A Categorical Approach to Probability Theory
A Categorical Approach to Probability Theory
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DOI：
10.1007/s11225-010-9232-z
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发表时间：
2010-03-01
期刊：
影响因子：
0.7
通讯作者：
Papco, Martin
中科院分区：
文献类型：
--
作者：
Fric, Roman;Papco, Martin
首先，我们从范畴论的观点讨论基本的概率概念。我们的方法是基于以下四个“sine quibus non”条件：1。（初等）范畴理论是有效的（和足够的）; 2。随机变量、可观测量、概率测度和状态是态射; 3.经典概率论和S. Gudder和S. Bugajski是更一般模型的特例; 4.其次，我们证明了模糊集的D-偏序集和序列连续D-同态的范畴ID可以将经典事件到模糊事件的传递刻画为具有非平凡量子特征的最小推广：退化状态可以转换为非退化状态。第三，我们描述了一个基于类别ID的概率论通用模型，使经典概率论和模糊概率论成为特例，并且该模型允许自然修改。最后，本文提出了一种改进方法，用一个合适的单纯形代替传统状态域的闭单位区间[0，1]。
First, we discuss basic probability notions from the viewpoint of category theory. Our approach is based on the following four "sine quibus non" conditions: 1. (elementary) category theory is efficient (and suffices); 2. random variables, observables, probability measures, and states are morphisms; 3. classical probability theory and fuzzy probability theory in the sense of S. Gudder and S. Bugajski are special cases of a more general model; 4. a good model allows natural modifications.Second, we show that the category ID of D-posets of fuzzy sets and sequentially continuous D-homomorphisms allows to characterize the passage from classical to fuzzy events as the minimal generalization having nontrivial quantum character: a degenerated state can be transported to a nondegenerated one.Third, we describe a general model of probability theory based on the category ID so that the classical and fuzzy probability theories become special cases and the model allows natural modifications.Finally, we present a modification in which the closed unit interval [0,1] as the domain of traditional states is replaced by a suitable simplex.