[Paper Review] Probabilistic Reasoning across the Causal Hierarchy

Duligur Ibeling, Thomas Icard|arXiv (Cornell University)|Jan 9, 2020

Bayesian Modeling and Causal Inference28 references4 citations

TL;DR

This paper formalizes the three-tier causal hierarchy—association, intervention, and counterfactuals—as progressively more expressive probabilistic logical languages ($\mathcal{L}_1$, $\mathcal{L}_2$, $\mathcal{L}_3$), each interpreted over structural causal models and probabilistic programs. It provides the first finitary, sound, and complete axiomatizations for each level, proving that satisfiability and validity are decidable in polynomial space, thereby establishing a foundational logical framework for probabilistic causal reasoning.

ABSTRACT

We propose a formalization of the three-tier causal hierarchy of association, intervention, and counterfactuals as a series of probabilistic logical languages. Our languages are of strictly increasing expressivity, the first capable of expressing quantitative probabilistic reasoning -- including conditional independence and Bayesian inference -- the second encoding do-calculus reasoning for causal effects, and the third capturing a fully expressive do-calculus for arbitrary counterfactual queries. We give a corresponding series of finitary axiomatizations complete over both structural causal models and probabilistic programs, and show that satisfiability and validity for each language are decidable in polynomial space.

Motivation & Objective

• To formalize the causal hierarchy—association, intervention, and counterfactuals—as distinct probabilistic logical languages with increasing expressivity.
• To develop finitary, sound, and complete axiomatizations for each level of the hierarchy.
• To show that satisfiability and validity for all three languages are decidable in polynomial space.
• To establish equivalence between probabilistic programs with causal interventions and a subclass of computable structural causal models.
• To clarify the logical structure of do-calculus and causal identification through formal syntax and semantics.

Proposed method

• Formalize three logical languages: $\mathcal{L}_1$ for probabilistic associations, $\mathcal{L}_2$ for intervention-based conditionals using $do$-operators, and $\mathcal{L}_3$ for arbitrary boolean combinations of counterfactuals.
• Define semantics over structural causal models and later over probabilistic programs, ensuring consistency across interpretations.
• Construct finitary axiomatizations ($\textsf{AX}_1$, $\textsf{AX}_2$, $\textsf{AX}_3$) for each language, proven sound and complete using techniques from semialgebraic geometry.
• Leverage Tarski's quantifier elimination for real closed fields to show that satisfiability and validity are in $\mathsf{PSPACE}$.
• Prove a small-model property that underpins the $\mathsf{PSPACE}$ decidability result for all three languages.
• Establish equivalence between probabilistic programs with causal interventions and a natural subclass of computable structural causal models, extending completeness to this procedural semantics.

Experimental results

Research questions

• RQ1 Can the three levels of the causal hierarchy—association, intervention, and counterfactuals—be formally captured as progressively more expressive probabilistic logical languages?
• RQ2 Is there a finitary, sound, and complete axiomatization for probabilistic reasoning at each level of the causal hierarchy?
• RQ3 Are satisfiability and validity decidable for these logical languages, and what is their computational complexity?
• RQ4 Can probabilistic programs with causal interventions serve as a procedural semantics equivalent to a subclass of structural causal models?
• RQ5 To what extent do logical languages like $\mathcal{L}_1$, $\mathcal{L}_2$, and $\mathcal{L}_3$ capture and clarify the principles of the do-calculus and causal identification?

Key findings

• The paper provides the first finitary, sound, and complete axiomatization for pure probabilistic logic ($\mathcal{L}_1$), resolving an open problem left by earlier work.
• It presents the first combined finitary axiomatization for probabilistic intervention and counterfactual reasoning ($\mathcal{L}_2$ and $\mathcal{L}_3$), unifying reasoning across the causal hierarchy.
• Satisfiability and validity for all three languages are decidable in polynomial space, with the complexity upper bound established via Tarski's quantifier elimination for real closed fields.
• A small-model property is proven, which implies that all models of a given formula can be bounded in size, enabling the $\mathsf{PSPACE}$ decidability result.
• The paper establishes a formal equivalence between probabilistic programs with causal interventions and a subclass of computable structural causal models, validating the axiomatizations under this procedural interpretation.
• The completeness of the $\textsf{AX}_3$ calculus implies that causal effect queries reducible to probabilistic expressions can be derived using only the do-calculus schemas (11) and (12), along with the axioms of $\textsf{AX}_3$.