Concept

Bias — where it appears

A persistent lean in a quantity that is equal in the limit, showing up in a lower-order term rather than in the leading one. The primes leaving remainder 3 on division by 4 lead those leaving 1 at almost every bound, and the two classes still have the same density.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Which regions each rule favours. Average seats above or below exact quota for the largest and the smallest region, under each of the five methods, over 400 generated instances.

The rule with no favourites

Over four hundred instances, Jefferson's method gives the largest region a third of a seat more than its exact share and the smallest a third of a seat less. Adams reverses both. Webster's average is a hundredth of a seat, and that is not luck.

applied · Apportionment

The primes 3 mod 4 against the primes 1 mod 4, out to 30,000. A plot of the difference between the counts of primes in two residue classes, against the bound, showing a persistent lead for one class and where it is lost.

Every class, and in equal shares

Euclid's argument aimed at a residue class reaches some classes and stalls at others. The theorem covering all of them is Dirichlet's, its proof abandons arithmetic entirely for analysis, and what it proves is stronger than infinitude — the classes are equal, though not at any point anybody has counted.

number · Infinitude of primes

A rectangle of the parity table, nearly balanced. A 32 by 32 grid of +1 and −1 entries — the parity of the inner product of row and column — with a 15 by 14 rectangle of chosen rows and columns highlighted; its entries sum to 8.

Two weak sources make one fair bit

No fixed rule can turn every weakly random source into fair bits: for any rule, some source with almost full unpredictability makes it constant. Two independent sources are different. Multiply their bits in pairs, add, and keep the parity — and if the two together carry more unpredictability than the length of one, the result is nearly fair, whatever else the sources do. The reason is that the table of those parities is balanced on every large rectangle.

computation · Pseudorandomness

How often each procedure finds the right verdict, with 3 judges of competence 0.7. both true: liable: premises 61.5%, conclusion 48.5%; first only: not liable: premises 83.1%, conclusion 88.6%; second only: not liable: premises 83.1%, conclusion 88.6%; neither: not liable: premises 95.3%, conclusion 97.7%; average 80.7% against 80.9%.

Which vote finds the truth more often

A court that must decide two premises and their conjunction can vote on the premises or on the conclusion, and the two votes sometimes disagree. Suppose there is a right answer and each judge is more often right than wrong. Then the question has an exact answer: for three judges the conclusion vote is more often right when competence is below √½ and the premise vote above it, and in a large court the premise vote wins at any competence — while the conclusion vote turns out to be a built-in standard of proof.

applied · Judgement aggregation

Named alongside it

The objects these essays reach for when they reach for this one.

Independence Apportionment Arithmetic progression Binomial distribution Condorcet jury theorem Density Dirichlet theorem Divisor method Entropy Expectation Geometric mean Inner product

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