Exactly Computing do-Shapley Values

TL;DR

This paper introduces an exact and efficient method for computing do-Shapley values in Structural Causal Models by reformulating the problem in terms of irreducible sets, significantly improving computational speed and estimation accuracy.

Contribution

It presents a novel reformulation of do-Shapley values that enables exact computation in linear time relative to irreducible sets, and develops estimators that outperform prior methods.

Findings

Exact computation time is linear in the number of irreducible sets

Estimators improve accuracy with limited query budgets

When the budget reaches the number of irreducible sets, results are up to machine precision

Abstract

Structural Causal Models (SCM) are a powerful framework for describing complicated dynamics across the natural sciences. A particularly elegant way of interpreting SCMs is do-Shapley, a game-theoretic method of quantifying the average effect of $d$ variables across exponentially many interventions. Like Shapley values, computing do-Shapley values generally requires evaluating exponentially many terms. The foundation of our work is a reformulation of do-Shapley values in terms of the irreducible sets of the underlying SCM. Leveraging this insight, we can exactly compute do-Shapley values in time linear in the number of irreducible sets $r$, which itself can range from $d$ to $2^d$ depending on the graph structure of the SCM. Since $r$ is unknown a priori, we complement the exact algorithm with an estimator that, like general Shapley value estimators, can be run with any query budget. As the query budget approaches $r$, our estimators can produce more accurate estimates than prior methods by several orders of magnitude, and, when the budget reaches $r$, return the Shapley values up to machine precision. Beyond computational speed, we also reduce the identification burden: we prove that non-parametric identifiability of do-Shapley values requires only the identification of interventional effects for the $d$ singleton coalitions, rather than all classes.