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 $X_1$, $\tilde{X}_2$ and $X_3$ are observed variables, where $\tilde{X}_2$ is a discretization of latent variables $X_2$. Applying existing test methods to the observations of $X_1$, $\tilde{X}_2$ and $X_3$ can lead to a false conclusion about the underlying conditional independence of variables $X_1$, $X_2$ and $X_3$. 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.

翻译：条件独立性检验在贝叶斯网络学习和因果发现等领域具有广泛应用，目前已提出多种检验方法。然而，现有方法通常无法处理仅有离散化观测数据的情况。具体而言，考虑观测变量$X_1$、$\tilde{X}_2$和$X_3$，其中$\tilde{X}_2$是潜在变量$X_2$的离散化结果。若将现有检验方法应用于$X_1$、$\tilde{X}_2$和$X_3$的观测数据，可能会导致对潜在变量$X_1$、$X_2$和$X_3$条件独立性的错误推断。为此，我们提出一种专门针对此类离散化场景设计的条件独立性检验方法。通过构建桥接方程恢复反映潜在连续变量统计信息的参数，我们推导了相应的检验统计量及其在条件独立性零假设下的渐近分布。理论证明与实证验证均表明，该检验方法具有有效性。