How Are Shap Values Calculated

Calculate Shapley Additive Explanations (SHAP) for machine learning model interpretability

Comprehensive Guide: How Are SHAP Values Calculated?

SHAP (SHapley Additive exPlanations) values represent a unified approach to explaining machine learning model outputs by connecting optimal credit allocation with local explanations. Developed by Lundberg and Lee in 2017, SHAP values provide a consistent framework for interpreting predictions across different model types while maintaining theoretical guarantees of fairness and consistency.

Mathematical Foundations of SHAP Values

SHAP values are based on the concept of Shapley values from cooperative game theory. For a machine learning model, each feature value of a data instance acts as a “player” in a coalition game where the “payout” is the prediction for that instance. The SHAP value for a feature represents its average marginal contribution across all possible feature coalitions.

The formal definition of SHAP value for feature i is:

φ_i(f, x) = ∑_S⊆F\{i} [|F|!/(|S|!(|F|-|S|-1)!)] × [f_x(S∪{i}) – f_x(S)]

Where:

• φ_i: SHAP value for feature i
• f: The prediction model
• x: The input instance being explained
• F: The set of all features
• S: A subset of features excluding feature i
• f_x(S): The model’s prediction using feature values from x for features in S and reference values for other features

Key Properties of SHAP Values

SHAP values satisfy three important properties that make them theoretically sound:

1. Efficiency: The sum of all SHAP values equals the difference between the model’s prediction for the instance and the average prediction (baseline).
2. Symmetry: Two features that contribute equally to the prediction receive equal SHAP values.
3. Dummy: Features that don’t affect the prediction receive a SHAP value of zero.
4. Additivity: For linear combinations of models, SHAP values are linear combinations of the individual models’ SHAP values.

Computational Approaches for SHAP Values

The exact computation of SHAP values requires evaluating all possible feature subsets (2^M for M features), which becomes computationally infeasible for models with more than 20-30 features. Several approximation methods have been developed:

Method	Best For	Complexity	Accuracy	Implementation
Exact SHAP	Small models (<20 features)	O(2^M × M)	100%	shap.Exact
Kernel SHAP	Any model type	O(T × M^2)	Good approximation	shap.KernelExplainer
Tree SHAP	Tree-based models	O(T × L × D^2)	Exact for trees	shap.TreeExplainer
Linear SHAP	Linear models	O(M)	Exact for linear	shap.LinearExplainer
Deep SHAP	Neural networks	O(T × M × L)	Good approximation	shap.DeepExplainer

The choice of method depends on the model type, number of features, and computational resources available. For production systems with large models, approximate methods like Kernel SHAP or model-specific optimizations (Tree SHAP, Deep SHAP) are typically used.

Practical Calculation Example

Let’s walk through a concrete example of calculating SHAP values for a simple model with 3 binary features (A, B, C) predicting house prices:

1. Define the model: f(A,B,C) = 100 + 50A + 30B + 20C + 10AB + 5AC
2. Choose an instance: x = (A=1, B=0, C=1)
3. Select a reference: Typically the mean prediction over the dataset (e.g., 150)
4. Calculate all coalitions:
   • f(∅) = 100 (baseline)
   • f(A) = 100 + 50 = 150
   • f(B) = 100 + 0 = 100
   • f(C) = 100 + 20 = 120
   • f(AB) = 100 + 50 + 0 + 0 = 150
   • f(AC) = 100 + 50 + 20 + 10 = 180
   • f(BC) = 100 + 0 + 20 = 120
   • f(ABC) = 100 + 50 + 0 + 20 + 0 + 5 = 175
5. Compute SHAP values:
   • φ_A = (1/3)[(150-100) + (1/2)(180-150) + (1/2)(175-120)] = 42.5
   • φ_B = (1/3)[(100-100) + (1/2)(150-100) + (1/2)(175-120)] = 12.5
   • φ_C = (1/3)[(120-100) + (1/2)(180-100) + (1/2)(175-150)] = 25
6. Verify efficiency: 42.5 + 12.5 + 25 = 80 = f(ABC) – f(∅) = 175 – 100 – 5 (baseline adjustment)

SHAP Values for Different Model Types

The computation of SHAP values varies by model type due to different internal structures:

Linear Models

For linear models (f(x) = w^Tx + b), SHAP values have a closed-form solution equal to the model weights multiplied by the feature values (relative to the reference):

φ_i = w_i(x_i – E[x_i])

Tree-Based Models

Tree SHAP leverages the tree structure to compute exact SHAP values in polynomial time (O(TLD²)) where T is trees, L is leaves, and D is depth. It propagates contributions through the tree paths.

Neural Networks

Deep SHAP extends the DeepLIFT approach by distributing contributions according to SHAP properties. It uses a reference distribution and computes multiplicative relationships between layers.

Kernel SHAP

Kernel SHAP is a model-agnostic method that uses a weighted linear regression to approximate SHAP values. The weights come from the Shapley equation and the “kernel” refers to the specific weighting scheme used.

Interpreting SHAP Values

SHAP values provide both local (instance-specific) and global (model-wide) interpretability:

• Local Interpretation:
  • Positive SHAP value: Feature contributes to increasing the prediction
  • Negative SHAP value: Feature contributes to decreasing the prediction
  • Magnitude shows the strength of contribution
• Global Interpretation:
  • SHAP summary plots show feature importance across the dataset
  • SHAP dependence plots reveal feature interactions
  • SHAP decision plots illustrate the prediction pathway

Visualization Type	Purpose	When to Use	Example Insight
Force Plot	Show individual prediction explanation	Explaining single predictions to stakeholders	“Age contributed +2.5 to the credit score prediction”
Summary Plot	Show feature importance and direction	Understanding global model behavior	“Income is the most important positive feature”
Dependence Plot	Show relationship between feature and prediction	Identifying non-linear relationships	“The effect of age on prediction is U-shaped”
Decision Plot	Show prediction pathway	Debugging model decisions	“The model first considered income, then debt ratio”

Limitations and Considerations

While SHAP values are powerful, they have some limitations to consider:

1. Computational Cost: Exact SHAP becomes impractical for models with >30 features. Approximation methods introduce tradeoffs between speed and accuracy.
2. Reference Dependence: Results depend on the chosen reference value (typically mean or median of the dataset).
3. Feature Correlation: SHAP values assume feature independence. Correlated features may lead to unstable explanations.
4. Model Complexity: For very complex models (e.g., large neural networks), even approximate methods can be computationally expensive.
5. Interpretation Challenges: SHAP values show contribution direction and magnitude but don’t explain why a feature has that effect.

Best practices for using SHAP values include:

• Using appropriate approximation methods for your model size
• Carefully selecting reference values that represent “typical” cases
• Combining SHAP with other interpretation methods for comprehensive understanding
• Validating explanations with domain experts
• Being transparent about approximation methods and their limitations

Advanced Topics in SHAP Values

Recent research has extended SHAP values in several directions:

Hierarchical SHAP

For features with hierarchical relationships (e.g., “location” → “city” → “neighborhood”), hierarchical SHAP distributes contributions according to the feature hierarchy while maintaining the SHAP properties.

Multi-output SHAP

Extended to models with multiple outputs (e.g., multi-class classification), where each output gets its own set of SHAP values that sum to the difference between that output’s prediction and the average.

Causal SHAP

Combines SHAP with causal inference to distinguish between correlational and causal feature contributions, addressing the “correlation ≠ causation” issue in interpretability.

Real-time SHAP

Optimizations for computing SHAP values in real-time for production systems, including model distillation techniques that create simpler “explanation models” that approximate the original model’s SHAP values.

Authoritative Resources on SHAP Values

For deeper understanding, consult these academic and government resources:

• Original SHAP Paper (NeurIPS 2017) – The foundational paper by Lundberg and Lee introducing SHAP values
• NIST AI Explainability Guidelines – U.S. government guidelines on AI interpretability including SHAP values
• SHAP for Neural Networks (arXiv 2019) – Extension of SHAP values to deep learning models
• U.S. Federal AI Guidance – Government policies on AI interpretability including SHAP

Implementing SHAP in Production Systems

For enterprise applications, consider these implementation strategies:

1. Pre-compute SHAP values for common prediction scenarios to reduce latency
2. Use model-specific explainers (TreeSHAP, DeepSHAP) when possible for better performance
3. Implement caching for frequently explained instances
4. Monitor explanation quality alongside model performance
5. Document approximation methods and their limitations for compliance
6. Consider explanation security – SHAP values may reveal sensitive information about the model

Popular libraries for implementing SHAP include:

• Python SHAP library (https://github.com/slundberg/shap) – The standard implementation with support for all major model types
• Alibi (https://github.com/SeldonIO/alibi) – Includes SHAP implementations optimized for production
• Captum (https://github.com/pytorch/captum) – PyTorch library with SHAP implementations for neural networks
• InterpretML (https://github.com/interpretml/interpret) – Microsoft’s interpretability library with SHAP support

The Future of SHAP Values

Emerging directions in SHAP research include:

• Causal SHAP: Integrating causal inference to distinguish between correlational and causal feature importance
• Counterfactual SHAP: Combining SHAP with counterfactual explanations for more actionable insights
• Uncertainty-aware SHAP: Quantifying uncertainty in SHAP values for more reliable interpretations
• Distributed SHAP: Scalable computation for very large models and datasets
• Regulatory compliance: Standardizing SHAP for compliance with AI regulations like the EU AI Act

As AI systems become more prevalent in high-stakes domains, methods like SHAP that provide rigorous, theoretically-grounded explanations will play an increasingly important role in ensuring transparency, fairness, and accountability.