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 μ ( 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 μ ( 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 hatDelta ( X ) is the estimated set of valid patterns of dichotomic independence, we estimate μ ( X ) as the intersection of all patterns of hatDelta ( 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.