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On the Granularity of Causal Effect Identifiability

Yizuo Chen, Adnan Darwiche

TL;DR

This work introduces state-based identifiability, a finer-grained notion of causal identifiability that targets specific states $\mathbf x$ and $\mathbf y$ of treatment and outcome variables, respectively. It shows that state-based identifiability can hold even when classical variable-based identifiability fails, provided additional knowledge such as context-specific independencies (CSIs), conditional functional dependencies (CFDs), or state constraints is available. The authors prove several results, including that CSIs and CFDs can create separations between variable-based and state-based identifiability, and that state constraints interact with CSIs/CFDs to widen identifiability under certain conditions. They extend the functional elimination approach to CFDs and demonstrate both enabling and limiting effects of constraints across different scenarios, illustrating that some causal effects are estimable from observational data only when these refined forms of knowledge are leveraged. Overall, the paper advocates a more nuanced, state-level view of identifiability with practical implications for when observational data can suffice for causal inference in the presence of domain knowledge.

Abstract

The classical notion of causal effect identifiability is defined in terms of treatment and outcome variables. In this note, we consider the identifiability of state-based causal effects: how an intervention on a particular state of treatment variables affects a particular state of outcome variables. We demonstrate that state-based causal effects may be identifiable even when variable-based causal effects may not. Moreover, we show that this separation occurs only when additional knowledge -- such as context-specific independencies and conditional functional dependencies -- is available. We further examine knowledge that constrains the states of variables, and show that such knowledge does not improve identifiability on its own but can improve both variable-based and state-based identifiability when combined with other knowledge such as context-specific independencies. Our findings highlight situations where causal effects of interest may be estimable from observational data and this identifiability may be missed by existing variable-based frameworks.

On the Granularity of Causal Effect Identifiability

TL;DR

This work introduces state-based identifiability, a finer-grained notion of causal identifiability that targets specific states and of treatment and outcome variables, respectively. It shows that state-based identifiability can hold even when classical variable-based identifiability fails, provided additional knowledge such as context-specific independencies (CSIs), conditional functional dependencies (CFDs), or state constraints is available. The authors prove several results, including that CSIs and CFDs can create separations between variable-based and state-based identifiability, and that state constraints interact with CSIs/CFDs to widen identifiability under certain conditions. They extend the functional elimination approach to CFDs and demonstrate both enabling and limiting effects of constraints across different scenarios, illustrating that some causal effects are estimable from observational data only when these refined forms of knowledge are leveraged. Overall, the paper advocates a more nuanced, state-level view of identifiability with practical implications for when observational data can suffice for causal inference in the presence of domain knowledge.

Abstract

The classical notion of causal effect identifiability is defined in terms of treatment and outcome variables. In this note, we consider the identifiability of state-based causal effects: how an intervention on a particular state of treatment variables affects a particular state of outcome variables. We demonstrate that state-based causal effects may be identifiable even when variable-based causal effects may not. Moreover, we show that this separation occurs only when additional knowledge -- such as context-specific independencies and conditional functional dependencies -- is available. We further examine knowledge that constrains the states of variables, and show that such knowledge does not improve identifiability on its own but can improve both variable-based and state-based identifiability when combined with other knowledge such as context-specific independencies. Our findings highlight situations where causal effects of interest may be estimable from observational data and this identifiability may be missed by existing variable-based frameworks.
Paper Structure (8 sections, 12 theorems, 47 equations, 6 figures)

This paper contains 8 sections, 12 theorems, 47 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Two Levels of Identifiability
  3. The Impact of CSI Constraints on Identifiability
  4. The Impact of Conditional Functional Dependencies on Identifiability
  5. The Impact of State Constraints on Identifiability
  6. Conclusion
  7. Functional Elimination Under Conditional Functional Dependencies
  8. Statements and Proofs of Propositions

Key Result

Theorem 1

Consider a CBN ${\cal M}$ that induces $\mathop{\rm Pr}\nolimits({\mathbf V})$ and satisfies the CFD $[{\mathbf P}_1, {\mathbf P}_2\texttt{=} {\mathbf p}_2] \rightarrow W.$ Let ${\cal M}'$ be the result of functionally eliminating $W$ from ${\cal M}$, which induces $\mathop{\rm Pr}\nolimits'({\mathb

Figures (6)

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  • ...and 1 more figures

Theorems & Definitions (27)

  • Definition 1: Classical Identifiability
  • Definition 2: State-Based Identifiability
  • Definition 3
  • Definition 4
  • Theorem 1
  • proof
  • Corollary 1
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 17 more