Normal Distribution
nothing is called 95% until it has been counted
Concept

Beta distribution — where it appears

The distribution on the unit interval with two shape parameters, and the law of the i-th of n ordered uniforms. It is the conjugate prior for a proportion, so a beta prior and binomial data give a beta posterior with the counts simply added on.

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

weakly informative — Beta(2, 2), updated by 5 of 20. The prior is worth 4 observations. With 20 observations the posterior mean is 0.292, against a data proportion of 0.250 and a prior mean of 0.500.

What a prior is worth

A prior is not a philosophical position, it is a component with a stated size. For a proportion it is worth exactly a + b observations, which turns "how much does the prior matter" from an argument into a subtraction.

bayes · Prior How many observations the smallest and largest of them need. The interval between the extremes of n draws holds at least 95% of the population with probability 1 - n p^(n-1) + (n-1) p^n, whatever the population is. Reaching 95% confidence takes 93 observations.

Ninety-three observations, and nothing assumed

The interval between the smallest and largest of a sample holds a share of the population whose distribution does not depend on the population — Beta(n − 1, 2), for anything continuous. Buying the 95/95 that normality buys at ten observations costs 93 of them, and that number is the exchange rate between an assumption and data.

estimated · Bands A prior centred on the truth's population and too confident about it: coverage across the population, and how often the check flags it. Truths drawn from a population of proportions centred at 0.2 and worth ten observations; twenty trials per study; the analyst's prior centred at 0.2 with the weight on the axis. Coverage of the 95% credible interval, exact over the population: 94.7% at 2, 96.0% at 5, 95.0% at 10, 88.1% at 20, 75.9% at 35, 61.3% at 60, 48.2% at 100, 34.0% at 200, 23.9% at 400. Share of studies whose count the prior-predictive check puts in its 5% tail: 0.1%, 1.8%, 3.2%, 14.2%, 17.3%, 17.3%, 21.9%, 21.9%, 21.9%.

A prior wrong about its own spread

A prior can be centred exactly where the truths lie and still be wrong about how far they spread. Ten times too confident — worth a hundred observations about proportions whose population is worth ten — its 95% credible intervals cover 48.2% of the population's studies at twenty trials, and nothing in the interval shows it: it is 0.456 of a flat prior's width, against 0.436 for a prior whose hundred observations' confidence is earned. A prior with tails too thin misses the one study in ten that comes from somewhere unusual a third of the time, and a flat component of two per cent brings that study's coverage from 65.6% to 84.0%.

bayes · Credible

Named alongside it

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

CoveragePriorSample sizeBinomial proportionConjugacyCredible intervalDistribution-freeExact enumerationFlat priorHeavy tailInterval widthMixture prior

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