Notes on Forr\'e's Notion of Conditional Independence and Causal Calculus for Continuous Variables

TL;DR

This paper discusses Forré's concept of transitional conditional independence, its connection to graphical models and causal calculus for continuous variables, and extends existing algorithms within this framework.

Contribution

It clarifies the motivations, explores subtleties in measure-theoretic causal calculus, and generalizes the ID algorithm for continuous variables.

Findings

Establishes a strong global Markov property for iDMGs.

Highlights subtleties in measure-theoretic causal calculus.

Extends the ID algorithm to a general measure-theoretic setting.

Abstract

Recently, Forr\'e (arXiv:2104.11547, 2021) introduced transitional conditional independence, a notion of conditional independence that provides a unified framework for both random and non-stochastic variables. The original paper establishes a strong global Markov property connecting transitional conditional independencies with suitable graphical separation criteria for directed mixed graphs with input nodes (iDMGs), together with a version of causal calculus for iDMGs in a general measure-theoretic setting. These notes aim to further illustrate the motivations behind this framework and its connections to the literature, highlight certain subtlies in the general measure-theoretic causal calculus, and extend the "one-line" formulation of the ID algorithm of Richardson et al. (Ann. Statist. 51(1):334--361, 2023) to the general measure-theoretic setting.