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Appears in Collections:Law and Philosophy Journal Articles
Peer Review Status: Refereed
Title: Bets and Boundaries: Assigning Probabilities to Imprecisely Specified Events
Author(s): Milne, Peter
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Keywords: probability
conditional probability
rough set theory
tolerance relations
Neural networks (Computer science)
Fuzzy systems
Fuzzy logic
Issue Date: Dec-2008
Date Deposited: 26-Jan-2010
Citation: Milne P (2008) Bets and Boundaries: Assigning Probabilities to Imprecisely Specified Events. Studia Logica, 90 (3), pp. 425-453.
Abstract: Uncertainty and vagueness/imprecision are not the same: one can be certain about events described using vague predicates and about imprecisely specified events, just as one can be uncertain about precisely specified events. Exactly because of this, a question arises about how one ought to assign probabilities to imprecisely specified events in the case when no possible available evidence will eradicate the imprecision (because, say, of the limits of accuracy of a measuring device). Modelling imprecision by rough sets over an approximation space presents an especially tractable case to help get one’s bearings. Two solutions present themselves: the first takes as upper and lower probabilities of the event X the (exact) probabilities assigned X’s upper and lower rough-set approximations; the second, motivated both by formal considerations and by a simple betting argument, is to treat X’s rough-set approximation as a conditional event and assign to it a point-valued (conditional) probability. With rough sets over an approximation space we get a lot of good behaviour. For example, in the first construction mentioned the lower probabilities are n-monotone, for every n ∈ N+. When we examine other models of approximation/imprecision/vagueness, and in particular, proximity spaces, we lose a lot of that good behaviour. In the literature there is not (even) agreement on the definition of upper and lower approximations for events (subsets) in the underlying domain. Betting considerations suggest one choice and, again, ways to assign upper and lower and point-valued probabilities, but nothing works well.
DOI Link: 10.1007/s11225-008-9160-3
Rights: Copyright Springer 2008; The publisher does not allow this work to be made publicly available in this Repository. Please use the Request a Copy feature at the foot of the Repository record to request a copy directly from the author; you can only request a copy if you wish to use this work for your own research or private study.
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