Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4615639 | Journal of Mathematical Analysis and Applications | 2015 | 39 Pages |
Abstract
Coherent lower previsions constitute a convex set that is closed and compact under the topology of point-wise convergence, and Maaà [13] has shown that any coherent lower prevision can be written as a 'countably additive convex combination' of the extreme points of this set. We show that when the possibility space has a finite number n of elements, these extreme points are either degenerate precise probabilities, or fully imprecise and in a one-to-one correspondence with Minkowski indecomposable non-empty convex compact subsets of Rnâ1. By exploiting this connection, we are able to prove that for n=3, fully imprecise extreme lower previsions are lower envelopes of at most three linear previsions. For nâ¥4, 'most' fully imprecise lower previsions are extreme. Finally, we show that in our context, MaaÃ's result can be strengthened as follows: any coherent lower prevision can be written as, or approximated arbitrarily closely by, a finite convex combination of finitely generated extreme lower previsions.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Jasper De Bock, Gert de Cooman,