Interpretation of Hellinger distance Given to discrete probability distribution $\mathbf{p}:=(p_1,p_2,\dots,p_n)$ and $\mathbf{q}:=(q_1,q_2,\dots,q_n)$, the Hellinger distance between $\mathbf{p}$ and $\mathbf{q}$ is defined as:
$$
d_H(\mathbf{p},\mathbf{q}):=\frac{1}{\sqrt{2}}\left\|\mathbf{p}^{1/2}-\mathbf{q}^{1/2}\right\|_2=\frac{1}{\sqrt{2}}\left(\sum_{i=1}^n \left(\sqrt{p_i}-\sqrt{q_i}\right)^2\right)^{1/2},
$$
Why is this distance extensively exploited in statistics and probability? What is the geometrical/statistical interpretation of this distance? Assuming that $\mathbf{p},\mathbf{q}$ represent vectors and not probability distributions, has this distance been studied in other areas different from statistics?
My questions are not technical, but I was not able to find references which clearly address them. 
Thank you for your help.
 A: From my understanding, the equation can be rewritten as (Cha, 2007)
$$
2\sqrt{1 - \sum_{i=1}^n\sqrt{p_iq_i}}
$$
Here we can see that the part below is basically the geometric mean. This mean is useful in comparing values with different ranges. It denotes a central value for the product of two probabilities, rather than the middle value in an arithmetic way. 
$$
\sum_{i=1}^n\sqrt{p_iq_i}
$$
In comparing two probability distributions, the probability of an event or outcome for both distributions are plugged in the formula and compared. When there is no overlap (either or both values are 0) the maximum distance is assigned. When components are non-empty there is a certain overlap and a distance is calculated.
From this point I do not understand why the geometric mean is subtracted from 1 and a second square root is taken. I do know that the formula is very powerful in high-dimensional data or skewed distributions through class imbalances. The distance metric should be insensitive the skewed date. For example, within computer sciences one application of hellinger distance is anomaly detection. 
Hopefully, someone else can contribute to answer these open questions.
Cha, Sung-Hyuk, Comprehensive Survey on Distance/Similarity Measures between Probability Density Functions, 2007.
A: I guess ultimately the geometric interpretation comes from the fact that the Hellinger distance is the 2-norm distance of the square roots of the point wise probability vectors. And as such, it corresponds to a scalar product:
$$\frac{d_H(p, q)}{\sqrt{2}} =\frac{||\sqrt{p} - \sqrt{q}||_2}{\sqrt{2}} = \sqrt{1 - \langle\sqrt{p}|\sqrt{q}\rangle}$$
Some notes that are obvious but still helpfull for the interpretation:

*

*the 1-norm resp. the TV distance does not correspond to a scalar product (through the parallelogram inequ.) but the 2-norm does

*probability vectors may have a vanishing (as a fct of the dimension) 2-norm, so this is not a good norm for measuring differences

*the 2-norm of the square root however is the 1-norm of the original vector. So for $p$ a probability vector $\sqrt{p}$ has 2-norm equal to 1

*it can be shown that the 1-norm or the TV distance is upper and lower bounded by the Hellinger distance somewhat as $c_1 d_H^2(p,q)\leq TV(p,q) \leq c_2 d_H(p,q)$ for some constants $c_1$ and $c_2$.

*So the Hellinger distance has a geometric interpretation (in terms of a vector space with scalar product and hence angles, where each point in the positiv unit sphere corresponds 1:1 to a probability vector) AND taking the TV distance as the 'natural' distance, the Hellinger distance can be used almost equivalently in the extrem cases ($d(p,q)\rightarrow 0$ or $1$)

