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Models which estimate main effects of individual variables alongside interaction effects have an identifiability challenge: effects can be freely moved between main effects and interaction effects without changing the model prediction. This is a critical problem for interpretability because it permits “contradictory" models to represent the same function. To solve this problem, we propose pure interaction effects: variance in the outcome which cannot be represented by any subset of features. This definition has an equivalence with the Functional ANOVA decomposition. To compute this decomposition, we present a fast, exact algorithm that transforms any piecewise-constant function (such as a tree-based model) into a purified, canonical representation. We apply this algorithm to Generalized Additive Models with interactions trained on several datasets and show large disparity, including contradictions, between the apparent and the purified effects. These results underscore the need to specify data distributions and ensure identifiability before interpreting model parameters. 

Black-box risk scoring models permeate our lives, yet are typically proprietary or opaque. We propose Distill-and-Compare, an approach to audit such models without probing the black-box model API or pre-defining features to audit. To gain insight into black-box models, we treat them as teachers, training transparent student models to mimic the risk scores assigned by the black-box models. We compare the mimic model trained with distillation to a second, un-distilled transparent model trained on ground truth outcomes, and use differences between the two models to gain insight into the black-box model. We demonstrate the approach on four data sets: COMPAS, Stop-and-Frisk, Chicago Police, and Lending Club. We also propose a statistical test to determine if a data set is missing key features used to train the black-box model. Our test finds that the ProPublica data is likely missing key feature(s) used in COMPAS.