  This work addresses testing the independence of two continuous and
finite-dimensional random variables from the design of a data-driven partition.
The empirical log-likelihood statistic is adopted to approximate the sufficient
statistics of an oracle test against independence (that knows the two
hypotheses). It is shown that approximating the sufficient statistics of the
oracle test offers a learning criterion for designing a data-driven partition
that connects with the problem of mutual information estimation. Applying these
ideas in the context of a data-dependent tree-structured partition (TSP), we
derive conditions on the TSP's parameters to achieve a strongly consistent
distribution-free test of independence over the family of probabilities
equipped with a density. Complementing this result, we present finite-length
results that show our TSP scheme's capacity to detect the scenario of
independence structurally with the data-driven partition as well as new
sampling complexity bounds for this detection. Finally, some experimental
analyses provide evidence regarding our scheme's advantage for testing
independence compared with some strategies that do not use data-driven
representations.
