A general definition of influence between stochastic processes

TL;DR

This paper generalizes the concept of influence between stochastic processes using weak and strong local conditional independence, providing a framework for causal modeling that clarifies direct and indirect influences.

Contribution

It introduces a broader class of processes D' for analyzing influence and establishes conditions under which different definitions of influence coincide, advancing causal inference methods.

Findings

Defines a larger class D' for influence analysis

Establishes equivalence of definitions under certain conditions

Clarifies the distinction between direct and indirect influence

Abstract

We extend the study of weak local conditional independence (WCLI) based on a measurability condition made by Commenges and G\'egout-Petit (2009) to a larger class of processes that we call D'. We also give a definition related to the same concept based on certain likelihood processes, using the Girsanov theorem. Under certain conditions, the two definitions coincide on D'. These results may be used in causal models in that we define what may be the largest class of processes in which influences of one component of a stochastic process on another can be described without ambiguity. From WCLI we can contruct a concept of strong local conditional independence (SCLI). When WCLI does not hold, there is a direct influence while when SCLI does not hold there is direct or indirect influence. We investigate whether WCLI and SCLI can be defined via conventional independence conditions and find that this is the case for the latter but not for the former. Finally we recall that causal interpretation does not follow from mere mathematical definitions, but requires working with a good system and with the true probability.