Introducing a notation that will be used throughout the remainder of this text, let an estimate of xn conditioned on all observations up to time tp be written as with
Thus, an estimate of xn that uses all observations, including the current one at time tn, is written as
while one that predicts xn based on all but the current observation is given by
The same can be done for the covariance matrix, writing
with prediction form of the covariance given by
In moving from ...