Under certain assumptions, frequently made for Bayes consistencyresults, we show that consistency in GeDTs and GeFs extend to any pattern of missing inputfeatures, ifmissing atrandom.

Then f (x) = p( Y | x), provided that p (x) > 0. Proof. Since the GeDT is deterministic, it has at most one non-zero child. Before proving Theorem 2 we need to introduce some background. We are now ready to prove Theorem 2. Proof. (see also proof of Theorem 1). Here, we assume for simplicity that all variables are continuous.

Decision Trees and Random Forests are among the most widely used machine learning models, and often achieve state-of-the-art performance in tabular, domain-agnostic datasets. Nonetheless, being primarily discriminative models they lack principled methods to manipulate the uncertainty of predictions. In this paper, we exploit Generative Forests (GeFs), a recent class of deep probabilistic models that addresses these issues by extending Random Forests to generative models representing the full joint distribution over the feature space. We demonstrate that GeFs are uncertainty-aware classifiers, capable of measuring the robustness of each prediction as well as detecting out-of-distribution samples.

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.