A Conditional Independence Test in the Presence of Discretization

TL;DR

This paper introduces a novel conditional independence test tailored for scenarios where some variables are observed only in discretized form, addressing limitations of existing methods in such settings.

Contribution

The paper proposes a new test that recovers latent continuous variable information from discretized observations, with derived asymptotic distribution and validated effectiveness.

Findings

The test accurately detects conditional independence with discretized data.

Theoretical derivation of the test's asymptotic distribution under null hypothesis.

Empirical validation confirms the test's effectiveness in practical scenarios.

Abstract

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.