We have suggested how we might adjust our prior expectation for any one
element of a collection 
using observations on a
collection 
. When we evaluate a collection of
adjusted expectations { 
, ..., 
}, we also implicitly
evaluate the adjusted value for each element of 
, the collection of
linear combinations of the elements of B, as, by the linearity of adjusted
expectation (equation 2),
We now analyse changes in beliefs over . We consider B, D as
vectors, of dimension r and k, respectively. We define the adjusted version
of the collection B given D, 
, to be the `residual' vector
The properties of the adjusted vector are as for a single quantity, namely
the r dimensional null vector,
the 
null matrix.
Therefore, just as for a single quantity X, we partition the vector B as the sum of two uncorrelated vectors, namely
so that we may partition the variance matrix of B into two variance components
We call
the resolved variance matrix, for B by D. We call
the adjusted variance matrix, for B by D.
, 
are calculated as in equations 1,
8, namely
