📂Probability Theory

Hellinger Distance of Probability Distributions

Definition

The following distance function defined on probability distributions themselves is called the Hellinger distance.

Discrete ¹

Let $p, q$ be a probability mass function. The Hellinger distance of $p, q$ is defined as: $$H \left( p , q \right) := \sqrt{ \frac{1}{2} \sum_{k} \left( \sqrt{p_{k}} - \sqrt{q_{k}} \right)^{2} }$$

Continuous ²

Let $f, g$ be a probability density function. The Hellinger distance of $f, g$ is defined as: $$\begin{align*} & H^{2} \left( f , g \right) \\ :=& {\frac{ 1 }{ 2 }} \int_{\mathbb{R}} \left( \sqrt{f(x)} - \sqrt{g(x)} \right)^{2} dx \\ =& 1 - \int_{\mathbb{R}} \sqrt{f(x)g(x)} dx \end{align*}$$

Explanation

The Hellinger distance is, by definition, a distance function that directly compares probability mass functions or probability density functions. It is bounded by $[0,1]$, being $0$ when they are identical and $1$ when they are completely different. While the Kullback-Leibler divergence is widely used for comparing probability distributions, the Hellinger distance has the distinguishing feature of being a proper distance function, allowing the discussion of metric spaces.

See Also

• Kullback-Leibler Divergence