Detection of Conditional Dependence Between Multiple Variables Using Multiinformation

We consider a problem of detecting the conditional dependence between multiple discrete variables. This is a generalization of well-known and widely studied problem of testing the conditional independence between two variables given a third one. The issue is important in various applications. For example, in the context of supervised learning, such test can be used to verify model adequacy of the popular Naive Bayes classifier. In epidemiology, there is a need to verify whether the occurrences of multiple diseases are dependent. However, focusing solely on occurrences of diseases may be misleading, as one has to take into account the confounding variables (such as gender or age) and preferably consider the conditional dependencies between diseases given the confounding variables. To address the aforementioned problem, we propose to use conditional multiinformation (CMI), which is a measure derived from information theory. We prove some new properties of CMI. To account for the uncertainty associated with a given data sample, we propose a formal statistical test of conditional independence based on the empirical version of CMI. The main contribution of the work is determination of the asymptotic distribution of empirical CMI, which leads to construction of the asymptotic test for conditional independence. The asymptotic test is compared with the permutation test and the scaled chi squared test. Simulation experiments indicate that the asymptotic test achieves larger power than the competitive methods thus leading to more frequent detection of conditional dependencies when they occur. We apply the method to detect dependencies in medical data set MIMIC-III.