Quantifying Uncertainty in Random Forests via Confidence Intervals and Hypothesis Tests

TL;DR

This paper introduces a statistical inference framework for random forests, enabling confidence intervals and hypothesis testing by modeling predictions as U-statistics, with practical estimators and variance estimation methods.

Contribution

It develops a novel inference framework for random forests using U-statistics, allowing confidence intervals and significance tests without extra computational cost.

Findings

Predictions are asymptotically normal, enabling confidence intervals.

The method provides significance testing for features.

Estimators work efficiently with incomplete U-statistics.

Abstract

This work develops formal statistical inference procedures for machine learning ensemble methods. Ensemble methods based on bootstrapping, such as bagging and random forests, have improved the predictive accuracy of individual trees, but fail to provide a framework in which distributional results can be easily determined. Instead of aggregating full bootstrap samples, we consider predicting by averaging over trees built on subsamples of the training set and demonstrate that the resulting estimator takes the form of a U-statistic. As such, predictions for individual feature vectors are asymptotically normal, allowing for confidence intervals to accompany predictions. In practice, a subset of subsamples is used for computational speed; here our estimators take the form of incomplete U-statistics and equivalent results are derived. We further demonstrate that this setup provides a framework for testing the significance of features. Moreover, the internal estimation method we develop allows us to estimate the variance parameters and perform these inference procedures at no additional computational cost. Simulations and illustrations on a real dataset are provided.