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

arXiv: https://arxiv.org/abs/2603.24333 · HTML: https://mlsanity.com/paper/2603.24333/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 25 Mar 2026.

## Builds on (most cited first, 11 of 11)

- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Nested Markov Properties for Acyclic Directed Mixed Graphs](https://mlsanity.com/paper/1701.06686/) (2017)
- [Causal Calculus in the Presence of Cycles, Latent Confounders and Selection Bias](https://mlsanity.com/paper/1901.00433/) (2019)
- [Causal models in string diagrams](https://mlsanity.com/paper/2304.07638/) (2023)
- [Revisiting the General Identifiability Problem](https://mlsanity.com/paper/2206.01081/) (2022)
- [The d-separation criterion in Categorical Probability](https://mlsanity.com/paper/2207.05740/) (2022)
- [Extended Conditional Independence and Applications in Causal Inference](https://mlsanity.com/paper/1512.00245/) (2015)
- [Transitional Conditional Independence](https://mlsanity.com/paper/2104.11547/) (2021)
- [De Finetti's Theorem in Categorical Probability](https://mlsanity.com/paper/2105.02639/) (2021)
- [The Aldousunicodex2013Hoover Theorem in Categorical Probability](https://mlsanity.com/paper/2411.12840/) (2024)
- [Empirical Measures and Strong Laws of Large Numbers in Categorical Probability](https://mlsanity.com/paper/2503.21576/) (2025)

## Built on by (most cited first, 1 of 1)

- [Complete Causal Identification from Ancestral Graphs under Selection Bias](https://mlsanity.com/paper/2603.26301/) (2026)

## Abstract (arXiv)

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.