全局相关量词的设计与统计充分性的连续概念
The Design of Global Correlation Quantifiers and Continuous Notions of Statistical Sufficiency
摘要：
使用第一性原理的推断，我们设计了一组泛函的目的\textit{ranking}联合概率分布相对于它们的相关性。从一个一般的功能，我们通过\textit{Principle of Constant Correlations}（PCC），这限制了相关功能的行为在统计独立的推理转换下，以一致的方式施加其所需的行为。PCC指导我们选择适当的设计准则来构造所需的泛函。由于推导依赖于将变量空间划分为$n$不相交子空间的选择，因此我们设计的一般泛函是$n$ -partite information（NPI），其中\textit{total correlation}和\textit{mutual information}是其特例。因此，这些泛函被发现是唯一能够确定是否一定类别的推理转换，$ρ\xrightarrow{*}ρ'$，保持，破坏或创建相关性。这通过排除其他可能的全局相关性量词来提供概念上的清晰性。最后，推导和结果使我们能够量化统计充分性的非二进制概念。我们的研究结果表达了在给定的推理变换或变量映射下保留的相关性的百分比。
英文摘要：
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, $ρ\xrightarrow{*}ρ'$, 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.
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