Inferring the finest pattern of mutual independence from data
This work addresses a foundational problem in statistical inference for researchers in machine learning and statistics, but it appears incremental as it builds on existing independence concepts.
The paper tackles the problem of blindly extracting the finest mutual independence pattern from data, proposing a method based on dichotomic independence and showing its estimation for multivariate normal distributions, with testing on simulated and experimental 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 $Δ( X )$ 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.