Discussion of ‘Multi-scale Fisher’s independence test for multivariate dependence’

原文摘要

The multi-scale Fisher’s independence test, MultiFIT, proposed by Gorsky & Ma (2022) is a novel method to test independence between two random vectors. By its design, this test is particularly useful in detecting local dependence. Moreover, by adopting a resampling-free approach, it can easily accommodate massive sample sizes. Another benefit of the proposed method is its ability to interpret the nature of dependency. We congratulate the authors for their very interesting and elegant work. In this comment, we would like to discuss a general framework unifying the MultiFIT and other tests and compare it with the binary expansion randomized ensemble test, BERET, proposed by Lee et al. (2022). We would also like to contribute our thoughts on potential extensions of the method. To understand the properties of MultiFIT, we analyse the test statistic under a similar multi-resolution approach in the binary expansion testing, BET, framework of Zhang (2019). Consider the test of independence of two continuous variables from the copula |$(U,V)$|⁠. The uniform consistency with respect to the total variation distance requires consistency for any alternative that is some distance from independence, i.e., |$H_0: P_{(U,\,V)} = {\rm Un}[-1,1]^2\text{ versus } H_1: {\small{\text{TV}}}(P_{(U,\,V)}, {\rm Un}[-1,1]^2) \geqslant \delta \text{ for some } 0<\delta\leqslant 1.$| Theorem 2.2 of Zhang (2019) shows the nonexistence of a test that is uniformly consistent with respect to the total variation distance. The key reason for nonuniform consistency is the unidentifiability in this problem. To avoid this issue and develop a nonparametric test statistic that is both powerful and robust, Zhang (2019) developed the BET framework to test approximate independence through a filtration approach. The filtration is constructed through the classical probability result of binary expansion: |$U=\sum_{d=1}^\infty {A_{1,\,d}/2^d}$|⁠, |$V=\sum_{d=1}^\infty {A_{2,\,d}/2^d}$| with |$A_{1,\,d}$| and |$A_{2,\,d}$| representing the |$d$|th bit of |$U$| and |$V$|⁠, respectively. It is known that if |$U$|⁠, or |$V$|⁠, is marginally distributed as |${\rm Un}[-1,1]$| if and only if |$A_{1,\,d}, \ {\rm or} \ A_{2,\,d}, \stackrel{{\rm i.i.d.}}{\sim} {\rm Rademacher}$|⁠. When we truncate the expansions at finite depths |$d_1$| and |$d_2$|⁠, |$U_{d_1}=\sum_{d=1}^{d_1} {A_{1,\,d}/2^d}$| and |$V_{d_2}=\sum_{d=1}^{d_2} {A_{2,\,d}/2^d}$| become discrete uniform variables that generate an analytically attractable filtration to approximate |$(U,V)$|⁠. At every depth of the binary expansion filtration, the probability model of |$(U_{d_1},V_{d_2})$| is a low resolution approximation of |$(U,V)$| providing global distributional information. The interactions of these binary variables are denoted by the |$A_\Lambda$|⁠, where |$\Lambda$| is a binary vector index with 1 or 0 indicating the presence of the bit of a variable in the interaction. The sums of observed interactions of the |$A_\Lambda$| with |$\Lambda \neq 0$| from |$n$| samples, which are referred to as symmetry statistics and are denoted by |$\bar{S}_\Lambda=\sum_{i=1}^{n}A_{\Lambda,\,i}/n$|⁠, are shown to be completely sufficient for dependence and form the building blocks of inference.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）