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Understanding variable importances in forests of randomized trees
Gilles Louppe · Louis Wehenkel · Antonio Sutera · Pierre Geurts

Sun Dec 08 12:16 PM -- 12:20 PM (PST) @ Harvey's Convention Center Floor, CC

Despite growing interest and practical use in various scientific areas, variable importances derived from tree-based ensemble methods are not well understood from a theoretical point of view. In this work we characterize the Mean Decrease Impurity (MDI) variable importances as measured by an ensemble of totally randomized trees in asymptotic sample and ensemble size conditions. We derive a three-level decomposition of the information jointly provided by all input variables about the output in terms of i) the MDI importance of each input variable, ii) the degree of interaction of a given input variable with the other input variables, iii) the different interaction terms of a given degree. We then show that this MDI importance of a variable is equal to zero if and only if the variable is irrelevant and that the MDI importance of a relevant variable is invariant with respect to the removal or the addition of irrelevant variables. We illustrate these properties on a simple example and discuss how they may change in the case of non-totally randomized trees such as Random Forests and Extra-Trees.

Author Information

Gilles Louppe (University of Liège)
Louis Wehenkel (Université de Liège)
Antonio Sutera (Université de Liège)
Pierre Geurts (Université de Liège)

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