NMI and AMI use inconsistent definitions of mutual information

There exist many defintions of NMI and AMI.

Vinh, N. X., Epps, J., & Bailey, J. (2010). Information theoretic measures for clusterings comparison: Variants, properties, normalization and correction for chance. Journal of Machine Learning Research, 11(Oct), 2837-2854.

mention 5 different definitions of NMI, and based on that give 4 different AMI.

The NMI implemented in sklearn uses sqrt(H(U), H(V)) for normalization. The AMI implemented in sklearn uses max(H(U), H(V)) for normalization.

There exists an NMI with the max normalization, and a AMI with the sqrt normalization, so this is inconsistent in sklearn. Ideally, they would both use the same definition by default, and allow using any of the others via an option.

[aryamccarthy]

Ooh, a twist. Sum is actually what V-measure uses—not sqrt. It seems we’ve covered the entire gamut. I’m going to take that as another argument in favor of sum. << Thought I hit ‘Comment’ on this some time ago.

[aryamccarthy]

I’ve created a PR; waiting for tests to pass. I think converging on sqrt is best for uniformity with V-measure. EDIT: Nope, it’s not sqrt. It’s sum.