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

Using first principles from inference, we design a set of functionals for the purposes of $\mathit{\text{ranking}}$ joint probability distributions with respect to their correlations. Starting with a general functional, we impose its desired behaviour through the $\mathit{\text{Principle}}\mathit{\text{of}}\mathit{\text{Constant}}\mathit{\text{Correlations}}$ (PCC), which constrains the correlation functional to behave in a consistent way under statistically independent inferential transformations. The PCC guides us in choosing the appropriate design criteria for constructing the desired functionals. Since the derivations depend on a choice of partitioning the variable space into $n$ disjoint subspaces, the general functional we design is the $n$-partite information (NPI), of which the $\mathit{\text{total}}\mathit{\text{correlation}}$ and $\mathit{\text{mutual}}\mathit{\text{information}}$ are special cases. Thus, these functionals are found to be uniquely capable of determining whether a certain class of inferential transformations, $\rho \stackrel{\ast }{\to }{\rho }^{\mathrm{\prime }}$, preserve, destroy or create correlations. This provides conceptual clarity by ruling out other possible global correlation quantifiers. Finally, the derivation and results allow us to quantify non-binary notions of statistical sufficency. Our results express what percentage of the correlations are preserved under a given inferential transformation or variable mapping.