To assess how much information about 
we expect to receive by observing
D, we may first identify the particular linear combination 
for which we expect the adjustment by D to be most informative in the sense
that 
maximises the resolution 
over all elements 
with non-zero prior variance. (Note from equation 10, that
maximising the resolution is equivalent to minimising the ratio of adjusted to
prior variance.) We may then proceed to identify directions for which we expect
progressively less information. This is equivalent to defining collections of canonical variables between and 
. We make the following definition.
DEFINITION The 
canonical direction for the adjustment of
B by D is the linear combination 
which maximises over
all elements with non-zero prior variance which are
uncorrelated a priori with 
. We scale each to
have prior expectation zero and prior variance one. The values
are termed the canonical resolutions. The number of canonical directions that we may define is equal to the rank, r(B), of the variance matrix of the elements of B.
The quantities 
are mutually incorrelated. It is also the case that the adjusted expectations, 
are also mutually uncorrelated, and each 
is uncorrelated with each 
The canonical resolutions may be calculated as the eigenvalues of the
resolution matrix, 
, defined as
We may calculate 
by finding the normed
eigenvectors, 
, ordered by eigenvalues 
, of , so that
The collection 
forms a ``grid'' of directions
over 
, summarising the effects of the adjustment. We expect to learn most
about those linear combinations of the elements of B which have large
correlations with those canonical directions with large resolutions. The exact
relation is as follows.
For any X in ,
where
