Inferring the finest pattern of mutual independence from data

Abstract

For a random variable $X$, we are interested in the blind extraction of its finest mutual independence pattern $μ( X )$. We introduce a specific kind of independence that we call dichotomic. If $Δ( X )$ stands for the set of all patterns of dichotomic independence that hold for $X$, we show that $μ( X )$ can be obtained as the intersection of all elements of $Δ( X )$. We then propose a method to estimate $Δ( X )$ when the data are independent and identically (i.i.d.) realizations of a multivariate normal distribution. If $\hatΔ ( X )$ is the estimated set of valid patterns of dichotomic independence, we estimate $μ( X )$ as the intersection of all patterns of $\hatΔ ( X )$. 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.