From global to local MDI variable importances for random forests and when they are Shapley values Supplementary materials Antonio Sutera A Proofs

A.1 Proof of Theorem 1 Theorem 1. (MDI are Shapley values) For all feature X Notice already the similarity with the intermediate formulation in the proof of Theorem 1 from [Louppe et al., 2013] where Equation 5 reduces the inner sum to a single term, the one corresponding to the given b = x This proof directly stems from the following intuitive observation: the irrelevance property considers all x while the local irrelevance one only considers one x. If local irrelevance is satisfied for all x, then irrelevance is satisfied.