A Conditional Independence Test in the Presence of Discretization

Testing conditional independence (CI) has many important applications, such as Bayesian network learning and causal discovery. Although several approaches have been developed for inferring CI relationships among observed variables, these existing methods generally fail when the variables of interest cannot be directly observed and only discretized values of those variables are available. For example, if X 1 , X2 and X 3 are the observed variables, where X2 is a discretization of the latent variable X 2 , applying the existing methods to the observations of X 1 , X2 and X 3 would lead to a false conclusion about the underlying CI of variables X 1 , X 2 and X 3 . Motivated by this, we propose a CI test specifically designed to accommodate the presence of discretization. To achieve this, a bridge equation and nodewise regression are used to recover the precision coefficients reflecting the conditional dependence of the latent continuous variables under the nonparanormal model. We propose a test statistic and derive its asymptotic distribution under the null hypothesis of CI. Theoretical analysis, along with empirical validation on various datasets, rigorously demonstrates the effectiveness of our testing methods. Our code implementation can be found in https:// github.com/boyangaaaaa/DCT .