Answering Causal Queries at Layer 3 with DiscoSCMs-Embracing Heterogeneity

TL;DR

This paper introduces Distribution-consistency Structural Causal Models (DiscoSCMs) to tackle Layer 3 counterfactuals in the presence of ubiquitous heterogeneity. By incorporating a unit selection variable $U$ and a distributional consistency rule $Y(x) \stackrel{d}{=} Y$, DiscoSCM unifies Potential Outcomes and Structural Causal Models and enables non-degenerate, individualized counterfactual analysis. Under Independent Counterfactual Noises (ICN), Layer 3 valuations factorize into products of Layer 2 valuations, facilitating identification and learning of individual counterfactual probabilities; without ICN, the work derives interpretable, heterogeneous bounds for quantities like the probability of causation. The framework thus provides a principled path to leverage heterogeneity in causal inference, with practical implications for personalized decision-making and causal bounds when cross-world information is limited.

Abstract

In the realm of causal inference, Potential Outcomes (PO) and Structural Causal Models (SCM) are recognized as the principal frameworks.However, when it comes to Layer 3 valuations -- counterfactual queries deeply entwined with individual-level semantics -- both frameworks encounter limitations due to the degenerative issues brought forth by the consistency rule. This paper advocates for the Distribution-consistency Structural Causal Models (DiscoSCM) framework as a pioneering approach to counterfactual inference, skillfully integrating the strengths of both PO and SCM. The DiscoSCM framework distinctively incorporates a unit selection variable $U$ and embraces the concept of uncontrollable exogenous noise realization. Through personalized incentive scenarios, we demonstrate the inadequacies of PO and SCM frameworks in representing the probability of a user being a complier (a Layer 3 event) without degeneration, an issue adeptly resolved by adopting the assumption of independent counterfactual noises within DiscoSCM. This innovative assumption broadens the foundational counterfactual theory, facilitating the extension of numerous theoretical results regarding the probability of causation to an individual granularity level and leading to a comprehensive set of theories on heterogeneous counterfactual bounds. Ultimately, our paper posits that if one acknowledges and wishes to leverage the ubiquitous heterogeneity, understanding causality as invariance across heterogeneous units, then DiscoSCM stands as a significant advancement in the methodology of counterfactual inference.

Figures (2)

• Figure 1: The simplistic DiscoSCM from Example \ref{['eg:noise']} presenting three different correlation patterns.
• Figure 2: DiscoSCM with heterogeneous causal effects, counterfactual noises correlations, and probabilities of consistency. The type of this DiscoSCM depends on the correlation pattern among counterfactual noises $E_{X_1}(t)$, see Fig. \ref{['fig:noise_corr']}: if the correlation coefficient is 1, it resembles an ordinary SCM; if there is some correlation, it is a general DiscoSCM with indeterminable heterogeneous counterfactuals; if the counterfactual noises are independent, it becomes a DiscoSCM where Layer 3 individual-level counterfactuals can be reduced to Layer 2 valuations. Specifically, for the counterfactual parameters $corr(Y^d_u(0), Y^d_u(1))$, the second row of Fig. \ref{['fig:indepDiscoSCM']} shows that it is always 1 in the SCM, while the third row reveals that its value lies between 0 and 1 in the general DiscoSCM, showing heterogeneity according to $X_1$. The fourth row of Fig. \ref{['fig:indepDiscoSCM']} demonstrates that this parameter is always 0 in a DiscoSCM with independent potential noise. To summarize, when the correlation between counterfactual noises is 1, as is the case in SCM, complete knowledge of structural equations is required to deducing for counterfactuals. When counterfactual noises exhibit some correlation, neither RCT or observational data can help recover related counterfactual parameters. Lastly, when counterfactual noises are independent, Layer 3 valuation can be reduced to Layer 2, typically allowing them to be learned from data.

Theorems & Definitions (18)

• definition 1
• definition 2
• definition 3: Counterfactual Outcome
• definition 4: Layer Valuation with DiscoSCM gong2024discoscm
• theorem 1: Individual-Level Valuations gong2024discoscm
• theorem 2: Population-Level Valuations gong2024discoscm
• definition 5
• theorem 3
• Lemma 1
• theorem 4