As with the mixture of Gaussians, we want to select to minimize . To do this, we must estimate the mean and variance of . With locally weighted regression, these are explicit: the mean is and the variance is . The estimate of is also explicit. Defining as the weight assigned to by the kernel we can compute these expectations exactly in closed form. For the LOESS model, the learner's expected new variance is
Note that, since , the new expectation of Equation 11 may be efficiently computed by caching the values of and . This obviates the need to recompute the entire sum for each new candidate point. The component expectations in Equation 11 are computed as follows:
Just as with the mixture of Gaussians, we can use the expectation in Equation 11 to guide active learning.