A computationally efficient estimator for the Bayes risk is one which achieves a desired accuracy with a minimum of computation. In many problems, for example speech recognition, point evaluations of the class conditional densities are computationally costly. Density evaluations are the single most important factor contributing to the computational effort in Bayes risk estimation, thus the amount of computation required by a bayes risk estimator is defined as the average number of conditional density evaluations it performs. The accuracy of a risk estimator is defined by its variance.