Explainable Mixture Models through Differentiable Rule Learning

Mixture models excel at decomposing complex, multi-modal distributions into simpler probabilistic components, but provide no insight into the conditions under which these components arise. We introduce explainable mixture models (XMM), a framework that pairs each mixture component with a human-interpretable rule over descriptive features. This enables mixtures that are not only statistically expressive but also transparently grounded in the underlying data. We formalize the problem and examine conditions under which an XMM exactly captures a target distribution. We then propose a scalable, differentiable learning procedure for discovering sets of rules. Experiments on synthetic and real-world datasets demonstrate that our method discovers interesting sub-populations in both univariate and multivariate settings, offering interpretable insights into the structure of complex distributions.