Simplex-valued probability

Rok, strany: 2010, 607 - 614

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 $S_n = \{(x₁,x₂, … ,x_n)\in Iⁿ: ∑_i=1ⁿ x_i≤ 1\}$, carrying a suitable difference structure; since $I$ and $S₁$ 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 $S_nD$, $n > 1$, and $ID$ are isomorphic and the generalized probability theories modeled in $ID$ and $S_nD$ are ``the same''. In the second case, when the morphisms map each maximal element to a maximal element, the resulting categories $WS_nD$, $n > 1$, lead to different models of generalized probability theories. We define basic notions of the corresponding simplex-valued probability theories and mention some applications.

ISO 690:

Frič, R. 2010. Simplex-valued probability. In Mathematica Slovaca, vol. 60, no.5, pp. 607-614. 0139-9918. DOI: https://doi.org/10.2478/s12175-010-0035-5

APA:

Frič, R. (2010). Simplex-valued probability. Mathematica Slovaca, 60(5), 607-614. 0139-9918. DOI: https://doi.org/10.2478/s12175-010-0035-5