# What is Hellinger Distance and when to use it?

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?

• 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
– Emre
Aug 31, 2017 at 5:43
• 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? Aug 31, 2017 at 5:59
• 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.
– Emre
Aug 31, 2017 at 6:06
• 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? Aug 31, 2017 at 6:11
• No, it has no inherent connection to LDA.
– Emre
Aug 31, 2017 at 6:12

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.

• (also for @Emre) where does the claim that hellinger distance is the probabilistic analog of euclidean distance comes from? why is that? May 21, 2020 at 16:06