  Decision Trees (DTs) and Random Forests (RFs) are powerful discriminative
learners and tools of central importance to the everyday machine learning
practitioner and data scientist. Due to their discriminative nature, however,
they lack principled methods to process inputs with missing features or to
detect outliers, which requires pairing them with imputation techniques or a
separate generative model. In this paper, we demonstrate that DTs and RFs can
naturally be interpreted as generative models, by drawing a connection to
Probabilistic Circuits, a prominent class of tractable probabilistic models.
This reinterpretation equips them with a full joint distribution over the
feature space and leads to Generative Decision Trees (GeDTs) and Generative
Forests (GeFs), a family of novel hybrid generative-discriminative models. This
family of models retains the overall characteristics of DTs and RFs while
additionally being able to handle missing features by means of marginalisation.
Under certain assumptions, frequently made for Bayes consistency results, we
show that consistency in GeDTs and GeFs extend to any pattern of missing input
features, if missing at random. Empirically, we show that our models often
outperform common routines to treat missing data, such as K-nearest neighbour
imputation, and moreover, that our models can naturally detect outliers by
monitoring the marginal probability of input features.
