Von Mises–Fisher distribution

Key takeaways

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

Reference excerpt

In directional statistics, the von Mises–Fisher distribution (named after Richard von Mises and Ronald Fisher), is a probability distribution on the ( p − 1 ) {\displaystyle (p-1)} -sphere in R p {\displaystyle \mathbb {R} ^{p}} . If p = 2 {\displaystyle p=2}

the distribution reduces to the von Mises distribution on the circle.

Definition The probability density function of the von Mises–Fisher distribution for the random p-dimensional unit vector x {\displaystyle \mathbf {x} } is given by:

f p ( x ; μ , κ ) = C p ( κ ) exp ⁡ ( κ μ T x ) , {\displaystyle f_{p}(\mathbf {x} ;{\boldsymbol {\mu }},\kappa )=C_{p}(\kappa )\exp \left({\kappa {\boldsymbol {\mu }}^{\mathsf {T}}\mathbf {x} }\right),}

where κ ≥ 0 , ‖ μ ‖ = 1 {\displaystyle \kappa \geq 0,\left\Vert {\boldsymbol {\mu }}\right\Vert =1} and the normalization constant C p ( κ ) {\displaystyle C_{p}(\kappa )} is equal to

C p ( κ ) = κ p / 2 − 1 ( 2 π ) p / 2 I p / 2 − 1 ( κ ) , {\displaystyle C_{p}(\kappa )={\frac {\kappa ^{p/2-1}}{(2\pi )^{p/2}I_{p/2-1}(\kappa )}},}

where I v {\displaystyle I_{v}} denotes the modified Bessel function of the first kind at order v {\displaystyle v} . If p = 3 {\displaystyle p=3} , the normalization constant reduces to

C 3 ( κ ) = κ 4 π sinh ⁡ κ = κ 2 π ( e κ − e − κ ) . {\displaystyle C_{3}(\kappa )={\frac {\kappa }{4\pi \sinh \kappa }}={\frac {\kappa }{2\pi (e^{\kappa }-e^{-\kappa })}}.}

The parameters μ {\displaystyle {\boldsymbol {\mu }}} and κ {\displaystyle \kappa } are called the mean direction and concentration parameter, respectively. The greater the value of κ {\displaystyle \kappa } , the higher the concentration of the distribution around the mean direction μ {\displaystyle {\boldsymbol {\mu }}} . The distribution is unimodal for κ > 0 {\displaystyle \kappa >0} , and is uniform on the sphere for κ = 0 {\displaystyle \kappa =0} . The von Mises–Fisher distribution for p = 3 {\displaystyle p=3} is also called the Fisher distribution. It was first used to model the interaction of electric dipoles in an electric field. Other applications are found in geology, bioinformatics, and text mining.

Support The support of the von Mises–Fisher distribution is the hypersphere, or more specifically, the ( p − 1 ) {\displaystyle (p-1)} -sphere, denoted as

S p − 1 = { x ∈ R p : ‖ x ‖ = 1 } {\displaystyle \mathbb {S} ^{p-1}=\left\{\mathbf {x} \in \mathbb {R} ^{p}:\left\|\mathbf {x} \right\|=1\right\}}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Von Mises–Fisher distribution

Start with the simplest possible case. Write down what Von Mises–Fisher distribution 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 Von Mises–Fisher distribution 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 Von Mises–Fisher distribution 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 Von Mises–Fisher distribution

In research: Von Mises–Fisher distribution 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 Von Mises–Fisher distribution 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: Von Mises–Fisher distribution is common in secondary-school and first-year university syllabi. It links to neighbouring topics Continuous distributions, Directional statistics, Exponential family distributions, so understanding it makes those chapters shorter.
In everyday life: Look for Von Mises–Fisher distribution 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 Von Mises–Fisher distribution in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Von Mises–Fisher distribution 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 Von Mises–Fisher distribution out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Von Mises–Fisher distribution in simple terms?

In directional statistics, the von Mises–Fisher distribution (named after Richard von Mises and Ronald Fisher), is a probability distribution on the ( p − 1 ) {\displaystyle (p-1)} -sphere in R p {\displaystyle \mathbb {R} ^{p}} . If p = 2 {\displaystyle p=2} the distribution reduces to the von Mis…

Why does Von Mises–Fisher distribution 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 Von Mises–Fisher distribution?

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 Von Mises–Fisher distribution.