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Counterfactual explainability and analysis of variance

Zijun Gao, Qingyuan Zhao

TL;DR

The paper introduces counterfactual explainability, a causal attribution framework that generalizes Sobol-like global sensitivity measures to dependent explanatory variables via DAGs. It links the concept to genetic heritability, uses comonotonicity to achieve point identification, and provides an axiomatic foundation that unifies existing variable-importance notions under an explanation algebra. Identification and estimation are discussed, with a practical estimation strategy based on conditional quantile functions and Monte Carlo sampling, demonstrated on income inequality data where education and gender show substantial explainability and interact in meaningful ways across age groups. The approach offers a principled, causally informed alternative to purely associational explainability methods and highlights both its potential and limitations, including partial identification under general settings and extensions to more complex causal structures.

Abstract

Existing tools for explaining complex models and systems are associational rather than causal and do not provide mechanistic understanding. We propose a new notion called counterfactual explainability for causal attribution that is motivated by the concept of genetic heritability in twin studies. Counterfactual explainability extends methods for global sensitivity analysis (including the functional analysis of variance and Sobol's indices), which assumes independent explanatory variables, to dependent explanations by using a directed acyclic graphs to describe their causal relationship. Therefore, this explanability measure directly incorporates causal mechanisms by construction. Under a comonotonicity assumption, we discuss methods for estimating counterfactual explainability and apply them to a real dataset dataset to explain income inequality by gender, race, and educational attainment.

Counterfactual explainability and analysis of variance

TL;DR

The paper introduces counterfactual explainability, a causal attribution framework that generalizes Sobol-like global sensitivity measures to dependent explanatory variables via DAGs. It links the concept to genetic heritability, uses comonotonicity to achieve point identification, and provides an axiomatic foundation that unifies existing variable-importance notions under an explanation algebra. Identification and estimation are discussed, with a practical estimation strategy based on conditional quantile functions and Monte Carlo sampling, demonstrated on income inequality data where education and gender show substantial explainability and interact in meaningful ways across age groups. The approach offers a principled, causally informed alternative to purely associational explainability methods and highlights both its potential and limitations, including partial identification under general settings and extensions to more complex causal structures.

Abstract

Existing tools for explaining complex models and systems are associational rather than causal and do not provide mechanistic understanding. We propose a new notion called counterfactual explainability for causal attribution that is motivated by the concept of genetic heritability in twin studies. Counterfactual explainability extends methods for global sensitivity analysis (including the functional analysis of variance and Sobol's indices), which assumes independent explanatory variables, to dependent explanations by using a directed acyclic graphs to describe their causal relationship. Therefore, this explanability measure directly incorporates causal mechanisms by construction. Under a comonotonicity assumption, we discuss methods for estimating counterfactual explainability and apply them to a real dataset dataset to explain income inequality by gender, race, and educational attainment.

Paper Structure

This paper contains 32 sections, 7 theorems, 69 equations, 26 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Why explanations need to be causal?
  3. Towards counterfactual explainability
  4. Overview of this article
  5. Functional ANOVA and global sensitivity analysis
  6. Literature review
  7. Unification of variable importance measures
  8. Axiomatization of total explainability
  9. Counterfactual explainability
  10. Motivation: Inconsistency of non-causal explanations
  11. Graphical causal model and counterfactual explainability
  12. Ancestral consistency of counterfactual explainability
  13. Identification and estimation
  14. Identification
  15. Estimation
  16. ...and 17 more sections

Key Result

Theorem 2

Let $W_1,\dots,W_K$ be independent and $f$ be a given function of $W = (W_1,\dots,W_K)$ such that $\mathop{\mathrm{\mathsf{Var}}}\nolimits(f(W)) < \infty$. Let $\xi_1$ be any probability measure on $\mathcal{E}(W)$ such that Let $\xi_2$ be any probability measure on $\mathcal{E}(W)$ such that Let $\xi_3$ be any probability measure on $\mathcal{E}(W)$ such that Let $\xi_4$ by any probability mea

Figures (26)

  • Figure 5: Venn diagrams for the estimated explainabilities from the US Income dataset across age groups.
  • Figure 7: Education distributions conditional on sex, race in age group [45, 50). Histograms for Female, Asian–Pac–Islander (left) and Male, White (right). Shapes are non-Gaussian and differ across groups.
  • Figure 8: Log of income distributions conditional on sex, race, education in age group [45, 50). Histograms for Male, White, education of value $10$ (top left), Male, Asian–Pac–Islander, education of value $10$ (top right), Female, White, education of value $10$ (bottom left), Male, White, education of value $6$ (bottom right). Shapes are non-Gaussian and differ across groups.
  • Figure 10: Comparison of the total explainability and Shapley value of sex.
  • Figure : (a) Linear.
  • ...and 21 more figures

Theorems & Definitions (22)

  • Example 1
  • Example 2
  • Example 3
  • Definition 1: Explanation algebra
  • Theorem 2
  • Example 4
  • Definition 3
  • Theorem 4
  • Example 5: Inconsistency of global sensitivity analysis
  • Definition 5: Causal Markov model
  • ...and 12 more