Abstract In order to trust the predictions of a machine learning algorithm, it is necessary to understand the factors that contribute to those predictions. In the case of probabilistic and uncertainty-aware models, it is necessary to understand not only the reasons for the predictions themselves, but also the reasons for the model’s level of confidence in those predictions. In this paper, we show how existing methods in explainability can be extended to uncertainty-aware models and how such extensions can be used to understand the sources of uncertainty in a model’s predictive distribution. In particular, by adapting permutation feature importance, partial dependence plots, and individual conditional expectation plots, we demonstrate that novel insights into model behaviour may be obtained and that these methods can be used to measure the impact of features on both the entropy of the predictive distribution and the log-likelihood of the ground truth labels under that distribution. With experiments using both synthetic and real-world data, we demonstrate the utility of these approaches to understand both the sources of uncertainty and their impact on model performance.

We present a theory of ensemble diversity, explaining the nature of diversity for a wide range of supervised learning scenarios. This challenge has been referred to as the “holy grail” of ensemble learning, an open research issue for over 30 years. Our framework reveals that diversity is in fact a hidden dimension in the bias-variance decomposition of the ensemble loss. We prove a family of exact bias-variance-diversity decompositions, for a wide range of losses in both regression and classification, e.g., squared, cross-entropy, and Poisson losses. For losses where an additive bias-variance decomposition is not available (e.g., 0/1 loss) we present an alternative approach: quantifying the effects of diversity, which turn out to be dependent on the label distribution. Overall, we argue that diversity is a measure of model fit, in precisely the same sense as bias and variance, but accounting for statistical dependencies between ensemble members. Thus, we should not be ‘maximising diversity’ as so many works aim to do---instead, we have a bias/variance/diversity trade-off to manage.

We introduce a novel bias-variance decomposition for a range of strictly convex margin losses, including the logistic loss (minimized by the classic LogitBoost algorithm) as well as the squared margin loss and canonical boosting loss. Furthermore we show that, for all strictly convex margin losses, the expected risk decomposes into the risk of a "central" model and a term quantifying variation in the functional margin with respect to variations in the training data. These decompositions provide a diagnostic tool for practitioners to understand model overfitting/underfitting, and have implications for additive ensemble models—for example, when our bias-variance decomposition holds, there is a corresponding "ambiguity" decomposition, which can be used to quantify model diversity.