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 2 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

Named alongside it

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

Coverage Sample size Conjugacy Distribution-free Flat prior Normality Order statistic Posterior Posterior mean Prior Prior weight Probability integral transform

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