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Concept

Large deviations

The study of how fast the chance of a fixed-size departure from the typical falls as the sample grows. The answer is exponential in the sample size, and the exponent is computed from the summand before any sample size is chosen.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The chance the average clears 0.75, against the number of draws. The exact probability that the average of n draws exceeds a fixed level, on a logarithmic scale, falling along a straight line whose slope is the rate function, with the normal approximation drawn beside it and diverging.

The tail is not a bell

The limit theorem describes a window of width one over the root of n around the mean; ask instead for the chance that an average lands a fixed distance away and the answer falls exponentially, at a rate computed from the summand before any n is chosen.

probability · Central limit The 91 histograms 12 draws can produce. A triangle whose points are the possible histograms of a fixed number of draws over three faces, each drawn as a dot shaded by how far it is from the true distribution.

When the whole histogram deviates

A rare average has a price, an exponent that grows with the number of trials. Ask instead for the chance that the whole tally of outcomes comes out wrong, and the exponent is no longer a function of one number — it is a distance between two distributions, and every rare-average rate is a shadow of it.

probability · Central limit The polynomial bound falls exponentially behind the space. A log-scale plot against the dimension of the number of points of the space, the polynomial method's bound on a cap, and the known largest caps: the bound's line is less steep than the space's and pulls away from it.

The polynomial that bounds the caps

For forty years the best bound on a set of SET cards with no set among them shrank only like one over the dimension. In 2016 a two-page argument made it shrink exponentially, and the whole proof is a count of monomials: a table that is diagonal on a cap, one polynomial that describes it, and the fact that three parts of a degree cannot all be large.

computation · Finite fields

Named alongside it

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

Central limit theoremCounting argumentRate functionTail boundCap setConcentration inequalityEmpirical distributionEntropyFinite fieldHeavy tailsLegendre transformMoment generating function

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