Imprecise Probability and Chance
- 219 Downloads
Understanding probabilities as something other than point values (e.g., as intervals) has often been motivated by the need to find more realistic models for degree of belief, and in particular the idea that degree of belief should have an objective basis in “statistical knowledge of the world.” I offer here another motivation growing out of efforts to understand how chance evolves as a function of time. If the world is “chancy” in that there are non-trivial, objective, physical probabilities at the macro-level, then the chance of an event e that happens at a given time is \(<1\) until it happens. But whether the chance of e goes to one continuously or not is left open. Discontinuities in such chance trajectories can have surprising and troubling consequences for probabilistic analyses of causation and accounts of how events occur in time. This, coupled with the compelling evidence for quantum discontinuities in chance’s evolution, gives rise to a “(dis)continuity bind” with respect to chance probability trajectories. I argue that a viable option for circumventing the (dis)continuity bind is to understand the probabilities “imprecisely,” that is, as intervals rather than point values. I then develop and motivate an alternative kind of continuity appropriate for interval-valued chance probability trajectories.
KeywordsInterval Function Temporal Index Ordinary Sense Imprecise Probability Probability Trajectory
Key parts of this work were done on sabbatical in Berlin 2012-13; I thank Dörte and Frieder Middelhauve and Jodi Melamed for helping make the sabbatical possible (and fun) along with Michael Pauen and the Berlin School of Mind and Brain for providing a place to work, and a stimulating and gemütlich community. A version of this paper was presented at the Imprecise Probabilities in Statistics and Philosophy Conference at Ludwig-Maximilians-Universität München, June 28, 2014. I am grateful for the comments and suggestions I received from participants. I thank also this journal's anonymous reviewers for their helpful comments.
- Anguelov, R., & Markov, S. (2007). Numerical computations with hausdorff continuous functions. In T. Boyanov, S. Dimova, K. Georgiev, & G. Nikolov (Eds.), Numerical methods and applications, lecture notes in computer science (Vol. 4310, pp. 279–286). Berlin: Springer.Google Scholar
- Aubin, J., & Frankowska, H. (1990). Set-valued analysis. Birkhäuser: Systems & Control.Google Scholar
- Flores-Franulic, A., Chalco-Cano, Y., & Roman-Flores, H. (2013). An ostrowski type inequality for interval-valued functions. In IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS), 2013 Joint, pp. 1459–1462.Google Scholar
- Glynn, L. (2014). Unsharp best system chances. http://philsci-archive.pitt.edu/10239/.
- Guerra, M., & Stefanini, L. (2011). A comparison index for interval ordering. In IEEE symposium on foundations of computational intelligence (FOCI), 2011, pp. 53–58.Google Scholar
- Hall, N. (2004). Two concepts of causation. In J. D. Collins, E. J. Hall, & L. A. Paul (Eds.), Causation and counterfactuals (pp. 181–276). Cambridge, MA: MIT Press.Google Scholar
- Hitchcock, C. (2004). Do all and only causes raise the probabilities of effects? In J. D. Collins, E. J. Hall, & L. A. Paul (Eds.), Causation and counterfactuals (pp. 403–418). Cambridge, MA: MIT Press.Google Scholar
- Kvart, I. (2004). Causation: Probabilistic and counterfactual analyses. In J. D. Collins, E. J. Hall, & L. A. Paul (Eds.), Causation and counterfactuals (pp. 359–386). Cambridge, MA: MIT Press.Google Scholar
- Kyburg, H.E. (1999). Interval-valued probabilities. In The society for imprecise probability: Theories and applications. http://www.sipta.org/documentation/interval_prob/kyburg.
- Lewis, D. (1986). Postscripts to ‘causation’. In D. Lewis (Ed.), Philosophical papers (Vol. II, pp. 172–213). Oxford: Oxford University Press.Google Scholar
- Li, F.C., & Li, J. (2010). Ordering method of interval numbers based on synthesizing effect. In International conference on machine learning and cybernetics (ICMLC), 2010, (vol. 1, pp. 108–112).Google Scholar
- Moore, R. (1966). Interval analysis. Prentice-Hall series in automatic computation. Englewood Cliffs: Prentice-Hall .Google Scholar
- Moore, R. (1979). Methods and Applications of Interval Analysis. Studies in Applied and Numerical Mathematics. Society for Industrial and Applied Mathematics.Google Scholar
- Moore, R., Kearfott, R., & Cloud, M. (2009). Introduction to Interval Analysis. SIAM e-books. Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104).Google Scholar
- Peressini, A. (forthcoming) Causation, probability, and the continuity bind.Google Scholar