While it is natural to view the variance inner product as describing our
uncertainties, we may choose any inner product over 
which describes
relevant aspects of our beliefs as the starting point for our analysis. One
particular choice that is frequently useful is the product inner
product,
This inner product does not set the unit constant 
to zero. We can
represent our original expectations by means of orthogonal projections onto the
subspace generated by the unit constant as
Within this belief structure, the covariance inner product is simply the
adjustment by the unit constant, so that the inner product space that we have
termed [B] above, under this representation is more fully expressed as
. Equivalently [B] is the orthogonal complement of in
under the product inner product. Usually, we suppress the prior
adjustment by for notational simplicity, illustrating our freedom to
choose whichever inner product is appropriate to emphasise the important
features of a particular analysis.
