Variance and bias for general loss functions

When using squared error loss, bias and variance and their decomposition of prediction error are well understood and widely used concepts. However, there is no universally accepted definition for other loss functions. Numerous attempts have been made to extend these concepts beyond squared error loss. Most approaches have focused solely on 0-1 loss functions and have produced significantly different definitions. These differences stem from disagreement as to the essential characteristics that variance and bias should display. This paper suggests an explicit list of rules that we feel any "reasonable" set of definitions should satisfy. Using this framework, bias and variance definitions are produced which generalize to any symmetric loss function. We illustrate these statistics on several loss functions with particular emphasis on 0-1 loss. We conclude with a discussion of the various definitions that have been proposed in the past as well as a method for estimating these quantities on real data sets.