Concept

Rate function

The exponent in a large-deviation estimate: the number for which an unlikely event's chance falls like e to the minus n times it. It is zero at the typical value and positive elsewhere, and its smallest value over a set is that set's own rate.

Named by 2 essays across one field — 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

Named alongside it

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

Central limit theorem Large deviations Tail bound Concentration inequality Counting argument Empirical distribution Entropy Heavy tails Legendre transform Moment generating function Multinomial Relative entropy

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