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 $\mathrm{\Delta }(X)$ 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 $\mathrm{\Delta }(X)$. We then propose a method to estimate $\mathrm{\Delta }(X)$ when the data are independent and identically (i.i.d.) realizations of a multivariate normal distribution. If $\widehat{\mathrm{\Delta }}(X)$ is the estimated set of valid patterns of dichotomic independence, we estimate $\mu (X)$ as the intersection of all patterns of $\widehat{\mathrm{\Delta }}(X)$. 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.