The implication problem for conditional independence (CI) asks whether the fact that a probability distribution obeys a given finite set of CI relations implies that a further CI statement also holds in this distribution. This problem has a long and fascinating history, cumulating in positive results about implications now known as the semigraphoid axioms as well as impossibility results about a general finite characterization of CI implications. Motivated by violation of faithfulness assumptions in causal discovery, we study the implication problem in the special setting where the CI relations are obtained from a directed acyclic graphical (DAG) model along with one additional CI statement. Focusing on the Gaussian case, we give a complete characterization of when such an implication is graphical by using algebraic techniques. Moreover, prompted by the relevance of strong faithfulness in statistical guarantees for causal discovery algorithms, we give a graphical solution for an approximate CI implication problem, in which we ask whether small values of one additional partial correlation entail small values for yet a further partial correlation.

翻译：条件独立性蕴含问题询问：某一概率分布满足给定有限集的条件独立性关系，是否意味着该分布中还存在另一个条件独立性陈述？该问题拥有悠久而迷人的研究历史，其正面成果集中于现称为半图拟阵公理的蕴含性质，而反面结果则表明条件独立性蕴含通常不存在有限的通用刻画。受因果发现中忠实性假设违反的驱动，我们研究了特殊设定下的蕴含问题：条件独立性关系来源于有向无环图模型外加一个条件独立性陈述。聚焦于高斯情形，我们利用代数技术给出了此类蕴含具有图刻画性的完整特征。此外，鉴于强忠实性在因果发现算法统计保证中的重要性，我们为近似条件独立性蕴含问题提供了图论解法，该问题探讨的是：某个额外偏相关系数的较小值是否蕴含另一偏相关系数的较小值。