From Local to Cluster: A Unified Framework for Causal Discovery with Latent Variables

Abstract

Latent variables pose a fundamental obstacle to both causal discovery and inference. Local approaches exploiting direct neighborhood relations provide little beyond immediate dependencies. Cluster-level methods, though capable of broader reasoning, generally require cluster assignments or causal sufficiency in advance, conditions that are rarely satisfied in practice. Applying single-variable discovery at cluster resolution violates these conditions and thereby produces systematic bias. L2C (Local to Cluster Causal Abstraction) learns cluster structure from local causal patterns without manual variable-to-cluster mapping. The framework rests on a cluster reduction theorem: each cluster can be represented by at most three micro-variables, with no loss of macro-level identifiability. In the presence of latent variables, local discovery on maximal ancestral graphs recovers direct causes, effects, and V-structures, followed by cluster-level inference through a corresponding calculus on the resulting graph. Macro causal effects are formulated as joint interventions on the micro-variables within each target cluster. The framework is sound, atomically complete, and polynomially bounded. L2C establishes a direct connection from local micro-level information to macro causal reasoning in settings where clusters are unknown and latent variables are unavoidable.