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

Credible interval — where it appears

A range containing a stated share of the posterior distribution, which is a statement about belief given the data rather than about repetition. Its frequentist coverage is a separate question with a separate answer, and the two agree only under conditions worth naming.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Coverage of four nominal 95% intervals, n = 20. Computed exactly by summing over all 21 possible counts, not simulated. The Wald interval drops to 18.2% and is jagged everywhere; Clopper–Pearson never falls below 95% and pays for it in width.

What the 95% refers to

An interval that claims 95% is making a checkable statement about a procedure, not about the interval in front of you. Build every possible sample and count, and the interval taught first turns out to cover 87.6% of the time.

intervals · Coverage What a 95% credible interval covers, n = 20. Computed by summing over all 21 possible counts rather than by simulating them. Jeffreys' prior covers close to 95% across the range; a confident prior centred in the wrong place covers almost nothing where the truth is far from it.

What a credible interval covers

A credible interval makes the statement everyone wants and does not claim to have a coverage. It has one anyway, it can be summed over the sample space exactly, and on a reasonable prior it beats the interval taught first.

bayes · Credible A normal mean with a flat prior — one interval, two readings. Both intervals are [1.878, 3.446]. The frequentist reading is that the procedure captures the truth 95% of the time; the Bayesian reading is that the parameter is in this interval with probability 0.95. The endpoints are identical to machine precision.

Where the two schools agree

With a flat prior on a normal mean, the credible interval and the confidence interval are the same interval, endpoint for endpoint. Knowing exactly when that stops being true is more useful than either camp's general argument.

bayes · Credible Counted coverage of two 95% intervals, over 2,000 datasets. Each point is one of the eight groups, at its own standard error. The integrated interval covers 95.2% overall against its stated 95%; the plug-in covers 78.8%, and its shortfall grows with the group's standard error — from 86.1% at se 0.5 to 77.0% at se 2.1. The mean widths are 3.48 and 2.66.

The interval that integrates

A credible interval for one group in a hierarchy has to average over every value the population spread might take. That averaging is what makes it cover — 95.2% against the plug-in's 78.8% — and it costs 31% more width, a heavier tail, and a mixture rather than a normal.

fullbayes · Credible Two 95% intervals for 2 of 20, under Jeffreys — Beta(½, ½). The equal-tailed interval runs from 0.0214 to 0.2839 and is 0.2625 wide; the shortest interval runs from 0.0093 to 0.2540 and is 0.2447 wide. Both hold 95% of the posterior, and the shorter one buys its 6.8% by moving its lower endpoint towards the denser side.

The shortest interval, and the one that does not move

Two 95% intervals come out of every posterior and they are not the same set. The shorter one is shorter by 4.86% on average and 22.41% at its best, it covers 86.72% where the other covers 95.68%, and it is not even the shortest once the parameter is written a different way.

bayes · Credible Four 95% intervals for the odds after 2 of 20. credible, transformed: 0.0218 to 0.3964. Wald, transformed: -0.0305 to 0.3012. delta method on the odds: -0.0512 to 0.2734. delta method on the log-odds: 0.0258 to 0.4789. The first two are the same intervals for the proportion with their endpoints put through the odds; the last two are fresh approximations made on the new scale.

An interval for something else

An interval for the odds is free — put the endpoints through the odds and the coverage does not move, exactly, for any interval at all. The method everyone uses instead computes a new standard error on the new scale, and at twenty trials that costs four points of coverage, produces negative odds, and has no value at all when nothing was observed.

bayes · Credible A prior worth 35 observations, moved across the range — truth 0.1, n = 20. The same prior weight centred at each of 33 places. Its interval covers 100.0% where the centre is near the truth and 0.0% at its worst, while the mean width where it covers least is 0.221 against a flat prior's 0.263 on the same data.

When the prior is confident and wrong

A prior worth thirty-five observations, centred in the wrong place, produces a 95% interval that covers nothing at all — and reports a width 5% narrower than an honest one. It takes seventeen thousand observations to repair, not thirty-five, and the worst study to run is the one whose sample size equals the prior's weight, exactly.

bayes · Credible

Named alongside it

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

CoveragePosteriorPriorFlat priorJeffreys' priorSample sizeConfidence intervalInterval widthLog-oddsMonotone transformationOddsPrior sensitivity

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