Hellinger distance

Key takeaways

Hellinger distance belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hellinger distance to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hellinger distance from memory before moving on to harder problems.

Reference excerpt

In probability and statistics, the Hellinger distance (closely related to, although different from, the Bhattacharyya distance) is used to quantify the similarity between two probability distributions. It is a type of f-divergence. The Hellinger distance is defined in terms of the Hellinger integral, which was introduced by Ernst Hellinger in 1909. It is sometimes called the Jeffreys distance.

Definition

Measure theory To define the Hellinger distance in terms of measure theory, let P {\displaystyle P} and Q {\displaystyle Q} denote two probability measures on a measure space X {\displaystyle {\mathcal {X}}} that are absolutely continuous with respect to an auxiliary measure λ {\displaystyle \lambda } . Such a measure always exists, e.g λ = ( P + Q ) {\displaystyle \lambda =(P+Q)} . The square of the Hellinger distance between P {\displaystyle P} and Q {\displaystyle Q} is defined as the quantity

H 2 ( P , Q ) = 1 2 ∫ X ( p ( x ) − q ( x ) ) 2 λ ( d x ) . {\displaystyle H^{2}(P,Q)={\frac {1}{2}}\displaystyle \int _{\mathcal {X}}\left({\sqrt {p(x)}}-{\sqrt {q(x)}}\right)^{2}\lambda (dx).}

Here, P ( d x ) = p ( x ) λ ( d x ) {\displaystyle P(dx)=p(x)\lambda (dx)} and Q ( d x ) = q ( x ) λ ( d x ) {\displaystyle Q(dx)=q(x)\lambda (dx)} , i.e. p {\displaystyle p} and q {\displaystyle q} are the Radon–Nikodym derivatives of P and Q respectively with respect to λ {\displaystyle \lambda } . This definition does not depend on λ {\displaystyle \lambda } , i.e. the Hellinger distance between P and Q does not change if λ {\displaystyle \lambda } is replaced with a different probability measure with respect to which both P and Q are absolutely continuous. For compactness, the above formula is often written as

H 2 ( P , Q ) = 1 2 ∫ X ( P ( d x ) − Q ( d x ) ) 2 . {\displaystyle H^{2}(P,Q)={\frac {1}{2}}\int _{\mathcal {X}}\left({\sqrt {P(dx)}}-{\sqrt {Q(dx)}}\right)^{2}.}

Probability theory using Lebesgue measure To define the Hellinger distance in terms of elementary probability theory, we take λ to be the Lebesgue measure, so that dP / dλ and dQ / dλ are simply probability density functions. If we denote the densities as f and g, respectively, the squared Hellinger distance can be expressed as a standard calculus integral

H 2 ( f , g ) = 1 2 ∫ ( f ( x ) − g ( x ) ) 2 d x = 1 − ∫ f ( x ) g ( x ) d x , {\displaystyle H^{2}(f,g)={\frac {1}{2}}\int \left({\sqrt {f(x)}}-{\sqrt {g(x)}}\right)^{2}\,dx=1-\int {\sqrt {f(x)g(x)}}\,dx,}

where the second form can be obtained by expanding the square and using the fact that the integral of a probability density over its domain equals 1. The Hellinger distance H(P, Q) satisfies the property (derivable from the Cauchy–Schwarz inequality)

0 ≤ H ( P , Q ) ≤ 1. {\displaystyle 0\leq H(P,Q)\leq 1.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hellinger distance

Start with the simplest possible case. Write down what Hellinger distance claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hellinger distance before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hellinger distance ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hellinger distance

In research: Hellinger distance appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hellinger distance in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hellinger distance is common in secondary-school and first-year university syllabi. It links to neighbouring topics F-divergences, Statistical distance, Theory of probability distributions, so understanding it makes those chapters shorter.
In everyday life: Look for Hellinger distance outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Hellinger distance in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hellinger distance means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hellinger distance out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hellinger distance in simple terms?

In probability and statistics, the Hellinger distance (closely related to, although different from, the Bhattacharyya distance) is used to quantify the similarity between two probability distributions. It is a type of f-divergence.

Why does Hellinger distance matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hellinger distance?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hellinger distance.