We develope the framework of transitional conditional independence. For this we introduce transition probability spaces and transitional random variables. These constructions will generalize, strengthen and unify previous notions of (conditional) random variables and non-stochastic variables, (extended) stochastic conditional independence and some form of functional conditional independence. Transitional conditional independence is asymmetric in general and it will be shown that it satisfies all desired relevance relations in terms of left and right versions of the separoid rules, except symmetry, on standard, analytic and universal measurable spaces. As a preparation we prove a disintegration theorem for transition probabilities, i.e. the existence and essential uniqueness of (regular) conditional Markov kernels, on those spaces. Transitional conditional independence will be able to express classical statistical concepts like sufficiency, adequacy and ancillarity. As an application, we will then show how transitional conditional independence can be used to prove a directed global Markov property for causal graphical models that allow for non-stochastic input variables in strong generality. This will then also allow us to show the main rules of causal/do-calculus, relating observational and interventional distributions, in such measure theoretic generality.
翻译:我们开发了过渡性有条件独立框架。 为此,我们引入了过渡性概率空间和过渡性随机变量。这些构造将概括、加强和统一(有条件)随机变量和非随机变量的先前概念,(延伸)随机有条件独立和某种形式的功能性有条件独立。过渡性有条件独立在总体上是不对称的,并将显示它满足了所有期望的关联关系,即单体规则的左版和右版,但对称性、标准、分析性和通用可计量空间除外。作为我们证明过渡性可能性的解体性理论的准备,即这些空间(定期)有条件的马尔科夫内核的存在和基本独特性。过渡性有条件独立将能够表达典型的统计概念,如充足性、适足性和相近性。然后作为应用,我们将展示过渡性有条件独立如何用来证明一种直接的全球马尔科夫属性,以用于用于为主的因果图形模型,允许在强烈的泛泛泛度中进行非随机输入变量。这将使我们能够展示(定期)有条件的马尔科夫骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质骨质