Asymptotic Theory for Random Forests

TL;DR

This paper establishes the asymptotic normality of random forest predictions based on subsampling and provides a method to estimate their error distribution, advancing random forests towards statistical inference.

Contribution

It proves asymptotic normality for subsampled random forests and introduces a consistent variance estimator using the infinitesimal jackknife.

Findings

Random forest predictions are asymptotically normal under certain subsampling conditions.

The asymptotic variance can be consistently estimated with the infinitesimal jackknife.

Results enable statistical inference for random forest predictions.

Abstract

Random forests have proven to be reliable predictive algorithms in many application areas. Not much is known, however, about the statistical properties of random forests. Several authors have established conditions under which their predictions are consistent, but these results do not provide practical estimates of random forest errors. In this paper, we analyze a random forest model based on subsampling, and show that random forest predictions are asymptotically normal provided that the subsample size s scales as s(n)/n = o(log(n)^{-d}), where n is the number of training examples and d is the number of features. Moreover, we show that the asymptotic variance can consistently be estimated using an infinitesimal jackknife for bagged ensembles recently proposed by Efron (2014). In other words, our results let us both characterize and estimate the error-distribution of random forest predictions, thus taking a step towards making random forests tools for statistical inference instead of just black-box predictive algorithms.