We have a collection, C, of random quantities, for which we have specified
prior means, variances and covariances. Suppose now that we observe the values
of a subset, 
, of the members of C. We intend to
modify our beliefs about the remaining quantities, 
, in C. A simple method by which we can modify our prior expectation
statements is to evaluate the adjusted expectation for each quantity.
The adjusted expectation of a random quantity 
,
given observation of a collection of quantities D, written 
, is
defined to be the linear combination
which minimises
over all collections 
, where 
.
is sometimes called the Bayes linear rule for X given D.
is determined by the prior mean, variance and covariance
specifications. If 
is of full rank
then
Adjusted expectation obeys the following properties:
, 
and constants 
we have,
