In particular, consider the reduced density functions 
, etc. Make the hierarchical expansion
and find
where 
means to take the product over unique ordered subsets of 
. Multiplying both sides of this hierarchy by 
and integrating over the variables gives a very suggestive information correlation hierarchy. The order-1 expressions in the hierarchy are the negative entropies of the individual random variables. The order-2 terms are the mutual informations. These are one measure of the degree of mutual dependence of two variables.The order-3 information correlation functions are less amenable to interpretation (see sections 3.13 and 3.14 for interpretations of the information correlation functions), but give an information correlation of three random variables, which is zero iff any two subsets are independent. These are one measure of the degree of mutual dependence of three variables.
Note that the full negentropy (information) of the system of variables is given by the permutation sum of the information correlations as in
It is useful to point out that we now have two quantities which are zero iff the density function factors, 
and 
. In later chapters we will analyze the information correlation structure of several physical systems and draw conclusions about how the correlation structure indicates underlying physical phenomena. Information correlation functions have attracted the attention of liquid theorists [26] and plasma theorists [39, 84].