The variance transform and the resolution transform are particular examples of
the general class of belief transforms. Suppose that we specify two inner
products 
, 
, over 
, derived perhaps from
alternative prior formulations or alternative sampling frames. Provided that
then we may define a bounded, self-adjoint transform T on , under inner product
, with norm 
, for which
T is termed the belief transform for , associated with
. For example, the variance transform 
is obtained by
selecting to be the inner product 
, and
to be the adjusted covariance inner product 
, so
that
Just as the eigenstructure of the variance transform summarises the comparison
between the prior and adjusted variance specification, so does the
eigenstructure of a general belief transform summarise the comparison between
any two inner products. The ratio 
will be large/ near
one / small according as whether X has large components corresponding to
eigenvectors with large/ near one / small eigenvalues.
Belief transforms provide a natural way to compare sequences of inner
products, as they are multiplicative. Let 
be the belief transform for
associated with 
. Then we have
(operator multiplication is by composition, namely
), as
This relation allows us to decompose a particular comparison into constituent
stages. For example, if we wish to adjust [B] by 
, then we
may decompose the overall variance transform 
, into the
product
where 
is the variance transform 
applied to the adjusted space
[B/D], so that
Such multiplicative forms offer a natural sequential construction for a
complicated belief transform. They also allow us to apply the collection of
interpretive and diagnostic tools that we have developed to each stage of a
belief comparison or adjustment.