I am trying to understand what exactly the distance between two distributions using Wasserstein distance means.
I have two samples coming from two distribution: a ground truth one and its empirical realization. I know that the Wasserstein distance can be used to quantify the difference between the two distributions. My question is when do we consider the distance between these distributions "small" enough? or what does this number mean ? say we obtain 0.25 for the distance. What does that tell us ?
I think the answer of this question comes down to understand what does the distance exactly quantify (and this question goes beyond the simple interpretation of the definition :the minimum cost if we want to obtain the first distribution by transporting the probability mass in second one )
I am including a python example here and I appreciate an answer with concrete examples
from scipy.stats import wasserstein_distance
wasserstein_distance([0, 1, 3], [5, 6, 8])
(note : the scipy implementation works only on 1d PDs)