  Random forests have been widely used for their ability to provide so-called
importance measures, which give insight at a global (per dataset) level on the
relevance of input variables to predict a certain output. On the other hand,
methods based on Shapley values have been introduced to refine the analysis of
feature relevance in tree-based models to a local (per instance) level. In this
context, we first show that the global Mean Decrease of Impurity (MDI) variable
importance scores correspond to Shapley values under some conditions. Then, we
derive a local MDI importance measure of variable relevance, which has a very
natural connection with the global MDI measure and can be related to a new
notion of local feature relevance. We further link local MDI importances with
Shapley values and discuss them in the light of related measures from the
literature. The measures are illustrated through experiments on several
classification and regression problems.
