In our example so far we have adjusted one belief structure, 
,
by another, 
. Have we exhausted our exploratory possibilities,
or are there extra insights to be had by approaching the problem in a
different way? Well, suppose that we consider the analogy of a
traditional multiple regression, where we try to predict a response
variable 
from a collection of regressors 
.
In terms of this analogy we may be interested not only in the predictive
power of the collection taken as a whole but also in whether every
is useful for the prediction; whether certain subsets of the
's are more useful than others; and so forth.
For our Bayes linear approach we are concerned with similar (but
broader) interests, which we investigate by applying the notion of
adjusting beliefs in stages. An illustration arises from queries raised
about our example so far: two of our analyses to date (the standardised
values for the adjusted expectations of both the original quantities and
the canonical directions) suggest actual changes in expectation
substantially larger than expected.
Simultaneously we expect to learn very little about 
(as its
resolution is only about 5%) and 
(its resolution is only some
2%), but their changes in expectation are relatively very large.
Various evidence points to a surprisingly large value of 
being
at fault. Suppose, then, that we consider 
and 
as
being two distinct sources of information, and suppose that we adjust
firstly by , and then by . (In what follows we
use the notation and 
synonymously.)