Inferring the finest pattern of mutual independence from data

TL;DR

This paper introduces a method to identify the finest mutual independence pattern in data by leveraging dichotomic independence, applicable to multivariate normal distributions, with validation on simulated and real data.

Contribution

It defines dichotomic independence, shows how to derive the finest pattern from it, and proposes an estimation method for this pattern from data.

Findings

Method successfully estimates independence patterns in simulated data.

Approach applied to toy and experimental data demonstrating practical utility.

Highlights limitations and advantages of the proposed estimation technique.

Abstract

For a random variable $X$, we are interested in the blind extraction of its finest mutual independence pattern $\mu ( X )$. We introduce a specific kind of independence that we call dichotomic. If $\Delta ( X )$ stands for the set of all patterns of dichotomic independence that hold for $X$, we show that $\mu ( X )$ can be obtained as the intersection of all elements of $\Delta ( X )$. We then propose a method to estimate $\Delta ( X )$ when the data are independent and identically (i.i.d.) realizations of a multivariate normal distribution. If $\hat{\Delta} ( X )$ is the estimated set of valid patterns of dichotomic independence, we estimate $\mu ( X )$ as the intersection of all patterns of $\hat{\Delta} ( X )$. The method is tested on simulated data, showing its advantages and limits. We also consider an application to a toy example as well as to experimental data.