Calculating distances: Using count data or discrete probabilities? I am looking into calculating distances between vectors for some data analysis.  One question I have is whether I should use actual count data or convert to discrete probabilities.
For some distances, the method is clear from the underlying theory (e.g. the Hellinger Distance).  However, for other distances, which approach to use is not so clear. I have different references that use one or the other approach. It seems to be quite a subjective call.
There are many examples I could provide so, for the sake of space and simplicity, let’s take the Soergel Distance $(d_s)$ here. (I understand this is a generalised version of the Jaccard Distance).
$$d_s(\mathbf{x},\mathbf{y})=1-{\frac{\sum_i min(x_i , y_i)}{\sum_i max(x_i , y_i)}}$$
Firstly, let’s play with the following vectors using count data (taken from survey data): $\mathbf{x}=(5,13,17,14,7)$ and $\mathbf{y}=(12,10,15,41,19)$. Completing the equation, we get $d_s = 0.500$
Now converting the count values to discrete probabilities (or, proportions, if one prefers), we have $\mathbf{\hat x}=(0.089,0.232,0.304,0.250,0.125)$ and $\mathbf{\hat y}=(0.124,0.103,0.155,0.423,0.196)$. Completing the equation again, we get $d_s = 0.435$
So which is the ‘true’ Soergel distance between the vectors? Or is the respective distance ‘valid’ for each approach, which means stating the context is as critical as stating the distance?

I know nothing of Soergel distance, but clearly $d_s$ is invariant to multiplicative constants, that is $d_s(x,y) = d_s(\alpha x , \alpha y)$ but in the case of the proportions you are not multiplying by the same constant $\alpha$. $x$ and $y$ are being multiplied by different constants. So it  is not surprising that the distance is different.
That is the math. What does it means? Distances are almost never invariant to multiplicative constants. Think about euclidian distances - the distance for the count is different that the distance for the proportions. So distances almost always depend on what is the measures of the components, and it is also true here. The Soergel distance for counts is different from the Soergel distance for proportions.
