Variation of information

Key takeaways

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

Reference excerpt

In probability theory and information theory, the variation of information or shared information distance is a measure of the distance between two clusterings (partitions of elements). It is closely related to mutual information; indeed, it is a simple linear expression involving the mutual information. Unlike the mutual information, however, the variation of information is a true metric, in that it obeys the triangle inequality.

Definition Suppose we have two partitions X {\displaystyle X} and Y {\displaystyle Y} of a set A {\displaystyle A} , namely X = { X 1 , X 2 , … , X k } {\displaystyle X=\{X_{1},X_{2},\ldots ,X_{k}\}} and Y = { Y 1 , Y 2 , … , Y l } {\displaystyle Y=\{Y_{1},Y_{2},\ldots ,Y_{l}\}} . Let:

n = ∑ i | X i | = ∑ j | Y j | = | A | {\displaystyle n=\sum _{i}|X_{i}|=\sum _{j}|Y_{j}|=|A|}

p i = | X i | / n {\displaystyle p_{i}=|X_{i}|/n} and q j = | Y j | / n {\displaystyle q_{j}=|Y_{j}|/n}

r i j = | X i ∩ Y j | / n {\displaystyle r_{ij}=|X_{i}\cap Y_{j}|/n}

Then the variation of information between the two partitions is:

V I ( X ; Y ) = − ∑ i , j r i j [ log ⁡ ( r i j / p i ) + log ⁡ ( r i j / q j ) ] {\displaystyle \mathrm {VI} (X;Y)=-\sum _{i,j}r_{ij}\left[\log(r_{ij}/p_{i})+\log(r_{ij}/q_{j})\right]} . This is equivalent to the shared information distance between the random variables i and j with respect to the uniform probability measure on A {\displaystyle A} defined by μ ( B ) := | B | / n {\displaystyle \mu (B):=|B|/n} for B ⊆ A {\displaystyle B\subseteq A} .

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Variation of information

Start with the simplest possible case. Write down what Variation of information 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 Variation of information 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 Variation of information 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 Variation of information

In research: Variation of information 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 Variation of information 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: Variation of information is common in secondary-school and first-year university syllabi. It links to neighbouring topics Clustering criteria, Entropy and information, Summary statistics for contingency tables, so understanding it makes those chapters shorter.
In everyday life: Look for Variation of information 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 Variation of information in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Variation of information 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 Variation of information out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Variation of information in simple terms?

In probability theory and information theory, the variation of information or shared information distance is a measure of the distance between two clusterings (partitions of elements). It is closely related to mutual information; indeed, it is a simple linear expression involving the mutual informa…

Why does Variation of information 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 Variation of information?

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 Variation of information.