Causal discovery from observational data alone in the presence of unobserved common causes is crucial yet challenging. We categorize the causal relationship between two random variables \(X\) and \(Y\) into the following four categories: two direct-case cases ( \(X \rightarrow Y\) or \(X \leftarrow Y\) ), a case in which \(X\) and \(Y\) are independent, and a case in which \(X\) and \(Y\) are confounded by unobserved common causes \(C\) , and aim to decide among them from observational data alone. We call this the Reichenbach problem, since this categorization is valid if the Reichenbach’s common cause principle is assumed to be true. Although many methods have been proposed to address causal structure learning in the presence of \(C\) , they typically impose assumptions on the nature of \(C\) that require knowledge of how unobserved confounders behave, and it is difficult to guarantee in many practical cases compared to assumptions about observed variables. In our previous study (Kobayashi et al. in: 2022 IEEE international conference on big data (Big Data). IEEE, pp 45–54, 2022), we proposed a causal discovery method without such assumptions regarding the nature of unmeasured confounders, named \({\mathbf{\mathsf{CLOUD}}}\) , for discrete data. Building on the Reichenbach’s common cause principle, if neither the two direct-cause models nor the independence model can explain the data well, we can infer the involvement of unobserved common causes in generating \(X\) and \(Y\) . To realize this idea in practice, \(\mathbf{\mathsf{CLOUD}}\) imposes structural assumptions on the direct-cause models such as additive noise models ( \(\mathbf{\mathsf{ANM}}\) s) (Peters et al. in: Proceedings of the thirteenth international conference on artificial intelligence and statistics. JMLR workshop and conference proceedings, pp 597–604, 2010), while allowing the confounded model to represent any joint distribution on ( \(X\) , \(Y\) ) by placing no assumptions on \(C\) . \(\mathbf{\mathsf{CLOUD}}\) then employs the normalized maximum likelihood (NML) codelength (Shtar’kov in Problemy Peredachi Informatsii 23(3):3–17, 1987) as an information criterion and compares models of varying capacities. This study extends \(\mathbf{\mathsf{CLOUD}}\) to include all data types. We provide NML formulations and theoretical guarantees for consistency in model selection for mixed and continuous cases. Through extensive experiments on both synthetic and real-world data, we demonstrated that \(\mathbf{\mathsf{CLOUD}}\) is more effective than existing methods in inferring causal relationships.
