Calculating Shapley Values

• $p!$: The total number of possible permutations (or orderings) of these variables.
• $J$: A possible permutation of the set of explanatory variables $\{1,2,\ldots,p\}$ included in the model $f()$.
• $\pi(J,j)$: Set of the indices of the variables that are positioned in $J$ before the $j$-th variable.
• $\Delta^{j|\pi(J,j)}(\underline{x}_*)$: The variable-importance measure of $j$ due the variables that have been used before (constant for all permutations $J$)

Average of the variable-importance measures across all possible orderings of explanatory variables

$$\varphi(\underline{x}_*,j) = \frac{1}{p!} \sum_{J} \Delta^{j|\pi(J,j)}(\underline{x}_*)$$

For a large $p$ we can use Monte Carlo estimator