The Bayes linear approach, which is based on partial belief specification with expectation as primitive, allows the straightforward construction of models reflecting second-order exchangeability. The approach requires only a small number of belief statements over observables.
Consider, for example, a simple case: a single infinite, exchangeable
sequence of vectors 
, for example, blood
pressure, temperature, etc., measured on a sequence of patients. All
that we must specify is the common expectation vector and variance
matrix, for each 
, and the common covariance matrix between
each pair and 
. From these three specifications, we
can generate an exchangeability representation for the sequence as a
combination of an underlying population vector 
and individual
variation vector 
, so that, for each i,
, where 
is
an uncorrelated sequence of vectors, each with zero mean vector and the
same variance matrix, all of which are uncorrelated with (see
[5]). Under this representation, we require as
specifications the
expectation vector and variance matrix over the mean components,
and 
; and also the residual vector variance matrix,
.
Learning about future values 
from observations
on a collection 
is equivalent to learning about
in the exchangeability representation, so that prediction and
estimation are equivalent. We may show further that the vector of sample
means from the observed sample is Bayes linear
sufficient for prediction of future observations, and that the canonical
directions for an adjustment under exchangeability do not depend on the
sample size. Therefore, the specification and adjustment of exchangeable
beliefs is particularly simple.
Discussions of the principles and practice of adjustment for exchangeable beliefs are given in [5] and [2], and in detail in [10].