I want to quantify the similarity between two probability mass functions (pmf) p and q, where q was noised with a function that changes the probability space of q. For instance if the following pmfs are given: the original distribution p = [(0,0.5),(1,0.25),(2,0.25)] and the noised pmf q = [(0,0.5), (2,0.25)], where in (x,f(x)), x is a value which can be generated, and f(x) is the probability of generating x.
Is it possible to calculate the similarity of these pmfs with the JS-divergence? Is it possible to expand the noised pmf q by inserting the value (1,0), so that both pmfs have the same support again?