Identifying Macro Causal Effects in a C-DMG over ADMGs

Causal effect identification using causal graphs is a fundamental challenge in causal inference. While extensive research has been conducted in this area, most existing methods assume the availability of fully specified directed acyclic graphs or acyclic directed mixed graphs. However, in complex domains such as medicine and epidemiology, complete causal knowledge is often unavailable, and only partial information about the system is accessible. This paper focuses on causal effect identification within partially specified causal graphs, with particular emphasis on cluster-directed mixed graphs (C-DMGs) which can represent many different acyclic directed mixed graphs (ADMGs). These graphs provide a higher-level representation of causal relationships by grouping variables into clusters, offering a more practical approach for handling complex systems. Unlike fully specified ADMGs, C-DMGs can contain cycles, which complicate their analysis and interpretation. Furthermore, their cluster-based nature introduces new challenges, as it gives rise to two distinct types of causal effects: macro causal effects and micro causal effects, each with different properties. In this work, we focus on macro causal effects, which describe the effects of entire clusters on other clusters. We establish that the do-calculus is both sound and complete for identifying these effects in C-DMGs over ADMGs when the cluster sizes are either unknown or of size greater than one. Additionally, we provide a graphical characterization of non-identifiability for macro causal effects in these graphs.