Sampling PermutationSHAP for Shapley Estimation

• The paper introduces Sampling PermutationSHAP, a Monte Carlo scheme that approximates Shapley values by averaging marginal contributions over sampled permutations.
• Paired sampling reduces variance and achieves exact recovery for value functions with pairwise interactions, differentiating it from kernel-based methods.
• Its asymptotic normality and additive group recovery enable statistical inference and confidence region construction in feature attribution tasks.

Sampling PermutationSHAP is a Monte Carlo approximation scheme for Shapley-value attribution that estimates feature credits by averaging marginal contributions over sampled feature permutations rather than enumerating all $q!$ orderings. In the permutation formulation, it is fully equivalent to the Shapley value definition itself; the approximation enters only through subsampling of permutations. The paired-sampling variant analyzed in "Shapley Values: Paired-Sampling Approximations" establishes three properties that are central to its modern characterization: unbiasedness with a multivariate central limit theorem, exact recovery for value functions with interactions of maximal order two, and an additive recovery property that is not shared by the paired kernel analogue (Mayer et al., 18 Aug 2025).

1. Permutation formulation of the Shapley value

PermutationSHAP expresses the Shapley value as an average over feature orderings. For feature index $j$, the value is

$$\phi_j = \frac{1}{q!} \sum_{\pi} \left[ \nu\left({\cal C}_{\pi, j} \cup \{j\}\right) - \nu\left({\cal C}_{\pi, j}\right) \right],$$

where ${\cal C}_{\pi, j}$ denotes the set of features preceding $j$ in permutation $\pi$ (Mayer et al., 18 Aug 2025). The quantity inside brackets is the marginal contribution of $j$ when the coalition is built according to $\pi$.

This representation matters because it separates the combinatorics of coalition averaging from the semantics of the value function $\nu$. Sampling PermutationSHAP does not redefine attribution; it approximates the same $\phi_j$ by replacing the full average over permutations with an empirical mean. A common misconception is therefore that permutation-based estimators and coalition-based estimators target different explainability objects. In fact, the permutation formulation is described as an alternative and fully equivalent formulation of Shapley values (Mayer et al., 18 Aug 2025).

2. Monte Carlo approximation and paired sampling

The basic sampling estimator draws i.i.d. permutations $j$0 and averages the corresponding marginal contributions:

$j$1

Its purpose is purely computational: exact evaluation over all permutations is often infeasible, whereas the estimator uses only $j$2 sampled orderings (Mayer et al., 18 Aug 2025).

The paired-sampling version augments each sampled permutation $j$3 with its reverse $j$4. The resulting estimator is

$j$5

The stated rationale is that paired sampling reduces variance and leverages symmetry; each pair $j$6 ensures that all positions for all players are considered symmetrically (Mayer et al., 18 Aug 2025). Operationally, for each sampled permutation one computes the marginal contribution of each feature under both the permutation and its reverse, sums the two contributions, and averages across sampled pairs.

This paired construction is the defining refinement of Sampling PermutationSHAP in the cited analysis. It is not merely a numerical trick: later sections show that it produces exactness properties unavailable to the unpaired Monte Carlo form.

3. Asymptotic distribution and statistical inference

A major theoretical contribution of the 2025 analysis is the asymptotic normality result for paired-sampling PermutationSHAP. The estimator is stated to be unbiased and to satisfy a multivariate central limit theorem:

$j$7

with covariance

$j$8

where

$j$9

(Mayer et al., 18 Aug 2025).

This result places paired-sampling PermutationSHAP within a standard asymptotic-inference regime: at scale $$\phi_j = \frac{1}{q!} \sum_{\pi} \left[ \nu\left({\cal C}_{\pi, j} \cup \{j\}\right) - \nu\left({\cal C}_{\pi, j}\right) \right],$$0, the estimation error converges to a Gaussian law. A plausible implication is that confidence regions, rank-stability diagnostics, and stopping rules can be built on asymptotic variance estimates rather than only on heuristic repetition.

That implication is reflected in adjacent work. "SHAP@k: Efficient and Probably Approximately Correct (PAC) Identification of Top-k Features" frames top-$$\phi_j = \frac{1}{q!} \sum_{\pi} \left[ \nu\left({\cal C}_{\pi, j} \cup \{j\}\right) - \nu\left({\cal C}_{\pi, j}\right) \right],$$1 SHAP identification as an Explore-$$\phi_j = \frac{1}{q!} \sum_{\pi} \left[ \nu\left({\cal C}_{\pi, j} \cup \{j\}\right) - \nu\left({\cal C}_{\pi, j}\right) \right],$$2 multi-armed-bandit problem and states that, for feature-wise estimators, overlap-based stopping and greedy allocation could be applied directly to Sampling PermutationSHAP (Kariyappa et al., 2023). "Statistical Significance of Feature Importance Rankings" then uses asymptotic normality of SHAP sample means to construct pairwise tests that guarantee correct top-$$\phi_j = \frac{1}{q!} \sum_{\pi} \left[ \nu\left({\cal C}_{\pi, j} \cup \{j\}\right) - \nu\left({\cal C}_{\pi, j}\right) \right],$$3 rankings with probability at least $$\phi_j = \frac{1}{q!} \sum_{\pi} \left[ \nu\left({\cal C}_{\pi, j} \cup \{j\}\right) - \nu\left({\cal C}_{\pi, j}\right) \right],$$4 for SHAP and PermutationSHAP (Goldwasser et al., 2024).

4. Exactness for maximal order-two interactions

The most striking non-asymptotic property proved for paired-sampling PermutationSHAP concerns bilinear value functions. If

$$\phi_j = \frac{1}{q!} \sum_{\pi} \left[ \nu\left({\cal C}_{\pi, j} \cup \{j\}\right) - \nu\left({\cal C}_{\pi, j}\right) \right],$$5

so that the value function contains interactions of at most order $$\phi_j = \frac{1}{q!} \sum_{\pi} \left[ \nu\left({\cal C}_{\pi, j} \cup \{j\}\right) - \nu\left({\cal C}_{\pi, j}\right) \right],$$6, then a single paired sample is already exact:

$$\phi_j = \frac{1}{q!} \sum_{\pi} \left[ \nu\left({\cal C}_{\pi, j} \cup \{j\}\right) - \nu\left({\cal C}_{\pi, j}\right) \right],$$7

The paper identifies this as a formal proof of a point that had been conjectured in SHAP documentation: one paired permutation suffices for exact Shapley values when only pairwise interactions are present (Mayer et al., 18 Aug 2025).

This exactness result is stronger than a generic variance-reduction statement. It identifies a concrete structural regime—maximal interaction order two—in which Monte Carlo approximation disappears entirely once reverse-permutation pairing is used. In the comparison reported by the same paper, paired KernelSHAP is also exact for order-two interactions, but it needs $$\phi_j = \frac{1}{q!} \sum_{\pi} \left[ \nu\left({\cal C}_{\pi, j} \cup \{j\}\right) - \nu\left({\cal C}_{\pi, j}\right) \right],$$8 independent samples, whereas paired PermutationSHAP is exact with one pair (Mayer et al., 18 Aug 2025).

A later theoretical development helps contextualize why paired constructions can be so effective more generally. "An Odd Estimator for Shapley Values" proves that the Shapley value depends only on the odd component of a set function and argues that paired sampling orthogonalizes the regression objective so that the irrelevant even component is filtered out (Fumagalli et al., 1 Feb 2026). This suggests a mechanism for the empirical and exactness advantages of paired estimators beyond the bilinear case, although the exact bilinear theorem for Sampling PermutationSHAP is already established directly in (Mayer et al., 18 Aug 2025).

5. Additive recovery and comparison with paired KernelSHAP

The additive recovery property distinguishes paired-sampling PermutationSHAP from its kernel counterpart. If the value function decomposes additively across possibly overlapping groups of features,

$$\phi_j = \frac{1}{q!} \sum_{\pi} \left[ \nu\left({\cal C}_{\pi, j} \cup \{j\}\right) - \nu\left({\cal C}_{\pi, j}\right) \right],$$9

then for each additive group ${\cal C}_{\pi, j}$0,

${\cal C}_{\pi, j}$1

The result is exact at the group level from a single paired sample (Mayer et al., 18 Aug 2025).

The significance of this property is conceptual as well as algorithmic. Local feature attributions are often interpreted in grouped form—subsystems, modalities, blocks of covariates, or semantically coherent fields. Paired-sampling PermutationSHAP exactly preserves the Shapley allocation summed over such groups under additive decomposition, whereas paired KernelSHAP does not possess this exact additive recovery property (Mayer et al., 18 Aug 2025).

The comparative picture reported in the paper is summarized below.

Property	Paired PermutationSHAP	Paired KernelSHAP
CLT (asymptotic normality)	Yes ${\cal C}_{\pi, j}$2	Yes ${\cal C}_{\pi, j}$3
Paired-sampling reduces variance	Yes	Yes
Exact for order-2 interactions	Yes, even for one pair	Yes, but needs ${\cal C}_{\pi, j}$4 independent samples
Additive recovery property	Yes, exact group recovery	No

A common simplification is to treat paired permutation and paired kernel estimators as interchangeable variance-reduction variants. The comparison above shows that this is incomplete: they share asymptotic normality and variance reduction, but they differ materially in finite-sample algebraic exactness and in additive group recovery (Mayer et al., 18 Aug 2025).

6. Broader developments in permutation-based Shapley sampling

Sampling PermutationSHAP is part of a broader line of work on how to choose permutations more effectively than plain i.i.d. Monte Carlo. "Sampling Permutations for Shapley Value Estimation" develops RKHS quadrature on permutation space using the Mallows kernel, kernel herding, and sequential Bayesian quadrature, and also proposes hypersphere-based constructions using orthogonal codes and Sobol sequences; the reported outcome is smaller RMSE than standard Monte Carlo for the same number of model evaluations (Mitchell et al., 2021). "Fast Approximation of the Shapley Values Based on Order-of-Addition Experimental Designs" uses Latin Squares and Component Orthogonal Arrays to obtain unbiased estimators with lower variance than simple random sampling, with special cases of deterministic recovery of the true Shapley value (Yang et al., 2023).

The paired-sampling theme has also been reinterpreted through regression-based approximation. "PolySHAP: Extending KernelSHAP with Interaction-Informed Polynomial Regression" proves that paired sampling in KernelSHAP produces exactly the same Shapley-value approximations as second-order PolySHAP, thereby giving a strong theoretical justification for paired sampling in that setting (Fumagalli et al., 26 Jan 2026). "An Odd Estimator for Shapley Values" generalizes the justification by showing that the Shapley value depends exclusively on the odd component of the set function and that paired sampling filters out the even component (Fumagalli et al., 1 Feb 2026). These results are not restatements of the permutation theorem in (Mayer et al., 18 Aug 2025), but they locate paired-sampling PermutationSHAP within a wider theoretical movement that treats pairing as a structural projection rather than only a variance heuristic.

Finally, the methodological lineage of Sampling PermutationSHAP extends into application-specific explainability frameworks. "TabSHAP" explicitly describes itself as generalizing the Sampling PermutationSHAP paradigm to LLM-based tabular classifiers, replacing scalar-output scoring with Jensen-Shannon divergence between full-input and masked-input class distributions and using key:value field-level masking rather than token-level masking (Chaudhary et al., 22 Apr 2026). This suggests that permutation-based Shapley sampling has become a reusable design pattern, not only a single estimator.

Taken together, these developments position Sampling PermutationSHAP as both a canonical Monte Carlo estimator for Shapley values and a locus of active theory on pairing, asymptotics, structured permutation design, and attribution under interaction. The 2025 paired-sampling analysis is central because it links these themes through exact theorems: unbiased Gaussian asymptotics, one-pair exactness for bilinear games, and exact additive group recovery (Mayer et al., 18 Aug 2025).