Shapley Values

Intuition

Features are players in a cooperative game whose payout is the prediction. A fair split gives each player their average marginal contribution over every order in which players could join.

Precise explanation

Value of a coalition $S$ (baseline/interventional version): evaluate the model on an input that takes $x_j$ for $j\in S$ and the baseline $x'_j$ otherwise: $v(S) = f(x_S, x'_{\bar S})$. Shapley value of feature $j$ among $d$ features:

Properties (Lundberg & Lee 2017; Molnar ch. "Shapley values"):

• Efficiency: $\sum_j\phi_j = f(x) - f(x')$.
• Symmetry: interchangeable features get equal values.
• Dummy: a feature that never changes $v$ gets 0. For a linear model, $\phi_j = w_j(x_j - x'_j)$ (XA-01). Exact computation needs $2^d$ model calls — feasible only for small d; SHAP approximates it for larger models.

Worked example

f(x) = x₀·x₁, x = (1, 1), baseline (0, 0): v(∅) = 0, v({0}) = 0, v({1}) = 0, v({0,1}) = 1 → φ₀ = ½(0 − 0) + ½(1 − 0) = 0.5, φ₁ = 0.5. The interaction is split equally.

Common mistakes

• Wrong weights (use $|S|!(d-|S|-1)!/d!$).
• Forgetting the empty coalition.
• Different baselines give different (both valid) explanations — report which you used.

Why this matters

SHAP is one of the most widely used explanation tools; implementing the exact formula shows what the approximations approximate.

References

• A Unified Approach to Interpreting Model Predictions — Scott Lundberg, Su-In Lee (S026)
• Interpretable Machine Learning (3rd edition) — Christoph Molnar (S024)