Consider the function 
. For instance, special cases of this function include all of the moments of a set of random variables, the average
the variance
and the other correlations and cumulants, etc. defined within chapter 3.
Here we demonstrate that the posterior moments of A are available in closed form with the results already available. This leads to the observation that the posterior moments are available for any function which may be expressed as a series in the form of A, and this includes all of the functions we consider in this chapter. However, great simplifications occur for many of these functions, as has already been seen in the case of the entropy, and as will be seen in the cases of the mutual information, chi-squared, and other functions to be considered in the next sections. For now, simply note that
If the components of 
range over the non-negative integers then the powers of A may be written as
where
Thus the posterior moments are available as
When 
then not only are all of the posterior moments available, but the complete posterior distribution is simply expressed (see equations 9.8, 9.12, and 9.10. Equation 9.10 becomes a vector equation, the delta-functions selecting a particular value of 
).