Abstract The interpretation of feature importance in machine learning models is challenging when features are dependent. Permutation feature importance (PFI) ignores such dependencies, which can cause misleading interpretations due to extrapolation. A possible remedy is more advanced conditional PFI approaches that enable the assessment of feature importance conditional on all other features. Due to this shift in perspective and in order to enable correct interpretations, it is beneficial if the conditioning is transparent and comprehensible. In this paper, we propose a new sampling mechanism for the conditional distribution based on permutations in conditional subgroups. As these subgroups are constructed using tree-based methods such as transformation trees, the conditioning becomes inherently interpretable. This not only provides a simple and effective estimator of conditional PFI, but also local PFI estimates within the subgroups. In addition, we apply the conditional subgroups approach to partial dependence plots, a popular method for describing feature effects that can also suffer from extrapolation when features are dependent and interactions are present in the model. In simulations and a real-world application, we demonstrate the advantages of the conditional subgroup approach over existing methods: It allows to compute conditional PFI that is more true to the data than existing proposals and enables a fine-grained interpretation of feature effects and importance within the conditional subgroups.

Abstract Beta coefficients for linear regression models represent the ideal form of an interpretable feature effect. However, for non-linear models such as generalized linear models, the estimated coefficients cannot be interpreted as a direct feature effect on the predicted outcome. Hence, marginal effects are typically used as approximations for feature effects, either as derivatives of the prediction function or forward differences in prediction due to changes in feature values. While marginal effects are commonly used in many scientific fields, they have not yet been adopted as a general model-agnostic interpretation method for machine learning models. This may stem from the ambiguity surrounding marginal effects and their inability to deal with the non-linearities found in black box models. We introduce a unified definition of forward marginal effects (FMEs) that includes univariate and multivariate, as well as continuous, categorical, and mixed-type features. To account for the non-linearity of prediction functions, we introduce a non-linearity measure for FMEs. Furthermore, we argue against summarizing feature effects of a non-linear prediction function in a single metric such as the average marginal effect. Instead, we propose to average homogeneous FMEs within population subgroups, which serve as conditional feature effect estimates.

Interpretable Machine Learning (IML) methods are used to gain insight into the relevance of a feature of interest for the performance of a model. Commonly used IML methods differ in whether they consider features of interest in isolation, e.g., Permutation Feature Importance (PFI), or in relation to all remaining feature variables, e.g., Conditional Feature Importance (CFI). As such, the perturbation mechanisms inherent to PFI and CFI represent extreme reference points. We introduce Relative Feature Importance (RFI), a generalization of PFI and CFI that allows for a more nuanced feature importance computation beyond the PFI versus CFI dichotomy. With RFI, the importance of a feature relative to any other subset of features can be assessed, including variables that were not available at training time. We derive general interpretation rules for RFI based on a detailed theoretical analysis of the implications of relative feature relevance, and demonstrate the method's usefulness on simulated examples.

Counterfactual explanations are one of the most popular methods to make predictions of black box machine learning models interpretable by providing explanations in the form of ‘what-if scenarios’. Most current approaches optimize a collapsed, weighted sum of multiple objectives, which are naturally difficult to balance a-priori. We propose the Multi-Objective Counterfactuals (MOC) method, which translates the counterfactual search into a multi-objective optimization problem. Our approach not only returns a diverse set of counterfactuals with different trade-offs between the proposed objectives, but also maintains diversity in feature space. This enables a more detailed post-hoc analysis to facilitate better understanding and also more options for actionable user responses to change the predicted outcome. Our approach is also model-agnostic and works for numerical and categorical input features. We show the usefulness of MOC in concrete cases and compare our approach with state-of-the-art methods for counterfactual explanations.