Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nondeterministic Causal Models

Sander Beckers

TL;DR

The paper introduces Nondeterministic Structural Causal Models (NSCMs) as a generalization of deterministic structural causal models by employing multi-valued structural equations and dropping exogenous-uniqueness assumptions. It develops three counterfactual logics—single-world (swc), single-context (scc), and single-model (smc)—with a formal actualized-refinement semantics to handle nondeterminism, and provides soundness and completeness results for the swc and scc logics in acyclic settings. The work extends NSCMs to probabilistic causal models (PNSCMs), connecting them to Causal Bayesian Networks and defining probabilities of counterfactuals in this nondeterministic framework, including a discussion of identifiability under partial knowledge. Overall, NSCMs preserve the core causal reasoning of Pearl’s framework while enabling richer counterfactual semantics and offering a promising middle ground for applications in personalized decision-making and causal discovery, with further developments in actual causation and probabilistic identification explored in companion work.

Abstract

I generalize acyclic deterministic structural causal models to the nondeterministic case and argue that this offers an improved semantics for counterfactuals. The standard, deterministic, semantics developed by Halpern (and based on the initial proposal of Galles & Pearl) assumes that for each assignment of values to parent variables there is a unique assignment to their child variable, and it assumes that the actual world (an assignment of values to all variables of a model) specifies a unique counterfactual world for each intervention. Both assumptions are unrealistic, and therefore I drop both of them in my proposal. I do so by allowing multi-valued functions in the structural equations. In addition, I adjust the semantics so that the solutions to the equations that obtained in the actual world are preserved in any counterfactual world. I provide a sound and complete axiomatization of the resulting logic and compare it to the standard one by Halpern and to more recent proposals that are closer to mine. Finally, I extend these models to the probabilistic case and show that they open up the way to identifying counterfactuals even in Causal Bayesian Networks.

Nondeterministic Causal Models

TL;DR

The paper introduces Nondeterministic Structural Causal Models (NSCMs) as a generalization of deterministic structural causal models by employing multi-valued structural equations and dropping exogenous-uniqueness assumptions. It develops three counterfactual logics—single-world (swc), single-context (scc), and single-model (smc)—with a formal actualized-refinement semantics to handle nondeterminism, and provides soundness and completeness results for the swc and scc logics in acyclic settings. The work extends NSCMs to probabilistic causal models (PNSCMs), connecting them to Causal Bayesian Networks and defining probabilities of counterfactuals in this nondeterministic framework, including a discussion of identifiability under partial knowledge. Overall, NSCMs preserve the core causal reasoning of Pearl’s framework while enabling richer counterfactual semantics and offering a promising middle ground for applications in personalized decision-making and causal discovery, with further developments in actual causation and probabilistic identification explored in companion work.

Abstract

I generalize acyclic deterministic structural causal models to the nondeterministic case and argue that this offers an improved semantics for counterfactuals. The standard, deterministic, semantics developed by Halpern (and based on the initial proposal of Galles & Pearl) assumes that for each assignment of values to parent variables there is a unique assignment to their child variable, and it assumes that the actual world (an assignment of values to all variables of a model) specifies a unique counterfactual world for each intervention. Both assumptions are unrealistic, and therefore I drop both of them in my proposal. I do so by allowing multi-valued functions in the structural equations. In addition, I adjust the semantics so that the solutions to the equations that obtained in the actual world are preserved in any counterfactual world. I provide a sound and complete axiomatization of the resulting logic and compare it to the standard one by Halpern and to more recent proposals that are closer to mine. Finally, I extend these models to the probabilistic case and show that they open up the way to identifying counterfactuals even in Causal Bayesian Networks.
Paper Structure (13 sections, 14 theorems, 8 equations)

This paper contains 13 sections, 14 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Nondeterministic Structural Causal Models
  3. The Causal Language
  4. Comparison to the Standard Semantics
  5. Axiomatization and Recent Related Work
  6. Related Work
  7. Probabilistic Causal Models
  8. Counterfactuals in Causal Bayesian Networks
  9. Conclusion
  10. Proofs of Theorems
  11. Details on the Markov condition
  12. Details on the well-definedness of Definition \ref{['def:pc']}
  13. Details on Example \ref{['ex:comp']}

Key Result

theorem 1

Given a nondeterministic causal model $M$, we have that for all $\vec{Y} \subseteq {\cal V}$, for all $\vec{y} \in {\cal R}(\vec{Y})$, and for all basic formulas $\varphi$:

Theorems & Definitions (23)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • theorem 1
  • theorem 2
  • proposition 1
  • definition 5
  • definition 6
  • definition 7
  • ...and 13 more