Imprecise Probability and Chance
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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.
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