Generalizations of the Bias/Variance Decomposition for Prediction Error

The bias and variance of a real valued random variable, using squared error loss, are well understood. However because of recent developments in classiication techniques it has become desirable to extend these concepts to general random variables and loss functions. The 0-1 (misclassiication) loss function with categorical random variables has been of particular interest. We explore the concepts of variance and bias and develop a decomposition of the prediction error into functions of the systematic and variable parts of our predictor. After providing some examples we conclude with a discussion of the various deenitions that have been proposed.