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Bets and boundaries: Assigning probabilities to imprecisely specified events. (English) Zbl 1159.03309
Summary: 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 \(n \in \mathbb{N}^{+}\). 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.

MSC:
03B48 Probability and inductive logic
03E70 Nonclassical and second-order set theories
68T37 Reasoning under uncertainty in the context of artificial intelligence
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