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（CI）的蕴含问题探讨的是：若一个概率分布满足给定的有限CI关系集，是否必然意味着该分布中还存在另一个CI陈述？这一问题历史悠久且引人入胜，最终既产生了关于CI蕴含的积极结论（如半图拟阵公理），也揭示了CI蕴含不存在通用有限特征化的不可能性结果。受因果发现中忠实性假设违反的驱动，我们研究了特殊场景下的CI蕴含问题：即CI关系来源于有向无环图（DAG）模型，并附加一个额外的CI陈述。聚焦于高斯情形，我们利用代数方法完整刻画了此类蕴含何时具有图论性质。此外，受强忠实性对因果发现算法统计保障的启发，我们为近似CI蕴含问题提供了图论解决方案——该问题询问：当某个额外偏相关系数取值较小时，是否必然导致另一个偏相关系数的取值也较小。