Here we cover the full specification of the experiments. Some details were omitted from the main text. If there were separate training and test sets, they were combined before creating the random 10-fold split. All attributes are normalized to mean 0 and standard deviation 1. Additional details for each model type follow.

The decision tree is a flexible machine learning model that finds its success in numerous applications. It is usually fitted in a recursively greedy manner using CART.

This work takes one step towards data encryption by incorporating some crucial ingredients of learning algorithm.

On one hand, the knowledge learned from public data is implicitly transferred into the private model.

's weights define an exponential family.

Probabilistic Regression Trees (PRTrees) generalize traditional decision trees by incorporating probability functions that associate each data point with different regions of the tree, providing smooth decisions and continuous responses. This paper introduces an adaptation of PRTrees capable of handling missing values in covariates through three distinct approaches: (i) a uniform probability method, (ii) a partial observation approach, and (iii) a dimension-reduced smoothing technique. The proposed methods preserve the interpretability properties of PRTrees while extending their applicability to incomplete datasets. Simulation studies under MCAR conditions demonstrate the relative performance of each approach, including comparisons with traditional regression trees on smooth function estimation tasks. The proposed methods, together with the original version, have been developed in R with highly optimized routines and are distributed in the PRTree package, publicly available on CRAN. In this paper we also present and discuss the main functionalities of the PRTree package, providing researchers and practitioners with new tools for incomplete data analysis.