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Notes on Forré's Notion of Conditional Independence and Causal Calculus for Continuous Variables

Leihao Chen

Abstract

Recently, Forré (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.

Notes on Forré's Notion of Conditional Independence and Causal Calculus for Continuous Variables

Abstract

Recently, Forré (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.

Paper Structure

This paper contains 18 sections, 9 theorems, 139 equations, 6 figures.

Table of Contents

  1. Preliminaries
  2. Motivating questions and examples
  3. Causal calculus
  4. Sufficiency, ancillarity and adequacy of statistics
  5. Forré's approach
  6. Forré's transitional conditional independence
  7. Causal identification results
  8. Concepts of statistics
  9. Why certain other approaches are unsatisfying
  10. Using classic stochastic conditional independence
  11. Other approaches
  12. On the asymmetry of transitional conditional independence
  13. Relation to Dawid's conditional independence for statistical operations
  14. More on causal calculus for continuous variables
  15. A positivity condition
  16. ...and 3 more sections

Key Result

Theorem 3.5

The transitional conditional independence (def:tran_ci) and the graphical separation rule (def:graph_sep) both satisfy the asymmetric separoid rules.

Figures (6)

  • Figure 1: String-diagrammatic representation of the probability calculus in \ref{['defthm:prob_calculus']}.
  • Figure 2: Causal graph $\mathfrak{D}$ of the CBN $\mathcal{M}$ in \ref{['ex:no_pointwise_ident']}.
  • Figure 3: Causal graph $\mathfrak{D}$ of the CBN $\mathcal{M}$ in \ref{['ex:fail_back_door']}.
  • Figure 4: Causal graph $\mathfrak{D}$ of the CBN $\mathcal{M}$ in \ref{['ex:pos_nonnecessary']}.
  • Figure 5: String-diagrammatic representation of transitional conditional independence.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Remark 1.2: String-diagrammatic representation of probability calculus
  • Definition 1.4: Causal Bayesian Network
  • Remark 1.5: Input nodes
  • Definition 1.6: Hard/soft manipulation on iADMGs
  • Definition 1.7: Hard/soft intervention on L-iCBN
  • Example 2.2: No pointwise identification in general
  • Example 2.3: Failure of back-door adjustment without appropriate positivity condition
  • Definition 3.1: Transitional probability space and transitional random variable
  • Definition 3.2: Transitional conditional independence
  • Remark 3.3: Essential uniqueness
  • ...and 28 more