Beyond Shapley: Efficient Computation of Asymmetric Shapley Values

Machine learning pipelines commonly flatten relational data into single-table representations, discarding structural constraints. Widely used Shapley value-based feature attributions then rely on feature independence, evaluating the model on combinations that could never arise in the underlying data, producing misleading explanations. We propose RelShap, a framework that incorporates relational constraints and data provenance into Shapley value computation, restricting both background data and coalition evaluation to relationally valid configurations. The framework is estimator-agnostic and composes with Kernel SHAP, Monte Carlo, and Leverage SHAP without altering their sampling or weighting properties. Functional dependencies further induce equivalence classes over feature coalitions, which RelShap exploits to reduce runtime without changing Shapley values; we provide a combinatorial characterization of the expected speedup. Experiments across multiple datasets, models, and estimators show that RelShap produces explanations that are more faithful to the data-generating process, correctly identifying the dominant feature in controlled settings where existing methods, including Conditional SHAP and ManifoldShap, do not. Our code is available at: https://github.com/duneag2/relshap.

We study the efficient computation of Shapley values for \emph{product games} -- cooperative games in which the coalition value factorizes as a product of per-player terms. Such games arise in machine learning explainability whenever the value function inherits a multiplicative structure from the underlying model, as in kernel methods with product kernels and tree-based models. Our key result is that the Shapley value of each player in a product game admits an exact one-dimensional integral representation: the weighted sum over exponentially many feature coalitions collapses to the integral of a degree-$(d-1)$ polynomial over $[0,1]$, where $d$ is the total number of features. This yields a Gauss--Legendre quadrature scheme that is \emph{provably exact} whenever the number of nodes satisfies $m_q \geq \lceil d/2 \rceil$, and otherwise provides a \emph{near-exact} approximation with error provably decaying geometrically in $m_q$. In practice, a few hundred nodes can achieve highly precise estimates even with thousands of features. Building on this formulation, we derive a numerically stable implementation via log-space evaluation, together with an efficient parallel implementation based on associative scan primitives that achieves $O(d\,m_q)$ total work and $O(\log d)$ parallel time. Experiments show that \textsc{QuadraSHAP} is the fastest numerically stable method across all tested configurations.

Explaining why a specific outcome occurred, and which inputs deserve the blame or credit, is central to philosophical, scientific, and policy analysis. Existing tools split into two camps. The theory of actual causality (AC) gives principled verdicts, but only for toy-sized models, because computing them requires enumerating counterfactual scenarios. Scalable attribution methods like SHAP (or even causal SHAP) at least partially ignore the causal structure that generated the data, and can give answers that conflict with a careful causal analysis. We close this gap with Probabilistic Causal Impact (PCI). PCI builds on actual causality and on Pearl's notions of probability of necessity and sufficiency, but recasts the question of explainability as an estimation problem on a probabilistic causal model that is easily approximated via Monte Carlo. By specifying a distribution over "candidate explanations," a distribution over counterfactual values, and a scoring function, PCI provides tractable, causally grounded, graded explanations, generalizing AC and Pearl's probability of causation as degenerate cases. We evaluate PCI in synthetic and real-world examples, spanning consistency checks with AC, scaling experiments, complex continuous-valued dynamical systems, and a real-world deployed causal machine learning model trained on millions of datapoints.