A Conditional Independence Test in the Presence of Discretization

Testing conditional independence has many applications, such as in Bayesian network learning and causal discovery. Different test methods have been proposed. However, existing methods generally can not work when only discretized observations are available. Specifically, consider X₁, X₂ and X₃ are observed variables, where X₂ is a discretization of latent variables X₂. Applying existing test methods to the observations of X₁, X₂ and X₃ can lead to a false conclusion about the underlying conditional independence of variables X₁, X₂ and X₃. Motivated by this, we propose a conditional independence test specifically designed to accommodate the presence of such discretization. To achieve this, we design the bridge equations to recover the parameter reflecting the statistical information of the underlying latent continuous variables. An appropriate test statistic and its asymptotic distribution under the null hypothesis of conditional independence have also been derived. Both theoretical results and empirical validation have been provided, demonstrating the effectiveness of our test methods.