On some properties of probability kernels
Hristo Sariev
TL;DR
This work analyzes when a probability kernel defines a proper regular conditional distribution (r.c.d.) by linking properness to fundamental properties such as stationarity, reversibility, and totality. It introduces and utilizes the generated σ-algebra $\sigma(R)$ to study the relationships among these properties, establishing implications and equivalences between stationarity, reversibility, totality, and compatibility. A central result shows that if a kernel satisfies $(S)$ and $(T)$ with respect to $\sigma(R)$, it is the r.c.d. of $\nu$ given some countably generated sub-σ-algebra $\mathcal{G}$, with $\mathcal{G}$ essentially unique; this is further refined in the g.c.p. (generated by a countable partition) case. The paper also examines the nuanced behavior of conditioning on atoms, addressing paradoxes and demonstrating consistency results when atoms have positive probability or when σ-algebras share atoms, thereby clarifying the interpretation and limitations of r.c.d.s in measure-theoretic conditioning contexts.
Abstract
Although regular conditional distributions (r.c.d.) are well-defined and widely used measure-theoretic objects, they can violate our intuition from the classical definition of a conditional probability given an event. For that purpose, the notion of a proper r.c.d. has been introduced. Here, we study how properness, viewed as a property of probability kernels in general, is related to stationarity, compatibility, reversibility and totality, revealing the effects these properties have on the structure of probability kernels. As a further development, we consider the inverse problem of characterizing certain classes of r.c.d.s in terms of the above properties. In particular, we derive necessary and sufficient conditions under which, for a given probability kernel, there exists a unique (in some sense) sub-$σ$-algebra such that the probability kernel is a proper r.c.d. given that sub-$σ$-algebra.