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Simplex-valued probability

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Mathematica Slovaca

Abstract

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.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, SK-040 01, Košice, Slovak Republic

    Roman Frič

  2. Catholic University in Ružomberok, Nám. A. Hlinku 60 Hrabovská cesta, SK-034 01, Ružomberok, Slovak Republic

    Roman Frič

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  1. Roman Frič
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Correspondence to Roman Frič.

Additional information

Communicated by Anatolij Dvurečenskij

Dedicated to Silvia Pulmannová

This work was supported by the Slovak Research and Development Agency (contract No. APVV-0071-06); and Slovak Scientific Grant Agency (VEGA project 2/6088/26).

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

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  • DOI: https://doi.org/10.2478/s12175-010-0035-5

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