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

## TL;DR

This paper introduces a principled framework for designing correlation quantifiers based on the Principle of Constant Correlations, unifying measures like total correlation and mutual information, and extends to continuous notions of statistical sufficiency.

## Contribution

It develops a set of correlation functionals grounded in first principles, clarifies their properties, and introduces a unified approach to quantify correlation preservation under transformations.

## Key findings

- The n-partite information generalizes total correlation and mutual information.
- Correlation quantifiers can determine if transformations preserve, destroy, or create correlations.
- Quantifies the percentage of correlations preserved under transformations.

## Abstract

Using first principles from inference, we design a set of functionals for the purposes of \textit{ranking} joint probability distributions with respect to their correlations. Starting with a general functional, we impose its desired behaviour through the \textit{Principle of Constant 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 \textit{total correlation} and \textit{mutual information} are special cases. Thus, these functionals are found to be uniquely capable of determining whether a certain class of inferential transformations, $\rho\xrightarrow{*}\rho'$, 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.