Carrara Nicholas, Vanslette Kevin
Department of Physics, University at Albany, Albany, NY 12222, USA.
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
Entropy (Basel). 2020 Mar 19;22(3):357. doi: 10.3390/e22030357.
Using first principles from inference, we design a set of functionals for the purposes of ranking joint probability distributions with respect to their correlations. Starting with a general functional, we impose its desired behavior through the 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 disjoint subspaces, the general functional we design is the -partite information (NPI), of which the total correlation and mutual information are special cases. Thus, these functionals are found to be uniquely capable of determining whether a certain class of inferential transformations, ρ → ∗ 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 sufficiency. Our results express what percentage of the correlations are preserved under a given inferential transformation or variable mapping.
利用推理的基本原理，我们设计了一组泛函，用于根据联合概率分布的相关性对其进行排序。从一个通用泛函开始，我们通过恒定相关性原理（PCC）来规定其期望的行为，该原理约束相关泛函在统计独立的推理变换下以一致的方式表现。PCC指导我们选择适当的设计标准来构建所需的泛函。由于推导依赖于将变量空间划分为不相交子空间的选择，我们设计的通用泛函是N - 部信息（NPI），其中总相关性和互信息是特殊情况。因此，发现这些泛函能够唯一地确定某类推理变换ρ → ∗ ρ ' 是保持、破坏还是创建相关性。通过排除其他可能的全局相关量化器，这提供了概念上的清晰性。最后，推导和结果使我们能够量化统计充分性的非二元概念。我们的结果表明在给定的推理变换或变量映射下，有多大比例的相关性得以保留。
