SHAP-Gated Inference Scheme

• The paper demonstrates how integrating SHAP attributions into inference yields interpretable and robust predictions using additive decomposition principles.
• It presents methodologies that leverage SHAP values to gate feature contributions, enabling hypothesis testing and effective model simplification.
• The approach scales to high-dimensional models with efficient computational strategies, supporting rigorous causal and fairness analyses.

A SHAP-gated inference scheme describes any predictive modeling approach in which inference is modulated—"gated"—using SHAP (SHapley Additive exPlanation) values or their generalizations, so that model predictions are explicitly conditioned on the explanatory strength or feature attribution computed via the Shapley framework. SHAP values represent a principled method for quantifying the contribution of each input feature to a prediction, originating from cooperative game theory and formalized by a unique additive decomposition that satisfies local accuracy, missingness, and consistency. The scheme leverages these explainer values post hoc or integrates them directly into the prediction pipeline, yielding interpretable, robust, and sometimes causal decision architectures.

1. Mathematical Foundation and Uniqueness of SHAP Attributions

SHAP is grounded in the framework of additive feature importance measures, which express an explanation model as a linear function $g(z') = \phi_0 + \sum_{i=1}^M \phi_i z'_i$ over binary feature presence indicators $z'$. Under this setup, the only attribution mechanism satisfying local accuracy ($f(x) = g(x')$), missingness ($z'_i=0 \implies \phi_i=0$), and consistency is provided by the Shapley value:

$$\phi_i(f, x) = \sum_{z' \subseteq x'} \frac{|z'|! (M - |z'| - 1)!}{M!} \left( f_x(z') - f_x(z' \setminus \{i\}) \right),$$

where $f_x(z')$ is the conditional expectation of the model output given the features present in $z'$ (Lundberg et al., 2017). This uniqueness result unifies disparate prior approaches and motivates gating schemes that rely on interpretable, fair attribution mechanisms.

2. Inference Schemes Based on SHAP Values

A SHAP-gated inference scheme refers to approaches where SHAP values are not just used for interpretation, but for directly influencing model output or modulating decisions. In one formalization, Shapley regressions decompose the prediction into additive Shapley components and structure inference as:

$$y_i = \phi_0^S + \sum_{k=1}^n \phi_k^S(\hat{f}, x_i) \cdot \beta_k^S + \epsilon_i,$$

where $\phi_k^S$ are the Shapley values for each input and $\beta_k^S$ are surrogate inference coefficients. Inference proceeds by hypothesis testing on these $\beta_k^S$; if a coefficient is close to zero, the corresponding feature is considered non-informative for prediction and can be effectively "gated" out (Joseph, 2019). This generalizes regression-style inference to arbitrary nonlinear models.

3. Integration with Causal and Path-Wise Explanations

The SHAP-gated approach can be refined for sensitivity analyses, mediation, and fairness diagnostics. Path-wise Shapley effects (PWSHAP) use on-manifold conditional reference distributions and a causal graph (DAG) to decompose the overall effect of a target predictor into contributions along specific causal paths:

$$^f_{C_i}(c) = {}^f_{T\to Y|C_{S^*}}(c) - {}^f_{T\to Y|C_{S^*\setminus \{i\}}}(c_{S^*\setminus \{i\}})$$

Here, the coalition-wise effect is computed by restricting conditioning to relevant subsets of covariates, with inference "gated" along paths specified by the DAG (Ter-Minassian et al., 2023). These methods support rigorous bias and mediation analyses at the local, individual-prediction level, resilient to distributional adversarial manipulation.

4. Computational Strategies and Scalability

Computational tractability is a central theme in SHAP-gated inference. Naïve SHAP computation requires exponentially many function evaluations; various strategies address this:

• Model structure exploitation: When the model admits an additive or low-order decomposition, the SHAP value for each feature can be computed via local summation over components or by using tail formulas with polynomial complexity. Order-$K$ explicit formulas further allow exact computation for arbitrary interaction order (Hu et al., 2023).
• Iterative convergence: For unknown model order, SHAP values are computed iteratively for increasing interaction complexity until convergence.
• Tensor network methods: For models expressible as tensor trains, SHAP computation becomes NC$^2$-parallelizable, allowing poly-logarithmic time evaluation with polynomial processors. Binarized neural networks and tree ensembles can be compiled into tensor networks, with tractability governed predominantly by width and sparsity rather than depth (Marzouk et al., 24 Oct 2025).
• Graph-local sampling: ShapG limits coalition construction to local graph neighborhoods based on feature correlations, drastically reducing the sample space and permitting efficient global explanations (Zhao et al., 29 Jun 2024).

The combination of these methods enables the practical deployment of SHAP-gated schemes for high-dimensional data and complex model architectures.

5. Trustworthiness, Feature Selection, and Aggregate Attribution

Feature selection and model simplification using aggregate SHAP values are common in practice. Theoretical analysis demonstrates that small average SHAP values over the extended support (i.e., sampling from product marginals via random column permutation) imply that the feature can be safely discarded without loss of predictive capacity. This is formalized using the Shapley Lie algebra framework, which rigorously connects vanishing attributions with feature irrelevancy:

If $\bar{\phi}_i(\mu^*, f) \approx 0$ for all $x\in\text{supp}(\mu^*)$, then $f$ is essentially $[d]\setminus\{i\}$-determined (Bhattacharjee et al., 29 Mar 2025). The adaptation of this logic to KernelSHAP via scrambled inputs decouples feature dependencies and ensures soundness of selection decisions.

6. Interaction-Aware and Multiplicative Extensions

Standard SHAP may obscure interaction effects. Interaction-aware schemes partition the feature set into groups based on statistically significant interactions, constructing a surrogate explanation model:

$f(x) \approx \sum_{S\in\Pi^*} v(S)$

where $\Pi^*$ is the optimal partition minimizing a combination of reconstruction error and interaction complexity (Xu et al., 8 Feb 2024). For domains where multiplicative effects dominate, such as insurance or biology, X-SHAP generalizes the SHAP mechanism to multiplicative decompositions:

$f(x) = \psi^0 \times \prod_{j=1}^m \psi^j(x_j)$

with $\psi^j(x)$ defined by coalition-wise ratios in log-space (Bouneder et al., 2020). Both approaches enable gating and aggregation at the interaction or multiplicative meta-feature level, adapting inference logic to the requirements of the application domain.

7. Practical Impact and End-User Applications

SHAP-gated inference has found direct applications in critical fields. An illustration from microseismic event detection combines raw event probabilities from a deep neural network with a gating criterion based on mean SHAP value across components/channels:

$$\hat{y} = \begin{cases} 1 & \text{if } S_6 \geq \text{SHAP}_t \text{ and } p_t \geq p^* \ 0 & \text{otherwise} \end{cases}$$

Experimental evaluations on seismic datasets demonstrate that this gating mechanism increases F1 score and robustness to noise compared to probability-only inference (Abdullin et al., 20 Oct 2025). More broadly, gating schemes informed by SHAP (or its extensions) can enhance trust, transparency, and robustness in automated decision systems used for finance, medicine, and policy.

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SHAP-gated inference schemes offer a theoretically grounded, computationally scalable, and empirically validated framework for controlling model decisions according to fair and interpretable feature attributions. By bridging classical and modern inference principles across arbitrary model families, these schemes ensure that predictive modeling remains both accurate and interpretable in complex, real-world applications.