We introduce and study the neighbourhood lattice decomposition of a distribution, which is a compact, non-graphical representation of conditional independence that is valid in the absence of a faithful graphical representation. The idea is to view the set of neighbourhoods of a variable as a subset lattice, and partition this lattice into convex sublattices, each of which directly encodes a collection of conditional independence relations. We show that this decomposition exists in any compositional graphoid and can be computed efficiently and consistently in high-dimensions. {In particular, this gives a way to encode all of independence relations implied by a distribution that satisfies the composition axiom, which is strictly weaker than the faithfulness assumption that is typically assumed by graphical approaches.} We also discuss various special cases such as graphical models and projection lattices, each of which has intuitive interpretations. Along the way, we see how this problem is closely related to neighbourhood regression, which has been extensively studied in the context of graphical models and structural equations.

翻译：我们引入并研究一个分布的相邻区域, 即一个紧凑的、非法律的、 有条件的独立代表, 在没有忠实的图形代表的情况下是有效的。 我们的想法是将一个变量的相邻区域视为一个子拉蒂, 并将这个 lattice 分割成一个相交点, 每一个都直接将一系列有条件的独立关系编码起来。 我们显示, 这种分解存在于任何组成图类中, 并且可以在高分层中高效和连贯地计算 。 { 特别是, 这使得能够将满足构成 exiom 的分布所隐含的所有独立关系编码起来, 这个分布绝对弱于通常由图形方法假设的忠诚假设。} 我们还讨论各种特殊案例, 如图形模型和投影等, 每一个都有直觉的解释。 顺便一看, 我们可以看到这个问题如何与邻的回归密切相关, 在图形模型和结构方程式中已经广泛研究过。