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 $$\mu (X)$$ μ ( X ) . We introduce a specific kind of independence that we call dichotomic. If $$\Delta (X)$$ Δ ( X ) stands for the set of all patterns of dichotomic independence that hold for X, we show that $$\mu (X)$$ μ ( X ) can be obtained as the intersection of all elements of $$\Delta (X)$$ Δ ( X ) . We then propose a method to estimate $$\Delta (X)$$ Δ ( X ) when the data are independent and identically (i.i.d.) realizations of a multivariate normal distribution. If $$\hat{\Delta } (X)$$ Δ ^ ( X ) is the estimated set of valid patterns of dichotomic independence, we estimate $$\mu (X)$$ μ ( X ) as the intersection of all patterns of $$\hat{\Delta } (X)$$ Δ ^ ( 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.