Shapley Curves: A Smoothing Perspective

TL;DR

This work defines population-level Shapley curves as local variable-importance functions $\phi_j(x)$ anchored by the true regression $m(x)$ and covariate distribution $F$, addressing the gap in statistical inference for Shapley-based explanations. It develops and analyzes two plug-in estimators—the component-based and integration-based approaches—proving minimax convergence rates and asymptotic normality under nonparametric smoothing, and introduces a novel wild bootstrap for reliable finite-sample inference. The theory reveals that while both estimators share the same asymptotic variance, the integration-based method incurs larger bias due to aggregating lower-dimensional components; the component-based approach is generally more accurate in finite samples. An empirical vehicle-price study demonstrates the practical utility of Shapley curves with confidence intervals, illustrating time-varying, interpretable contributions of horsepower, weight, and length to price.

Abstract

This paper fills the limited statistical understanding of Shapley values as a variable importance measure from a nonparametric (or smoothing) perspective. We introduce population-level \textit{Shapley curves} to measure the true variable importance, determined by the conditional expectation function and the distribution of covariates. Having defined the estimand, we derive minimax convergence rates and asymptotic normality under general conditions for the two leading estimation strategies. For finite sample inference, we propose a novel version of the wild bootstrap procedure tailored for capturing lower-order terms in the estimation of Shapley curves. Numerical studies confirm our theoretical findings, and an empirical application analyzes the determining factors of vehicle prices.

Figures (7)

• Figure 1: Estimated component-based Shapley curves for horsepower in dependence of horsepower (in hp) for a vehicle length of $190$ inches and vehicle weight of $3500$ pounds. The time periods are 2001--2007, 2008--2013, and 2014--2020. Estimated SHAP values (black crosses) and smoothed curve based on these values (green curve).
• Figure 2: First row: Estimated component-based Shapley curves for vehicle weight in dependence of weight (in lbs) for a vehicle length of $190$ inches and $190$ horsepower. The time periods are 2001--2007, 2008--2013, and 2014--2020. Second row: Estimated KernelSHAP values (black crosses) and smoothed curve based on these values (green curve).
• Figure 3: Heatmaps for $m(x) = -\text{sin}(2x_1) + \text{cos}(2x_2) + x_3$ with Gaussian error terms with $n=2000$ for the first variable at $x_3=0$. Left: Population Shapley Curve. Centre: Componend-based estimated Shapley Curve. Right: Squared residuals between estimated and population curve.
• Figure 4: Population Shapley Curve (black) and component-based estimated Shapley Curve (blue) with $95\%$ bootstrap confidence intervals (red) for the first variable (left) and second variable (right). $x_1$ varies on [-2,2] and $x_2=-0.5$. The DGP 3 is used with Gaussian error terms. $B=10000$ bootstrap replications and $n=2000$.
• Figure 5: Estimated component-based Shapley curve for horsepower in dependence of vehicle weight (in lbs) and horsepower (in hp) for a vehicle length of $190$ inches for the pooled data set from the year 2001--2020.

Theorems & Definitions (8)

• Proposition 1
• Theorem 2
• Proposition 3
• Remark 4
• Theorem 5
• Remark 6
• Remark 7
• Corollary 8