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Incorporating Information into Shapley Values: Reweighting via a Maximum Entropy Approach

ICML 2024 · Graph learning › Graph algorithms and clustering · approach: Continuous DAG structure learning

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Reinterprets Shapley values through a maximum-entropy lens and proposes Markov Blanket Shapley values and Ancestor Shapley values that reweight orderings or causal graphs using prior structural information.

Problem

Shapley value feature attribution averages marginal contributions uniformly over all orderings, while causal discovery (Pearl-Verma, faithfulness) selects orderings yielding sparsest graphs; the two approaches are not reconciled and uniform weighting ignores prior knowledge about the model's causal

Approach

Standard Shapley values equal the maximum-entropy weighting over orderings; incorporating prior information means zeroing out non-sparsest orderings (Markov Blanket Shapley values, restricted to parents, children and spouses) and, for interventional explanations, averaging per-graph interventional Shapley values over a class of plausible causal graphs (Ancestor Shapley values).

Key results

• In a minimalistic 3-variable linear SEM, Ancestor Shapley values under the three graph sets give X1,X2 attributions of (0.2,0.19), (0.1,0.29) and (0.07,0.19) with intervenability 0.39, 0.39 and 0.26.
• Theorems show that sparsest-ordering weighting over orderings starting from parents and children yields non-zero attributions only for Markov blanket members (parents, children, spouses).
• Markov Blanket Shapley values satisfy efficiency, missingness, symmetry and dummy, while Ancestor Shapley values lose linearity and the dummy property.

Measured effects reported in the paper

Surprising findings

• Blindly applying Occam's razor (sparsest orderings) to Shapley values excludes spouses, which are needed for least-squares prediction, so the sparsest explanation is not fully satisfactory.

Limitations

• Results are theoretical and demonstrated only on a minimalistic linear SEM example.
• Ancestor Shapley values lose linearity and dummy properties.
• Computation requires enumerating all orderings, which is expensive, though efficient algorithms are mentioned for Markov Blanket values in the appendix.

Experimental setup

Metrics: Shapley values intervenability

Baselines: Standard Shapley values Asymmetric Shapley values Causal Shapley values

Builds on: Shapley values Weighted Shapley values (Kalai & Samet) Asymmetric Shapley values (Frye et al.) Causal Shapley values (Heskes et al.) Pearl-Verma theorem

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