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：本文探讨了几种关于相关性如何影响信息的常见模型。主要考虑的是香农互信息 $I(S:R_1,\cdots, R_i)$，其分布具有固定的边际分布 $P_{S,R_i}$（其中 $S$ 代表刺激，$R_i$ 代表神经元）。我们利用代数几何工具详细推导了基本模型，写出判别式以将概率单纯形中具有不同定性行为的分布划分为托里奇区域，并代数计算这些区域的体积。作为副产品，我们提供了由级数展开启发的互信息分解与部分信息分解（PID）问题中分解之间的直接转换，并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不同领域在该主题上的交流。关键词：信息论，代数统计，数学神经科学，部分信息分解