  We study the variability of predictions made by bagged learners and random
forests, and show how to estimate standard errors for these methods. Our work
builds on variance estimates for bagging proposed by Efron (1992, 2012) that
are based on the jackknife and the infinitesimal jackknife (IJ). In practice,
bagged predictors are computed using a finite number B of bootstrap replicates,
and working with a large B can be computationally expensive. Direct
applications of jackknife and IJ estimators to bagging require B on the order
of n^{1.5} bootstrap replicates to converge, where n is the size of the
training set. We propose improved versions that only require B on the order of
n replicates. Moreover, we show that the IJ estimator requires 1.7 times less
bootstrap replicates than the jackknife to achieve a given accuracy. Finally,
we study the sampling distributions of the jackknife and IJ variance estimates
themselves. We illustrate our findings with multiple experiments and simulation
studies.
