogle Scholar - Cozman, F. (2012). Sets of probability distributions, independence, and convexity.
*Synthese, 186*(2), 577–600.CrossRefGoogle Scholar - de Cooman, G., & Miranda, E. (2007). Symmetry of models versus models of symmetry. In W. Harper, & G. Wheeler (Eds.),
*Probability and inference: Essays in honor of Henry E. Kyburg, Jr*. (pp. 67–149). London: King’s College Publications.Google Scholar - de Cooman, G., & Miranda, E. (2009). Forward irrelevance.
*Journal of Statistical Planning, 139*, 256–276.CrossRefGoogle Scholar - de Cooman, G., Miranda, E., & Zaffalon, M. (2011). Independent natural extension.
*Artificial Intelligence, 175*, 1911–1950.CrossRefGoogle Scholar - de Finetti, B. (1974a).
*Theory of probability*(vol. I). Wiley, 1990 edition.Google Scholar - de Finetti, B. (1974b).
*Theory of probability*(vol. II). Wiley, 1990 edition.Google Scholar - Elga, A. (2010). Subjective probabilities should be sharp.
*Philosophers Imprint, 10*(5).Google Scholar - Ellsberg, D. (1961). Risk, ambiguity and the savage axioms.
*Quarterly Journal of Economics, 75*, 643–669.CrossRefGoogle Scholar - Good, I. J. (1952). Rational decisions.
*Journal of the Royal Statistical Society. Series B, 14*(1), 107–114.Google Scholar - Good, I. J. (1967). On the principle of total evidence.
*The British Journal for the Philosophy of Science, 17*(4), 319–321.CrossRefGoogle Scholar - Good, I. J. (1974). A little learning can be dangerous.
*The British Journal for the Philosophy of Science, 25*(4), 340–342.CrossRefGoogle Scholar - Grünwald, P., & Halpern, J. Y. (2004). When ignorance is bliss. In J. Y. Halpern (Ed.),
*Proceedings of the 20th conference on uncertainty in artificial intelligence (UAI ’04)*(pp. 226–234). Arlington, VA: AUAI Press.Google Scholar - Haenni, R., Romeijn, J.-W., Wheeler, G., & Williamson, J. (2011).
*Probabilistic logics and probabilistic networks. Synthese library*. Dordrecht: Springer.CrossRefGoogle Scholar - Halmos, P. R. (1950).
*Measure theory*. New York: Van Nostrand Reinhold Company.CrossRefGoogle Scholar - Harper, W. L. (1982). Kyburg on direct inference. In R. Bogdan (Ed.),
*Henry E. Kyburg and Isaac Levi*(pp. 97–128). Dordrecht: Kluwer.Google Scholar - Herron, T., Seidenfeld, T., & Wasserman, L. (1994). The extent of dilation of sets of probabilities and the asymptotics of robust bayesian inference. In
*PSA 1994 proceedings of the Biennial meeting of the philosophy of science association*(vol. 1, pp. 250–259).Google Scholar - Herron, T., Seidenfeld, T., & Wasserman, L. (1997). Divisive conditioning: Further results on dilation.
*Philosophy of Science, 64*, 411–444.CrossRefGoogle Scholar - Horn, A., & Tarski, A. (1948). Measures in boolean algebras.
*Transactions of the AMS, 64*(1), 467–497.CrossRefGoogle Scholar - Joyce, J. (2011). A defense of imprecise credences in inference and decision making.
*Philosophical Perspectives, 24*(1), 281–323.CrossRefGoogle Scholar - Koopman, B. O. (1940). The axioms and algebra of intuitive probability.
*Annals of Mathematics, 41*(2), 269–292.CrossRefGoogle Scholar - Kyburg, H. E., Jr. (1961).
*Probability and the logic of rational belief*. Middletown, CT: Wesleyan University Press.Google Scholar - Kyburg, H. E., Jr. (1974).
*The logical foundations of statistical inference*. Dordrecht: D. Reidel.Google Scholar - Kyburg, H. E., Jr. (2007). Bayesian inference with evidential probability. In W. Harper, & G. Wheeler (Eds.),
*Probability and inference: Essays in honor of Henry E. Kyburg, Jr*. (pp. 281–296). London: King’s College.Google Scholar - Kyburg, H. E., Jr., & Teng, C. M. (2001).
*Uncertain inference*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Levi, I. (1974). On indeterminate probabilities.
*Journal of Philosophy, 71*, 391–418.CrossRefGoogle Scholar - Levi, I. (1977). Direct inference.
*Journal of Philosophy, 74*, 5–29.CrossRefGoogle Scholar - Levi, I. (1980).
*The enterprise of knowledge*. Cambridge, MA: MIT Press.Google Scholar - Rao, K. B., & Rao, M. B. (1983).
*Theory of charges: A study of finitely additive measures*. London: Academic Press.Google Scholar - Romeijn, J.-W. (2006). Analogical predictions for explicit similarity.
*Erkenntnis, 64*, 253–80.CrossRefGoogle Scholar - Savage, L. J. (1972).
*Foundations of statistics*. New York: Dover.Google Scholar - Schlosshauer, M., & Wheeler, G. (2011). Focused correlation, confirmation, and the Jigsaw puzzle of variable evidence.
*Philosophy of Science, 78*(3), 276–92.CrossRefGoogle Scholar - Seidenfeld, T. (1994). When normal and extensive form decisions differ. In D. Prawitz, B. Skyrms, & D. Westerstahl (Eds.),
*Logic, methodology and philosophy of science*. Amsterdam: Elsevier.Google Scholar - Seidenfeld, T. (2007). Forbidden fruit: When epistemic probability may not take a bite of the Bayesian apple. In W. Harper, & G. Wheeler (Eds.),
*Probability and inference: Essays in honor of Henry E. Kyburg, Jr*. London: King’s College Publications.Google Scholar - Seidenfeld, T., Schervish, M. J., & Kadane, J. B. (2010). Coherent choice functions under uncertainty.
*Synthese, 172*(1), 157–176.CrossRefGoogle Scholar - Seidenfeld, T., & Wasserman, L. (1993). Dilation for sets of probabilities.
*The Annals of Statistics, 21*, 1139–154.CrossRefGoogle Scholar - Shogenji, T. (1999). Is coherence truth conducive?
*Analysis, 59*, 338–345.CrossRefGoogle Scholar - Smith, C. A. B. (1961). Consistency in statistical inference (with discussion).
*Journal of the Royal Statistical Society, 23*, 1–37.Google Scholar - Sturgeon, S. (2008). Reason and the grain of belief.
*Noûs, 42*(1), 139–165.CrossRefGoogle Scholar - Sturgeon, S. (2010). Confidence and coarse-grain attitudes. In T. S. Gendler, & J. Hawthorne (Eds.),
*Oxford studies in epistemology*(vol. 3, pp. 126–149). Oxford: Oxford University Press.Google Scholar - Walley, P. (1991).
*Statistical reasoning with imprecise probabilities*. London: Chapman and Hall.CrossRefGoogle Scholar - Wayne, A. (1995). Bayesianism and diverse evidence.
*Philosophy of Science, 62*(1), 111–121.CrossRefGoogle Scholar - Wheeler, G. (2006). Rational acceptance and conjunctive/disjunctive absorption.
*Journal of Logic, Language and Information, 15*(1–2), 49–63.CrossRefGoogle Scholar - Wheeler, G. (2009a). Focused correlation and confirmation.
*The British Journal for the Philosophy of Science, 60*(1), 79–100.CrossRefGoogle Scholar - Wheeler, G. (2009b). A good year for imprecise probability. In V. F. Hendricks (Ed.),
*PHIBOOK*. New York: VIP/Automatic Press.Google Scholar - Wheeler, G. (2012). Objective Bayesianism and the problem of non-convex evidence.
*The British Journal for the Philosophy of Science, 63*(3), 841–850.CrossRefGoogle Scholar - Wheeler, G. (2013). Character matching and the envelope of belief. In F. Lihoreau, & M. Rebuschi (Eds.),
*Epistemology, context, and formalism, synthese library*(pp. 185–194). Berlin: Springer. Presented at the 2010 APA Pacific division meeting.Google Scholar - Wheeler, G., & Scheines, R. (2013). Coherence, confirmation, and causation.
*Mind, 122*(435), 135–170.CrossRefGoogle Scholar - White, R. (2010). Evidential symmetry and mushy credence. In T. S. Gendler, & J. Hawthorne (Eds.),
*Oxford studies in epistemology*(vol. 3, pp. 161–186). Oxford: Oxford University Press.Google Scholar - Williamson, J. (2007). Motivating objective Bayesianism: From empirical constraints to objective probabilities. In W. Harper, & G. Wheeler (Eds.),
*Probability and inference: Essays in honor of Henry E. Kyburg, Jr*. London: College Publications.Google Scholar - Williamson, J. (2010).
*In defence of objective Bayesianism*. Oxford: Oxford University Press.CrossRefGoogle Scholar - Yule, U. (1911).
*Introduction to the theory of statistics*. London: Griffin.CrossRefGoogle Scholar