I am interested in knowing what really happens in Hellinger Distance (in simple terms). Furthermore, I am also interested in knowing what are types of problems that we can use Hellinger Distance? What are the benefits of using Hellinger Distance?

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    $\begingroup$ The Hellinger distance is a probabilistic analog of the Euclidean distance. A salient property is its symmetry, as a metric. Such mathematical properties are useful if you are writing a paper and you need a distance function that possesses certain properties to make your proof possible. In application, someone might discover that one metric produces nicer or better results than another for a certain task; e.g., the Wasserstein distance is all the rage in generative adversarial networks $\endgroup$
    – Emre
    Commented Aug 31, 2017 at 5:43
  • $\begingroup$ Thank you for the comment. I came across this question, which is quite similar to the question I have now. datascience.stackexchange.com/questions/22324/… Please let me know, why the answer says Hellinger Distance is suitable? $\endgroup$ Commented Aug 31, 2017 at 5:59
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    $\begingroup$ Probably to visualize the topics in a metric space. Another nice property is that the Hellinger distance is finite for distributions with different support. It is good that you are asking these questions. I suggest trying different metrics for yourself and observing the results. $\endgroup$
    – Emre
    Commented Aug 31, 2017 at 6:06
  • $\begingroup$ Thanks. its a good link. helps a lot. But is Hellinger distance only limited to topics derived from Latent Dirichlet Allocation (LDA) as mentioned in the link? $\endgroup$ Commented Aug 31, 2017 at 6:11
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    $\begingroup$ No, it has no inherent connection to LDA. $\endgroup$
    – Emre
    Commented Aug 31, 2017 at 6:12

1 Answer 1


Hellinger distance is a metric to measure the difference between two probability distributions. It is the probabilistic analog of Euclidean distance.

Given two probability distributions, $P$ and $Q$, Hellinger distance is defined as:

$$h(P,Q) = \frac1{\sqrt2}\cdot \|\sqrt{P}-\sqrt{Q}\|_2$$

It is useful when quantifying the difference between two probability distributions. For example, if you estimate a distribution for users and non-users of a service. If the Hellinger distance is small between those groups for some features, then those features are not statistically useful for segmentation.

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    $\begingroup$ (also for @Emre) where does the claim that hellinger distance is the probabilistic analog of euclidean distance comes from? why is that? $\endgroup$
    – carlo
    Commented May 21, 2020 at 16:06

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