What eight groups say about τ, when the truth is 1

The posterior density for the population spread after eight groups whose standard errors run from 0.5 to 2.1. The shaded band is the central 95% interval, from 1.02 to 4.44; the posterior median is 1.99 and the mean 2.18. The vertical mark at 1.07 is the moment estimate that empirical Bayes substitutes and then treats as known.

The posterior for τ under a flat prior (mean 2.18), a half-Cauchy of scale 1 (1.78) and one of scale 0.25 (1.69). The three answers differ by 23% of the widest. The prior does visible work when eight groups cannot separate a small spread from none, and almost none when they can.

Four curves and only two calculations. Under a flat prior the two grids give the same answer — posterior mean 1.389 against 1.389, a difference of 0.0%. Under a 1/τ prior they do not: 1.172 against 1.031, a fall of 12.0%, with the density at the smallest τ on the grid rising from 0.53 to 0.72. There is nothing to converge to: the likelihood is finite at τ = 0 and 1/τ is not integrable there.

Two posteriors for one group's value. The narrow one conditions on the moment estimate τ̂ = 1.07 and treats it as known; the wide one integrates over everything the eight groups leave open about τ. The intervals are 4.11 and 6.03 wide, a factor of 1.47, and their centres differ by only 0.257.

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.

Every dataset here was generated with a real population spread of 1. The moment estimator is the difference between the observed spread and what noise alone would produce, clamped at zero, and the difference comes out negative often: at eight groups it reports exactly zero on 32.6% of datasets, which is an instruction to pool completely and give all eight groups the same estimate. The rate falls to 4.2% at 48 groups.

The eight groups' own means as hollow circles, the integrated posterior's interval as a bar, and the plug-in estimate as a filled mark. The moment estimate of τ came out exactly zero on this dataset, so every plug-in estimate is the same number, 0.14. The integrated posterior, on the same observations, still gives the groups different values, because its interval for τ runs from 0.02 to 1.82 and it never has to pick one.