We explore a few common models on how correlations affect information. The main model considered is the Shannon mutual information $I(S:R_1,\cdots, R_i)$ over distributions with marginals $P_{S,R_i}$ fixed for each $i$, with the analogy in which $S$ is the stimulus and $R_i$'s are neurons. We work out basic models in details, using algebro-geometric tools to write down discriminants that separate distributions with distinct qualitative behaviours in the probability simplex into toric chambers and evaluate the volumes of them algebraically. As a byproduct, we provide direct translation between a decomposition of mutual information inspired by a series expansion and one from partial information decomposition (PID) problems, characterising the synergistic terms of the former. We hope this paper serves for communication between communities especially mathematics and theoretical neuroscience on the topic. KEYWORDS: information theory, algebraic statistics, mathematical neuroscience, partial information decomposition

翻译：我们探讨了关联性如何影响信息的几个常见模型。主要考虑的模型是在每个i的边际分布P_{S,R_i}固定的条件下，香农互信息I(S:R_1,⋯,R_i)在分布上的表现形式，其中类比S为刺激，R_i为神经元。我们详细研究了基础模型，利用代数几何工具写出判别式，将概率单纯形中具有不同定性行为的分布划分为环面区域，并代数地计算这些区域的体积。作为副产品，我们提供了从级数展开启发的互信息分解与部分信息分解问题之间的直接转换，刻画了前者的协同项。希望本文能促进数学与理论神经科学等社区之间的交流。关键词：信息论；代数统计；数学神经科学；部分信息分解