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.
