We propose here a generalization of regression trees, referred to as Probabilistic Regression (PR) trees, that adapt to the smoothness of the prediction function relating input and output variables while preserving the interpretability of the prediction and being robust to noise. In PR trees, an observation is associated to all regions of a tree through a probability distribution that reflects how far the observation is to a region. We show that such trees are consistent, meaning that their error tends to 0 when the sample size tends to infinity, a property that has not been established for similar, previous proposals as Soft trees and Smooth Transition Regression trees. We further explain how PR trees can be used in different ensemble methods, namely Random Forests and Gradient Boosted Trees. Lastly, we assess their performance through extensive experiments that illustrate their benefits in terms of performance, interpretability and robustness to noise.

Regression (PR) trees, that adapt to the smoothness of the prediction function relating input and output variables while preserving the interpretability of the prediction and being robust to noise. In PR trees, an observation is associated to all regions of a tree through a probability distribution that reflects how far the observation is to a region.

We propose here a generalization of regression trees, referred to as Probabilistic Regression (PR) trees, that adapt to the smoothness of the prediction function relating input and output variables while preserving the interpretability of the prediction and being robust to noise. In PR trees, an observation is associated to all regions of a tree through a probability distribution that reflects how far the observation is to a region. We show that such trees are consistent, meaning that their error tends to 0 when the sample size tends to infinity, a property that has not been established for similar, previous proposals as Soft trees and Smooth Transition Regression trees. We further explain how PR trees can be used in different ensemble methods, namely Random Forests and Gradient Boosted Trees. Lastly, we assess their performance through extensive experiments that illustrate their benefits in terms of performance, interpretability and robustness to noise.

Tree-based ensemble methods such as random forests, gradient-boosted trees, and Bayesianadditive regression trees have been successfully used for regression problems in many applicationsand research studies. In this paper, we study ensemble versions of probabilisticregression trees that provide smooth approximations of the objective function by assigningeach observation to each region with respect to a probability distribution. We prove thatthe ensemble versions of probabilistic regression trees considered are consistent, and experimentallystudy their bias-variance trade-off and compare them with the state-of-the-art interms of performance prediction.