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

Improper posterior — where it appears

A posterior that cannot be normalised, because the prior was improper and the data do not carry enough information to fix the integral. It usually arises for a between-group spread estimated from very few groups, and a sampler run on it returns numbers that describe nothing.

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

Three priors on the spread, at a true τ of 0.5. The posterior for τ under a flat prior (mean 1.66), a half-Cauchy of scale 1 (1.32) and one of scale 0.25 (1.15). The three answers differ by 30% of the widest. The prior does visible work when eight groups cannot separate a small spread from none, and almost none when they can.

A prior on the spread

Integrating over the population spread means putting a prior on it, which sounds like the objection rather than the repair. The prior's effect is measurable, it is invisible where the groups are clearly different, and the reflex choice for a scale parameter turns out not to have a posterior at all.

fullbayes · Prior How often a trial that borrows its control from an earlier one declares a treatment with no effect a success, by the earlier control's drift. Fifty patients an arm, an earlier control of two hundred, a one-sided 2.5% threshold. Without borrowing the rate is 2.5% at every drift. Pooling outright reaches 73.1% at a drift of half a standard deviation. A half-normal prior of scale 0.05 on the between-trial spread peaks at 35.2% and is still at 31.8% at a drift of 1.5; a half-Cauchy of the same scale peaks at 11.0% near a drift of 0.4 and falls back to 3.4%.

A control borrowed from the last trial

A trial of fifty patients an arm can borrow its control group from an earlier trial of two hundred by treating the two control means as draws from one population with a spread between trials. With two trials there is one difference to learn that spread from, so the prior on it decides how much is borrowed — and its tail decides whether the borrowing stops when the two trials disagree. A half-normal prior of scale 0.05 buys 17.4 points of power and, at a drift of 1.5 standard deviations, still declares a treatment with no effect a success 31.8% of the time. A half-Cauchy of the same scale buys 13.6 points, peaks at 11.0% and falls back to 3.4%.

fullbayes · Prior

Named alongside it

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

The half-Cauchy priorHierarchical modelPriorPrior sensitivityEmpirical BayesError rateFlat priorMarginal likelihoodPartial poolingPosteriorPosterior meanStatistical power

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