When BK is available in addition to observational data, a fundamental problem is: what causal relations are identifiable in the presence of latent variables? This problem is fundamental for its implication on the maximally identifiable causal knowledge with the observational data and BK.

In finding causal effects, researchers may want to know the effect across an entire population, sometimes called a total or unconditional causal effect. For example, does free access to pre-kindergarten (pre-K) improve children's socio-emotional skills throughout elementary school (Moffett et al., 2023)? However, researchers may want to know the effect within subgroups of the population, or a conditional causal effect. For instance, is there a subgroup of children who particularly benefit from free access to pre-K? Our research considers identifying these conditional effects from observational data.

Identifying causal relations is crucial for a variety of downstream tasks. In additional to observational data, background knowledge (BK), which could be attained from human expertise or experiments, is usually introduced for uncovering causal relations. This raises an open problem that in the presence of latent variables, what causal relations are identifiable from observational data and BK. In this paper, we propose two novel rules for incorporating BK, which offer a new perspective to the open problem. In addition, we show that these rules are applicable in some typical causality tasks, such as determining the set of possible causal effects with observational data. Our rule-based approach enhances the state-of-the-art method by circumventing a process of enumerating block sets that would otherwise take exponential complexity.