The formal structure which is described by our belief specification is as
follows. We have a collection of random quantities 
,
each with finite prior variance. We construct the linear space
consisting of all finite linear combinations
of the elements of C, where 
is the unit constant. We view as a
vector space in which each 
is a vector, and linear combinations of
vectors are the corresponding linear combinations of the random quantities.
is in general the largest structure over which expectations are defined
once we have defined expectations for the elements of C.
Covariance defines an inner product 
and norm over ,
defined, for 
to be
The vector space, , with the covariance inner product 
,
defines an inner product space, which we denote 
. We call
a belief structure with base 
.
In this space, the
`length' of any vector is equal to the standard deviation of the random
quantity.
A belief structure provides the minimal formal structuring for a belief specification which is sufficient for our general analyses. A traditional discrete probability space is represented within this formulation by a base consisting of indicator functions over a partition, so that the vectors are the linear combinations of the indicator functions, or, equivalently, the random variables over the probability space. A continuous probability specification is expressed as the Hilbert space of square integrable functions over the space with respect to the prior measure. In the probability specification, all covariances between all such pairs of random quantities over the space must be specified. The belief structure allows us to restrict, by our choice of base, the specification to any linear subspace of this collection, so that we may specify only those aspects of our beliefs which we are both able and willing to quantify. Therefore, the formal properties of our approach follow from the linearity underlying the inner product structure, which is why we term our approach Bayes linear.
In the following sections, we describe various general properties of belief adjustment. In the final section, we return to the geometry underlying this approach, and describe the formal structure of the analysis.