Tuesday, July 17, 2012

Permutation tests, graph non-isomorphism and kernels

I was reading Larry Wasserman's post on the modern two-sample test and a few thoughts came to mind.

1. I've known about permutation tests for a long time, but it only just occurred to me that the permutation test is exactly what you do in the AM protocol for Graph non-isomorphism. The principle is that if the two distributions are identical, your test statistic should not be able to tell them apart, and the way to test this is by randomly changing labels. Replace distributions by graphs and test by Merlin, and it looks the same. Is this a trivially trite observation, or are there other examples of protocols that use standard statistical tests in disguise ? (hmm I see a cstheory question here)

2. The kernel test he mentions, if you look closely, is merely computing the kernel distance between the two distributions. And the energy test he mentions later is doing something akin to earthmover. The kernel distance strikes again !!!

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