Satosi Watanabe’s ugly duckling theorem (1969) says that an ugly duckling and a swan are exactly as similar as two swans – if every possible property counts equally. Classification and machine learning are only possible because we choose which features matter (an inductive bias).
How to use the Ugly Duckling Theorem Calculator
- Number of objects n
- Group size k
Results update as you type, and everything is calculated in your browser.
How it works
Enter the number of objects n and the size k of the group you compare. Each predicate is a subset of the objects (the ones for which it is true); a group shares the predicates whose subsets contain all of its members.
Inputs explained
| Field | What to enter |
|---|---|
| Number of objects n | A number |
| Group size k | A number |
What each result means
- Number of possible predicates = 2^[Number of objects n]
- Predicates shared by any two objects = 2^([Number of objects n] – 2)
- Share of all predicates = 100÷2^[Group size k]
Example values
How the main result changes when Number of objects n changes and everything else stays at the example values above:
| Number of objects n | Predicates shared by any two objects |
|---|---|
| 2 | 1 |
| 3 | 2 |
| 4 | 4 |
| 5 | 8 |
| 6 | 16 |
| 8 | 64 |
Formula
Predicates on n objects: 2ⁿ · shared by any 2 objects: 2ⁿ⁻² · shared by any k objects: 2ⁿ⁻ᵏ
Worked example
Inputs — Number of objects n: 4; Group size k: 2.
Results — Number of possible predicates = 16; Predicates shared by any two objects = 4; Predicates shared by any k objects = 4; Share of all predicates = 25 %.
Key terms
- Predicate
- A property that is either true or false of each object.
Frequently asked questions
What does the theorem say?
Without weighting features, every pair of objects shares the same number of properties.
Why is it called the ugly duckling theorem?
An ugly duckling is as similar to a swan as two swans are to each other.
Does it make classification impossible?
No – it shows that classification requires choosing relevant features.
What is an inductive bias?
The assumptions a learner uses to generalise.
Sources
- Watanabe, S. (1969). Knowing and Guessing. Wiley