Shapley Values

Game-theoretic method assigning contribution of individual players in achieving a certain payout.

Definition

The average marginal contribution of a feature value across all possible Feature Subsets.

$\mathrm{val}x(S)=\int \hat{f}(x_1,\dots ,x_p),d\mathbb{P}x\notin S-E_X(\hat{f}(X))$

$$\mathrm{val}x(S)=\int \hat{f}(x_1,\dots ,x_p),d\mathbb{P}x\notin S$$

todo add all math here

Shapley Additive Explanations

Efficiency

• shapley values add up to prediction difference to average prediction

Symmetry

• equal shapley value ←→ equal contributions in all coalitions

Dummy

• feature that doesnt change prediction must have value of zero

Additivity

• linear operation

Pseudocode

featureA = "temp"
X_local = ... # local datapoint

shapley_value = 0.0
for _ in range(N):
		feature_subset = random_subset_that_contains(featureA)
		for _ in range(1_000):
			X = X_local.copy()
			X[not feature_subset] = random_feature(not feature_subset)
			X_withoutA = X.copy()
			X_withoutA[featureA] = random_feature(featureA)

			shapley_value += prefactor(feature_subset)*(f(X) – f(X_withoutA))
	shapley_value = shapley_value / (N*1_000)

Pros

• Model-Agnostic
• “Fair payout” properties
• solid theory

Cons

• only approximations feasible
• not selective, explanations always contain all features
• access to training data needed
• unrealistic datapoints if data is correlated because of random marginalization of features

References

https://www.youtube.com/watch?v=oCIybnawLdg