Inferring the finest pattern of mutual independence from data

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.