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On Shapley value for measuring importance of dependent inputs
Art B. Owen, Clémentine Prieur
TL;DR
Dependent inputs expose conceptual problems in ANOVA-based importance measures, while the paper argues that Shapley value provides a more general alternative. Through examples, it shows that Shapley importance avoids negative values and restrictive distributional assumptions, while computation remains challenging.
Problem
ANOVA-based importance measures can face conceptual problems when input variables are dependent, including restrictive distributional assumptions and negative importance values.
Method
The paper uses Shapley value to attribute explanatory power among input variables and studies its properties through simple dependent-input examples.
Results
Shapley value avoids the two conceptual problems: it is defined without onerous assumptions on the input distribution and never gives negative importance.
Takeaways & Limitations
The paper concludes that Shapley value is appropriate for measuring importance with dependent inputs, including cases where functionally equivalent variables receive equal importance and invertible transformations preserve importance.
Takeaways & Limitations
Computing Shapley values remains a serious challenge outside special cases because the definition involves 2^d − 1 models.
Abstract
from arXiv · showhide
This paper makes the case for using Shapley value to quantify the importance of random input variables to a function. Alternatives based on the ANOVA decomposition can run into conceptual and computational problems when the input variables are dependent. Our main goal here is to show that Shapley value removes the conceptual problems. We do this with some simple examples where Shapley value leads to intuitively reasonable nearly closed form values.
1 Introduction
Dependent inputs make conventional ANOVA-based importance measures difficult to define and interpret. The paper argues that Shapley value avoids these conceptual problems, while computational challenges remain.
- Motivation: Sobol’ indices rely on ANOVA, which is conventionally defined for statistically independent inputs, although dependence is common in applications and observational data.Dependence also arises in constrained computer experiments and moderately high-dimensional designed experiments.
- Problems with dependent-input ANOVA: Generalized ANOVA methods for dependent inputs require restrictive distributional conditions and can assign negative importance.The required condition fails for correlated Gaussian inputs, simplex-constrained inputs, and empirical distributions with impossible or unobserved combinations.
- Problems with dependent-input ANOVA: Negative importance is conceptually problematic because a variable irrelevant to the function would receive zero importance and thereby appear more important than a genuinely used variable with negative importance.
- Shapley value: Shapley value offers an alternative that is available for any function in L2 of the appropriate domain and never gives negative importance.It attributes the value created by a team to individual members, treating input variables as team members.
- Scope: The paper focuses on the conceptual appropriateness of Shapley value rather than solving its computational cost, which involves 2^d − 1 models for d inputs.Monte Carlo estimation has been proposed, but computation remains a serious challenge.
2 Notation
The paper introduces functional ANOVA and Sobol’ importance for independent inputs, then explains why dependent-input generalizations lose orthogonality and require restrictive support conditions.
- Notation: The framework considers a real-valued function f on a d-component input x distributed according to user-chosen P, with f(x) having finite second moment.
- Independent-input ANOVA: For independent inputs, functional ANOVA decomposes f into effects indexed by variable subsets, whose orthogonality supports variance-based importance measures.Sobol’ indices measure the importance of individual variables or variable sets through explained variance.
- Dependent-input ANOVA: In dependent-input settings, the Stone-Hooker ANOVA generalization does not generally produce orthogonal effects.Consequently, the effects’ mean squares need not sum to the variance of f.
- Dependent-input ANOVA: For correlated Gaussian inputs and linear f, the generalized effects contain correlation-dependent terms and are not orthogonal under the joint distribution.The example gives f_{1}(x) = (β1 + β2ρ)x1, f_{2}(x) = (β2 + β1ρ)x2, and f_{1,2}(x) = −β2ρx1 −β1ρx2.
- Support conditions: Dependent-input ANOVA requires a joint density bounded below by a product of marginal densities, ruling out holes in the support.This condition fails for non-rectangular regions such as triangles, correlated normal inputs, and distributions with impossible combinations.
- Computational implications: Sobol’ pick-freeze methods can fail under dependence because fixing one input may determine another and prevent valid resampling.In the cited discrete example, setting x1 = 1 means x2 cannot be changed.
3 Shapley value
Shapley value attributes a subset’s explanatory power among individual input variables using axioms that uniquely determine the allocation. For dependent inputs, it remains broadly defined and nonnegative, addressing conceptual problems of dependent-input ANOVA importance measures.
- Shapley value attributes a team’s economic output to individual members; here, the team is the set of input variables and subset value is explanatory power.
- The attribution satisfies efficiency, symmetry, dummy, and additivity, with Shapley showing that these axioms uniquely determine the valuation.
- Shapley value is defined whenever var(E(f(x) | x_u)) is defined, including functions with non-real-valued components and domains containing holes.
- Each marginal explanatory-power gain is nonnegative, so every variable has φ_j ≥ 0 and Shapley value avoids negative importance.
- For independent inputs, using variance explained as subset value yields Shapley values bracketed by two Sobol’ indices.
4 Special cases
The paper develops closed or nearly closed-form Shapley importance results for dependent and independent inputs, showing sensible behavior across regular and corner cases.
- 4.1 Linear functions: Dependence complicates linear Shapley expressions, but a variable with zero regression coefficient can still receive positive importance through dependence.The paper also notes that Gaussian conditional covariances make the calculations convenient.
- 4.1 Linear functions: Shapley importance remains interpretable for correlated Gaussian linear inputs, whereas the Stone-Hooker ANOVA is unavailable when the covariance matrix is non-diagonal.The result applies to f(x)=β0+x^Tβ with x∼N(μ,Σ) and full-rank Σ.
- 4.1 Linear functions: For f(x)=x1, the second variable has importance φ2=ρ^2/2 and the first has φ1=1−ρ^2/2, reaching equal importance when |ρ|=1.The equality at perfect correlation follows from the bijection between variables.
- 4.2 Transformations, bijections and invariance: Under invertible coordinate transformations, Shapley importance is unchanged; bijective variables therefore receive equal importance.This invariance also explains why the extreme linear example with x1=10^6x2 assigns φ1=φ2.
- 4.3 Dependent-variable examples: In the Farlie-Gumbel-Morgenstern example, φ1/σ^2 decreases as dependence increases toward θ=1 but remains above 1/2 when β1>β2>0.For Gaussian inputs, increasing |ρ| generally first raises x1's relative importance before a rapid drop to 1/2 at high correlation; the lognormal case converges monotonically to 1/2.
- 4.4 Corner cases: For three-state binary inputs with an unreachable state, Shapley importance is 1/2 when p0=0 and favors the more probable input when the two nonzero outcomes are equal.The p0=0 case gives a bijection between x1 and x2.
- 4.5 Maximum of exponential random variables: For independent exponential inputs, the paper generalizes a closed-form Shapley calculation for the value E(max_j∈u xj) to any dimension d≥1.The authors include and prove the formula because the earlier result lacked a proof.
5 Conclusions
The paper concludes that Shapley value resolves conceptual difficulties in measuring importance for dependent inputs while retaining useful invariance properties. It also identifies applications and emphasizes that computation remains challenging outside special cases.
- Shapley value avoids negative importance and onerous distributional assumptions that create conceptual problems for dependent-input ANOVA.
- Functionally equivalent variables receive equal Shapley values, while invertible transformations preserve a variable’s Shapley value.
- The paper supports using Shapley value for dependent inputs, following the proposal of Song et al. (2016).
- Computing Shapley values remains challenging outside special cases such as those analyzed in the paper.
- A potential Bayesian application measures how parameter uncertainty contributes to posterior uncertainty in a prediction under an approximate Gaussian posterior.
- The paper also identifies order statistics as a potential application because they naturally lie in a non-rectangular set.
6 Proofs
The proofs derive Shapley values for dependent Gaussian linear models and several discrete or exponential examples. They use conditional covariance, partitioned-matrix identities, and expected maxima to obtain the stated results.
- 4.1 Linear functions: For Gaussian inputs and a linear function, the proof assumes a full-rank covariance matrix and derives conditional variance expressions.
- The proof partitions variables into a subset, the focal variable, and its complement to express the relevant conditional covariance terms.
- The Shapley value of variable j is obtained by combining these conditional quantities in the Shapley-value expression.
- For independent exponential inputs, the proof defines value as the expected maximum and evaluates it for non-identically distributed variables.
- Binomial-coefficient identities and algebraic rearrangement complete the derivation of the exponential-input Shapley result.