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Using first principles from inference, we design a set of functionals for thepurposes of \textit{ranking} joint probability distributions with respect totheir correlations. Starting with a general functional, we impose its desiredbehaviour through the \textit{Principle of Constant Correlations} (PCC), whichconstrains the correlation functional to behave in a consistent way understatistically independent inferential transformations. The PCC guides us inchoosing the appropriate design criteria for constructing the desiredfunctionals. Since the derivations depend on a choice of partitioning thevariable space into $n$ disjoint subspaces, the general functional we design isthe $n$-partite information (NPI), of which the \textit{total correlation} and\textit{mutual information} are special cases. Thus, these functionals arefound to be uniquely capable of determining whether a certain class ofinferential transformations, $\rho\xrightarrow{*}\rho'$, preserve, destroy orcreate correlations. This provides conceptual clarity by ruling out otherpossible global correlation quantifiers. Finally, the derivation and resultsallow us to quantify non-binary notions of statistical sufficency. Our resultsexpress what percentage of the correlations are preserved under a giveninferential transformation or variable mapping.

The Design of Global Correlation Quantifiers and Continuous Notions of Statistical Sufficiency