The graphical structure of Probabilistic Graphical Models (PGMs) encodes the conditional independence (CI) relations that hold in the modeled distribution. Graph algorithms, such as d-separation, use this structure to infer additional conditional independencies, and to query whether a specific CI holds in the distribution. The premise of all current systems-of-inference for deriving CIs in PGMs, is that the set of CIs used for the construction of the PGM hold exactly. In practice, algorithms for extracting the structure of PGMs from data, discover approximate CIs that do not hold exactly in the distribution. In this paper, we ask how the error in this set propagates to the inferred CIs read off the graphical structure. More precisely, what guarantee can we provide on the inferred CI when the set of CIs that entailed it hold only approximately? It has recently been shown that in the general case, no such guarantee can be provided. We prove that such a guarantee exists for the set of CIs inferred in directed graphical models, making the d-separation algorithm a sound and complete system for inferring approximate CIs. We also prove an approximation guarantee for independence relations derived from marginal CIs.

翻译：概率图形模型(PGM) 的图形结构将模型分布中维持的有条件独立关系(CI) 编码为在模型分布中维持的有条件独立关系。 图表算法, 如 d- 分离, 使用此结构来推断附加的有条件依赖性, 并询问某个特定的 CIS 在分布中是否持有 。 目前所有用于在 PGM 中产生 CIS 的系统推论的前提, 是用于构建 PGM 的一组 CIS 精确地维持着。 实际上, 从数据中提取 PGM 结构的算法, 发现分布中并不完全持有的大约 CIM 。 在本文中, 我们询问该集的错误如何传播到推断的 CI 结构 。 更确切地说, 当包含该集的光学序列只维持大约一点时, 我们能够向推断出什么保证? 最近, 在一般情况下, 无法提供这种保证。 我们证明, 在直接的图形模型中推断出一套 CIS 的数据集存在这样的保证, 使 d- 算算算算算算算算算得上离 。 我们还验证了离 CI 的系统 。