Distribution-consistency Structural Causal Models

TL;DR

This work addresses the degenerative counterfactual problem that arises when strict consistency is imposed in modeling the joint distribution of counterfactuals, such as $(Y(0), Y(1))$. It introduces distribution-consistency and the Distribution-consistency Structural Causal Model (DiscoSCM), which explicitly incorporates a unit-selection variable $U$ and counterfactual noises $\mathbf{E}(\mathbf{x})$ to model probabilistic counterfactuals. The authors prove that Layer-1 and Layer-2 valuations under DiscoSCM coincide with those from SCM, while Layer-3 valuations generally differ, and they introduce the probability of consistency PC$(u)$ as a new, identifiable parameter within DiscoSCM. Through theoretical results and synthetic examples, the paper demonstrates how DiscoSCM enables more nuanced personalizable counterfactual reasoning (e.g., personalized incentives) and provides a Ladder of Causation framework tailored to counterfactuals, offering a practical path for future work in causal modeling and decision-making.

Abstract

In the field of causal modeling, potential outcomes (PO) and structural causal models (SCMs) stand as the predominant frameworks. However, these frameworks face notable challenges in practically modeling counterfactuals, formalized as parameters of the joint distribution of potential outcomes. Counterfactual reasoning holds paramount importance in contemporary decision-making processes, especially in scenarios that demand personalized incentives based on the joint values of $(Y(0), Y(1))$. This paper begins with an investigation of the PO and SCM frameworks for modeling counterfactuals. Through the analysis, we identify an inherent model capacity limitation, termed as the ``degenerative counterfactual problem'', emerging from the consistency rule that is the cornerstone of both frameworks. To address this limitation, we introduce a novel \textit{distribution-consistency} assumption, and in alignment with it, we propose the Distribution-consistency Structural Causal Models (DiscoSCMs) offering enhanced capabilities to model counterfactuals. To concretely reveal the enhanced model capacity, we introduce a new identifiable causal parameter, \textit{the probability of consistency}, which holds practical significance within DiscoSCM alone, showcased with a personalized incentive example. Furthermore, we provide a comprehensive set of theoretical results about the ``Ladder of Causation'' within the DiscoSCM framework. We hope it opens new avenues for future research of counterfactual modeling, ultimately enhancing our understanding of causality and its real-world applications.

Figures (1)

• Figure 1: Causal model for personalized incentives: this diagram illustrates the causal relationships among group assignment $S$, incentive treatment $T$, pre-treatment features $\mathbf{X}$, and the outcome variable $Y$. The model integrates a user representation $U$, capturing all relevant endogenous information (excluding $T$) that determines the Layer valuations regarding to $(T, Y)$. $\pi_B$ stands for a personalized incentive policy that is designed to optimize returns under total budget constraint $B$.

Theorems & Definitions (35)

• definition 1: Structural Causal Models pearl2009causality
• definition 2: Submodel-"Interventional SCM" pearl2009causality
• definition 3: Layer Valuation bareinboim2022pearl
• example 1
• lemma 1
• proof
• definition 4: Distribution-consistency Structural Causal Model (DiscoSCM)
• definition 5: Do-operator
• definition 6: Counterfactual Outcome
• example 2