Causal Discovery beyond Conditional Independence: Identifiability, Algorithms, and Statistical Inference for Directed Acyclic and Cyclic Graphs

Learning causal structure from data requires assumptions about mechanisms, noises, observation schemes, and interventions. Historically, causal discovery has relied primarily on conditional independence. This review focuses on modern causal discovery methods that exploit properties of distributions beyond conditional independence. Our organizing principle separates the source of identification from the statistical estimator and its computational implementation. This perspective connects conditional-independence methods with distributional asymmetries, conditional-moment restrictions, and variation across environments or individuals. We examine discovery under latent confounding and feedback, distinguishing causal information about observed variables from reconstruction of latent structure and emphasizing the additional semantics required by cyclic models. We also distinguish population identification, frequentist and Bayesian consistency, uncertainty quantification, and algorithmic guarantees. Throughout, examples show why the same observational distribution can support different conclusions under different model classes. We conclude with guidance for comparing methods at their identifiable targets and a research agenda for uncertainty, heterogeneous populations, non-scalar variables, and the intersection of discrete data, feedback, and hidden confounding.