We have defined a (partial) belief structure as follows:
We have a collection 
, finite or infinite,
of random quantities, each with finite prior variance. We construct the vector
space 
consisting of all finite linear combinations
of the elements of C, where 
is the unit constant. Covariance defines an
inner product 
and norm, over the closure of the equivalence classes of
random quantities which differ by a constant in , defined, for 
to be
The space with covariance inner product is denoted as 
, the
(partial) belief structure with base 
.
Belief adjustment is represented within this structure as follows:
We have a collection 
, the base for our analysis. We construct [C] as above. We
construct the two subspaces [B] and [D] corresponding to bases 
and 
. We define P 
to be the orthogonal
projection from [B] to [D]. Thus, for any 
,
is the element of [D] which is closest to X in the
variance norm. This orthogonal projection is therefore equivalent to the
adjusted expectation, i.e.
Thus the adjusted version of X is
namely the perpendicular vector from X to the subspace [D]. The adjustment
variance 
is therefore equal to the squared perpendicular distance
from X to [D]. Further, as
and [X/D] is perpendicular to , we have
which is the variance partition expressed in equation 9.
If we adjust each member of 
by D, we obtain a new base
, which we write as 
. We use [B/D] to
represent both the vector of elements of and the adjusted belief structure of B by D.
Alternately, it is often useful to identify [B/D] as a subspace of
the overall inner product space 
, namely the orthogonal
complement of [D] in .
Note from this latter representation that
for any bases D and F we may write a direct sum decomposition of 
into orthogonal subspaces as
Therefore, we may write
where the two projections on the right hand side of equation 43 are mutually orthogonal. The variance partition for a partial belief adjustment follows directly from this representation.
