The ideas of aleatoric and epistemic uncertainty are widely used to reason about the probabilistic predictions of machine-learning models. We identify incoherence in existing discussions of these ideas and suggest this stems from the aleatoric-epistemic view being insufficiently expressive to capture all the distinct quantities that researchers are interested in. To address this we present a decision-theoretic perspective that relates rigorous notions of uncertainty, predictive performance and statistical dispersion in data. This serves to support clearer thinking as the field moves forward. Additionally we provide insights into popular information-theoretic quantities, showing they can be poor estimators of what they are often purported to measure, while also explaining how they can still be useful in guiding data acquisition.