From global to local MDI variable importances for random forests and when they are Shapley values

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