Equating different forms of Hellinger Distance

Question

For a research report, I want to show that the standard expression of the Hellinger Distance between two discrete distributions,

$$H(p,q)={\sqrt{\frac{1}{2} \sum_{x \in X} \left[\sqrt{p(x)}-\sqrt{q(x)} \right]^{2}}}$$

is equivalent to the alternative expression

$$H(p,q)={\sqrt{1-\sum_{x \in X} \sqrt{p(x)q(x)}}}$$

I have tried the expand the square but can't seem to equate the two expressions. Can anyone help?

@User1865345 - Bingo! I see it now - I got it to equate! Thanks so much. — anna6931, Commented Oct 3, 2022 at 9:19
A remark. There exist different versions of the formula of Hellinger distance, they differ by a constant factor. The version I've seen more often uses $2$ instead of $\frac{1}{2}$ under the root. Note also that if you remove this coefficient (set it to $1$), that will be distance sometimes called Matusita distance. Ref: Michel Deza, Elena Deza. Encyclopedia of Distances. Also: Sung-Hyuk Cha. Comprehensive Survey on Distance/Similarity Measures between Probability Density Functions // INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES — ttnphns, Commented Oct 3, 2022 at 15:30
@ttnphns - I have a few references on the Hellinger Distance (43 at last count) and I have only come across two references where the constant of $2$ is used (which are the two you list) hence this seems to be a non-standard version. The version with the $\frac{1}{2}$ factor, as given by the OP, seems to be more common, most likely because the metric is bounded by [0,1]. (Note: Re-posted comment to correct a typographical error). — Mari153, Commented Oct 4, 2022 at 8:09

User1865345 · Accepted Answer · 2022-10-03 09:23:43Z

7

Hint: what would $\sum_{x\in \mathcal X} p(x) $ be equal to?

answered Oct 3, 2022 at 9:23

User1865345

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