Identifying macro conditional independencies and macro total effects in summary causal graphs with latent confounding

Understanding causal relationships in dynamic systems is essential for numerous scientific fields, including epidemiology, economics, and biology. While causal inference methods have been extensively studied, they often rely on fully specified causal graphs, which may not always be available or practical in complex dynamic systems. Partially specified causal graphs, such as summary causal graphs (SCGs), provide a simplified representation of causal relationships, omitting temporal information and focusing on high-level causal structures. This simplification introduces new challenges concerning the types of queries of interest: macro queries, which involve relationships between clusters represented as vertices in the graph, and micro queries, which pertain to relationships between variables that are not directly visible through the vertices of the graph. In this paper, we first clearly distinguish between macro conditional independencies and micro conditional independencies and between macro total effects and micro total effects. Then, we demonstrate the soundness and completeness of the d-separation to identify macro conditional independencies in SCGs. Furthermore, we establish that the do-calculus is sound and complete for identifying macro total effects in SCGs. Conversely, we also show through various examples that these results do not hold when considering micro conditional independencies and micro total effects.