格上用于高阶交互的信息论测度
Information-Theoretic Measures on Lattices for High-Order Interactions
AI总结：
本文提出基于格理论的系统框架推导高阶信息论测度，并引入定义在完整划分格上、使用广义散度的 Streitberg Information，以完整刻画 d 个变量间的所有交互，并在股票、神经信号和特征选择中验证其有效性。
摘要：
仅基于成对关联的传统测度往往无法捕捉多元数据复杂的统计结构。现有用于识别 d>3 个变量之间共享信息的方法通常在计算上不可行、相对于目标变量不对称，或者无法说明联合概率分布所有可能的因式分解方式。本文提出了一个基于格理论的系统框架，用于推导多元数据的高阶信息论测度。我们的构造使用格与算子函数对，其中算子函数作用于表示变量间代数关系的格上。我们表明，许多常用测度都可以在该框架内推导出来，但它们在 d>3 时无法捕捉所有交互，原因要么是它们定义在受限子格上，要么是因为使用 KL 散度作为算子函数这一典型选择会导致对交互组的不当忽略。为了完整刻画 d 个变量之间的所有交互，我们引入了 Streitberg Information，它定义在完整划分格上，并使用广义散度（超出 KL）作为算子函数。我们在合成数据上验证了 Streitberg Information，并展示了其在检测股票之间的复杂交互、解码神经信号以及机器学习中的特征选择方面的应用。
英文摘要：
Traditional measures based solely on pairwise associations often fail to capture the complex statistical structure of multivariate data. Existing approaches for identifying information shared among $d>3$ variables are frequently computationally intractable, asymmetric with respect to a target variable, or unable to account for all the ways in which the joint probability distribution can be factorised. Here we present a systematic framework based on lattice theory to derive higher-order information-theoretic measures for multivariate data. Our construction uses lattice and operator function pairs, whereby an operator function is applied over a lattice that represents the algebraic relationships among variables. We show that many commonly used measures can be derived within this framework, yet they fail to capture all interactions for $d>3$, either because they are defined on restricted sublattices, or because the use of the KL divergence as an operator function, a typical choice, leads to undesired disregard of groups of interactions. To fully characterise all interactions among $d$ variables, we introduce the Streitberg Information, which is defined over the full partition lattice and uses generalised divergences (beyond KL) as operator functions. We validate the Streitberg Information on synthetic data, and illustrate its application in detecting complex interactions among stocks, decoding neural signals, and performing feature selection in machine learning.
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