Simplex-valued probability

Mathematica Slovaca

Roman Frič¹ 

关键词: categorical probability theory;    cogenerator;    probability domain;    D-poset of fuzzy sets;    state;    simple S n D-domain;    SnD-observable;    SnD-valued state;    weak S n D-observable;    weak S n D-valued state;    simplex-valued probability;   

DOI  :  10.2478/s12175-010-0035-5

学科分类：数学（综合）

来源: Slovenska Akademia Vied * Matematicky Ustav / Slovak Academy of Sciences, Mathematical Institute

【 摘 要 】

We continue our study of generalized probability from the viewpoint of category theory. Assuming that each generalized probability measure is a morphism, we model basic probabilistic notions within the category cogenerated by its range. It is known that the closed unit interval I = [0, 1], carrying a suitable difference structure, cogenerates the category ID in which the classical and fuzzy probability theories can be modeled. We study generalized probability theories modeled within two different categories cogenerated by a simplex S n = {(x 1, x 2, …, x n) ∈ I n: $$mathop sum limits_{i = 1}^n $$ x i ≤ 1}, carrying a suitable difference structure; since I and S 1 coincide, for n = 1 we get the fuzzy probability theory as a special case. In the first case, when the morphisms preserve the so-called pure elements, the resulting category S n D, n > 1, and ID are isomorphic and the generalized probability theories modeled in ID and S n D are “the same”. In the second case, when the morphisms map each maximal element to a maximal element, the resulting categories WS n D, n > 1, lead to different models of generalized probability theories. We define basic notions of the corresponding simplex-valued probability theories and mention some applications.

【 授权许可】

Unknown