A Kernel Test for Three-Variable Interactions
We introduce kernel nonparametric tests for Lancaster three-variable interaction and for total independence, using embeddings of signed measures into a reproducing kernel Hilbert space. The resulting test statistics are straightforward to compute, and are used in powerful interaction tests, which are consistent against all alternatives for a large family of reproducing kernels. We show the Lancaster test to be sensitive to cases where two independent causes individually have weak influence on a third dependent variable, but their combined effect has a strong influence. This makes the Lancaster test especially suited to finding structure in directed graphical models, where it outperforms competing nonparametric tests in detecting such V-structures.
This paper provides a very interesting nonparametric 
      approach to test for three-variable interactions (and independence) using 
      embedding properties of ISPD kernels. The paper is overall well-written, 
      clear and exhaustive in its coverage of the underlying algebraic proofs. 
      With the need for efficient statistical testing tools to detect complex 
      interaction effects in bioinformatics, this work seems promising in its 
      applications. Given the stated illustrative purpose of identifying 
      gene interactions, it could have been nice to see experiments on 
      non-synthetic data and/or comparison of performances to other genomics 
      methods (such as based on partial correlation). Also, the possibility 
      raised in the concluding paragraph that the test could be extended to more 
      than 3 variables seems to be somewhat contradicted by the point raised in 
      section 4.3. Finally, it would be of great interest to see how the method 
      behaves in large-dimensional spaces. Minor issue: There is no 
      reference to Figure 3 in the main text