Data-Driven Representations for Testing Independence: A Connection with Mutual Information Estimation

Abstract

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.

Studying the Interplay between Information Loss and Operation Loss in Representations for Classification

“…The first inequality in (19) comes from the fact that ( Ũ , U i ) is a deterministic function of X and the second comes from (17).…”

“…Therefore, a non-zero operation loss on using U i instead of X (stated in (18)) implies a respective non-zero weak information loss as stated in (19).…”

“…These results offer concrete conditions for finite-size representations that, in light of Theorem 2 (see also Theorem 6), will be operationally sufficient for classification (details presented in Section VI-A). In this context, it is worth highlighting the family of data-driven partitions studied in [33], [31], [32], [38], [13], [19]. These data-driven representations are information sufficient (with probability one) and, consequently, they are operationally sufficient (with probability one) as a direct corollary of Theorem 2 (details presented in Section VI-B).…”

“…l n is designed to guarantee that the statistically equivalent splitting approach has a sufficient number of samples 19 . Therefore, after iterating this principle and meeting the stopping criterion in every resulting cell, we have a partition π(ln) n with a binary tree-structure that has almost the same number of samples in every cell (balanced), and in addition the number of samples per cell is lower bounded by l n .…”

“…Importantly, this new condition will be derived from Theorem 2 and specific assumptions on the structure (or richness) of Λ. 19 A systematic exposition of the statistical properties of this scheme and its implementation and use can be found in [14] for pattern recognition and in [33], [32] for estimating information measures.…”