# Compare KS test and Wasserstein distance or Earth mover's distance

I have tried the following question in couple of exchange sites but I did not get any views or reply. I am asking here as I am kind of desperate for the answer, please be considerate. Any suggestion how to approach the problem would be really appreciated.

Consider two sets of data points A and B. Both these data set are from mixture of unknown number of Gaussians. The mean of the Gaussians are little different for each set (there may have few overlap or very close separated mean values). However for both cases the variance of all the Gaussians are small. Now, if we give a set of data points say C, how to estimate C is from A or from from B? I understand there are many methods to do so: is there a way tell the most efficient method? This is a very board question, so specifically can we compare the KS test https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test and https://en.wikipedia.org/wiki/Wasserstein_metric for this problem? Is there a way to prove that KS test/Wasserstein metric would give better estimate?

It appears to me that Cumulative distribution is not smooth so Wasserstein metric would be better, is it true?

In summary,A and B are known to generate from different distribution (physically different but close); I want to know from where C was generated, it must be generated either form A or B.

• Why would you not use a likelihood ratio or Bayes factor approach to consider this? – Henry Sep 23 '18 at 10:05
• @Henry First I need an criteria to choose how sharp each measure is. – Creator Sep 23 '18 at 21:14