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Bagging is a device intended for reducing the prediction error of learningalgorithms. In its simplest form, bagging draws bootstrap samples from thetraining sample, applies the learning algorithm to each bootstrap sample, andthen averages the resulting prediction rules.  We extend the definition of bagging from statistics to statisticalfunctionals and study the von Mises expansion of bagged statisticalfunctionals. We show that the expansion is related to the Efron-Stein ANOVAexpansion of the raw (unbagged) functional. The basic observation is that abagged functional is always smooth in the sense that the von Mises expansionexists and is finite of length 1 + resample size $M$. This holds even if theraw functional is rough or unstable. The resample size $M$ acts as a smoothingparameter, where a smaller $M$ means more smoothing.

Smoothing Effects of Bagging: Von Mises Expansions of Bagged Statistical Functionals