To calculate SHAP (SHapley Additive exPlanations) values from Shapley values, we need to understand the foundational principles of Shapley values and their application in machine learning, particularly in explainable AI (XAI). Here is a summary of the relevant information from the provided research papers:

Shapley Value Fundamentals:
 The Shapley value is a solution concept in cooperative game theory that distributes the total gains to the players based on their marginal contributions .
 It is characterized by axioms such as efficiency, symmetry, the null player axiom, and additivity .

Shapley Values in Explainable AI:
 Shapley values have been adapted to explain the output of machine learning models by attributing the contribution of each feature to the prediction .
 SHAP values are a specific implementation of Shapley values in the context of feature importance in machine learning models.

Computational Methods:
 Calculating Shapley values directly can be computationally expensive, but various approximation methods have been developed to make this feasible for large datasets and complex models .
 FastSHAP is one such method that estimates Shapley values efficiently using a learned explainer model.

Applications and Interpretations:
 SHAP values provide a way to interpret the importance of features in a model, offering insights into how different features contribute to the predictions.
 They are particularly useful in regression analysis to understand the intrinsic meaning of predictor variables.
Key Steps to Calculate SHAP from Shapley Values:

Define the Characteristic Function:
 The characteristic function assigns a value to each subset of features, representing the contribution of that subset to the prediction.

Compute Marginal Contributions:
 For each feature, calculate its marginal contribution by considering the difference in the model's output with and without the feature.

Aggregate Contributions:
 Aggregate the marginal contributions across all possible subsets of features to obtain the Shapley value for each feature.

Normalize and Interpret:
 Normalize the Shapley values to ensure they sum up to the total prediction. Interpret these values as the SHAP values, which indicate the importance of each feature in the model's prediction.
By following these steps, you can effectively calculate SHAP values from Shapley values, leveraging the theoretical foundations and computational methods discussed in the provided research papers.
Citations:
 "The Shapley Value"
 "Uniqueness of the Shapley Value"
 "A linear approximation method for the Shapley value"
 "A new axiomatization of the shapley value"
 "FaithShap: The Faithful Shapley Interaction Index"
 "Shapley values for feature selection: The good, the bad, and the axioms"
 "FastSHAP: RealTime Shapley Value Estimation"
 "Intrinsic Meaning of Shapley Values in Regression"