From interpretability to inference: an estimation framework for universal approximators

TL;DR

The paper bridges interpretability and econometric inference for universal function approximators by decomposing predictions into Shapley values and analyzing their bias and variance. By extending to Shapley-Taylor interactions and coupling with cross-fitting and training-bootstrap, it derives asymptotic unbiasedness of Shapley components and introduces Shapley regressions to test whether components reflect the true DGP, with coefficients that are 0 or 1 in the limit. The methodology is demonstrated on both simulated heterogeneous treatment effects and a real Bank of England information-treatment experiment, showing how to identify true treatment channels such as age and quantify uncertainty. This framework provides a practical, model-agnostic path from interpretability to statistically valid inference for modern ML models, while highlighting local rather than global validity and suggesting avenues for further extensions.

Abstract

We present a novel framework for estimation and inference with the broad class of universal approximators. Estimation is based on the decomposition of model predictions into Shapley values. Inference relies on analyzing the bias and variance properties of individual Shapley components. We show that Shapley value estimation is asymptotically unbiased, and we introduce Shapley regressions as a tool to uncover the true data generating process from noisy data alone. The well-known case of the linear regression is the special case in our framework if the model is linear in parameters. We present theoretical, numerical, and empirical results for the estimation of heterogeneous treatment effects as our guiding example.

Figures (5)

• Figure 1: The principle behind Shapley regression (\ref{['eq:shap_reg']}): Shapley values project the learned functional forms of prediction components on the left-hand side (Shapley value output space) into a linear space with respect to the target space (right-hand side), where the true coefficient values $\beta^S$ can either be zero (noise) or one (signal).
• Figure 2: Inference analysis on simulated DGP (\ref{['eq:sim_dgp']}) using RF, SVM and ANN (columns) for learning selected real and spurious components (rows). Left-hand side axes: Shapley regression coefficients $\hat{\beta}^S$ (dashed lines) and 99$\%$ confidence intervals (shaded areas). Right-hand side axes: True ($\Gamma^S$, dotted lines) and learned ($\hat{\Gamma}^S$ dashed-dotted lines) Shapley predictive shares. We excluded the terms $\phi_{0}$ and $\phi_{t12}$ for better presentation, for which the results are analogous to the other spurious components.
• Figure 3: True (dashed lines) versus learned (dots) treatment interaction effects $\phi_{t*1}$ for RF, SVM and ANN (columns) for different sample sizes: 100 (upper row), 1000 (lower row). Best-fit linear treatment functions are given by the solid lines.
• Figure 4: Distributions of $\hat{\tau}_i$ for different models using box plots: mean (center line), interquartile range (IQR; box), $95\%$ quantile range (whiskers). The linear ATE estimate is shown for reference (horizontal dashed line) with the corresponding $95\%$ confidence intervals (shaded area) from Table \ref{['tab:att']}.
• Figure 5: Estimated treatment function component $\phi_{t*age}$ for different models as a function of age for the treated based on the training bootstrap distribution: individual means (dots), best fits (solid lines, degree-4 polynomials), $95\%$ confidence intervals (CI; inner), sample split adjusted CI (outer). Three individuals older than 80 have been excluded for clearer presentation.

Theorems & Definitions (11)

• Lemma 3.1
• Lemma 4.1
• Theorem 4.2
• Theorem 4.3
• Corollary 4.4
• Proposition 4.5
• Proposition 4.6
• Theorem 4.7
• Lemma 4.8
• Theorem 4.9