Information-theoretic quantities reveal dependencies among variables in the structure of joint, marginal, and conditional entropies, but leave some fundamentally different systems indistinguishable. Furthermore, there is no consensus on how to construct and interpret a higher-order generalisation of mutual information (MI). In this manuscript, we show that a recently proposed model-free definition of higher-order interactions amongst binary variables (MFIs), like mutual information, is a M\"obius inversion on a Boolean algebra, but of surprisal instead of entropy. This gives an information-theoretic interpretation to the MFIs, and by extension to Ising interactions. We study the dual objects to MI and MFIs on the order-reversed lattice, and find that dual MI is related to the previously studied differential mutual information, while dual interactions (outeractions) are interactions with respect to a different background state. Unlike mutual information, in- and outeractions uniquely identify all six 2-input logic gates, the dy- and triadic distributions, and different causal dynamics that are identical in terms of their Shannon-information content.

翻译：信息论量通过联合熵、边缘熵和条件熵的结构揭示了变量间的依赖性，但无法区分本质上不同的系统。此外，关于如何构建和解释互信息的高阶推广尚未达成共识。本文表明，近期提出的二元变量间高阶相互作用的无模型定义（MFIs）与互信息类似，是基于布尔代数的默比乌斯反演，但其对象是惊奇度而非熵。这一发现为MFIs提供了信息论解释，并进一步推广至伊辛相互作用。我们在逆序格上研究了互信息与MFIs的对偶对象，发现对偶互信息与先前研究的微分互信息相关，而对偶相互作用（外作用）则是相对于不同背景状态的相互作用。与互信息不同，内作用和外作用能唯一识别所有六种二输入逻辑门、二元和三元分布，以及具有相同香农信息内容的不同因果动力学过程。