# How to Gain on Power: Novel Conditional Independence Tests Based on Short Expansion of Conditional Mutual Information

Mariusz Kubkowski, Jan Mielniczuk, Paweł Teisseyre.

Year: 2021, Volume: 22, Issue: 62, Pages: 1−57

#### Abstract

Conditional independence tests play a crucial role in many machine learning procedures such as feature selection, causal discovery, and structure learning of dependence networks. They are used in most of the existing algorithms for Markov Blanket discovery such as Grow-Shrink or Incremental Association Markov Blanket. One of the most frequently used tests for categorical variables is based on the conditional mutual information ($CMI$) and its asymptotic distribution. However, it is known that the power of such test dramatically decreases when the size of the conditioning set grows, i.e. the test fails to detect true significant variables, when the set of already selected variables is large. To overcome this drawback for discrete data, we propose to replace the conditional mutual information by Short Expansion of Conditional Mutual Information (called $SECMI$), obtained by truncating the Möbius representation of $CMI$. We prove that the distribution of $SECMI$ converges to either a normal distribution or to a distribution of some quadratic form in normal random variables. This property is crucial for the construction of a novel test of conditional independence which uses one of these distributions, chosen in a data dependent way, as a reference under the null hypothesis. The proposed methods have significantly larger power for discrete data than the standard asymptotic tests of conditional independence based on $CMI$ while retaining control of the probability of type I error.