Bernoulli Unit
The unit type, denoted by (), is the singleton type with exactly one value. This document explores the Bernoulli approximation B_{()} of the unit type.
Theoretical Properties
Unit Type Characteristics:
- Cardinality: |()| = 1
- Values: Just ()
- No uncertainty possible in a singleton
Bernoulli Unit Analysis
Confusion Matrix:
For B_{()}, we have a 1×1 confusion matrix:
| latent \ observed | () |
|---|---|
() |
1 |
Fundamental Measures: - Order: 0 (no free parameters) - Rank: 1 (trivial 1×1 identity matrix) - Entropy: H = 0 (no uncertainty)
Why B_{()} = ()
Since there's only one possible value, there's no room for approximation or error:
Any "noise" in the unit type would have to map to... what? There are no other values available.
Degenerate Case: The unit type represents the degenerate case where Bernoulli approximation becomes trivial: - No approximation possible - Perfect inference always - Information-theoretically trivial
Connection to Other Types
Compositional Role:
While B_{()} itself is trivial, it plays important roles in:
- Sum types:
B_{X + ()}≈ optional types with noise - Product types:
B_{() × Y}≈B_Y(unit contributes no information) - Function types:
B_{() → Y}≈B_Y(constant functions)
Algebraic Identity:
The unit type serves as the multiplicative identity in type algebra:
- X × () ≅ X
- B_X × B_{()} ≅ B_X
Implementation Note
In practice, B_{()} is useful for:
- Theoretical completeness of the Bernoulli type system
- Compositional building blocks where units appear in complex types
- Degenerate case handling in generic Bernoulli type implementations
The unit type demonstrates that Bernoulli approximation gracefully degenerates to perfect accuracy when no uncertainty is possible.