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Fairness Analysis with Shapley-Owen Effects

Harald Ruess

TL;DR

A spectral decomposition of the Shapley-Owen effects is a spectral decomposition of the Shapley-Owen effects, which decomposes the computation of these indices into a model-specific and a model-independent part.

Abstract

We argue that relative importance and its equitable attribution in terms of Shapley-Owen effects is an appropriate one, and, if we accept a small number of reasonable imperatives for equitable attribution, the only way to measure fairness. On the other hand, the computation of Shapley-Owen effects can be very demanding. Our main technical result is a spectral decomposition of the Shapley-Owen effects, which decomposes the computation of these indices into a model-specific and a model-independent part. The model-independent part is precomputed once and for all, and the model-specific computation of Shapley-Owen effects is expressed analytically in terms of the coefficients of the model's \emph{polynomial chaos expansion} (PCE), which can now be reused to compute different Shapley-Owen effects. We also propose an algorithm for computing precise and sparse truncations of the PCE of the model and the spectral decomposition of the Shapley-Owen effects, together with upper bounds on the accumulated approximation errors. The approximations of both the PCE and the Shapley-Owen effects converge to their true values.

Fairness Analysis with Shapley-Owen Effects

TL;DR

A spectral decomposition of the Shapley-Owen effects is a spectral decomposition of the Shapley-Owen effects, which decomposes the computation of these indices into a model-specific and a model-independent part.

Abstract

We argue that relative importance and its equitable attribution in terms of Shapley-Owen effects is an appropriate one, and, if we accept a small number of reasonable imperatives for equitable attribution, the only way to measure fairness. On the other hand, the computation of Shapley-Owen effects can be very demanding. Our main technical result is a spectral decomposition of the Shapley-Owen effects, which decomposes the computation of these indices into a model-specific and a model-independent part. The model-independent part is precomputed once and for all, and the model-specific computation of Shapley-Owen effects is expressed analytically in terms of the coefficients of the model's \emph{polynomial chaos expansion} (PCE), which can now be reused to compute different Shapley-Owen effects. We also propose an algorithm for computing precise and sparse truncations of the PCE of the model and the spectral decomposition of the Shapley-Owen effects, together with upper bounds on the accumulated approximation errors. The approximations of both the PCE and the Shapley-Owen effects converge to their true values.
Paper Structure (27 sections, 8 theorems, 73 equations)

This paper contains 27 sections, 8 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relative Importance
  4. Shapley-Owen Effects and Fairness
  5. Shapley-Owen Effects
  6. Fairness
  7. Fair Attribution
  8. Polynomial Chaos Expansion
  9. Orthonormal Basis
  10. PCE Coefficients
  11. Truncated PCE
  12. Sparse PCE
  13. Sensitivity
  14. PCE for Dependent Inputs
  15. Computing Shapley-Owen Effects
  16. ...and 12 more sections

Key Result

Proposition 5.1

For $\mathcal{M}(X)$ with variance $\sigma^2 < \infty$, the truncation $\widehat{\mathcal{M}}_l(X) \overset{\mathrm{def}}{=\joinrel=} \sum_{\alpha=0}^l y_{\alpha} \Psi_{\alpha}(X)$ of the PCE of $\mathcal{M}(X)$, and $\omega > 0$: $\mathbb{V}({\epsilon_l})< \omega$ if and only if $\sigma^2 - \sum_{\

Theorems & Definitions (17)

  • Example 4.1: huang2023inadequacyshapleyvaluesexplainability
  • Example 4.2
  • Example 4.3
  • Example 4.4
  • Proposition 5.1
  • Proposition 5.2
  • Lemma 5.1
  • Corollary 5.1
  • Lemma 5.2
  • Lemma 6.1
  • ...and 7 more