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.

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.

Wald: 87.6% coverage on a mean width of 0.212. Wilson: 95.7% coverage on a mean width of 0.263. Clopper–Pearson: 98.9% coverage on a mean width of 0.292. credible, Jeffreys: 95.7% coverage on a mean width of 0.249. credible, flat: 95.7% coverage on a mean width of 0.263. All computed by the same exact sum over the 21 possible counts.

A prior is worth exactly a + b observations: Jeffreys — Beta(½, ½) is worth 1, weakly informative — Beta(2, 2) is worth 4, confident, centred at 0.5 — Beta(20, 20) is worth 40, confident and wrong — Beta(30, 5) is worth 35. Each curve is the posterior mean as the data accumulates, and every one of them converges on 0.25.

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.

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.

Three 95% intervals at each of the 21 possible counts. The shortest interval is shorter than the equal-tailed one at 20 of them, by 4.86% on average and 22.41% at most.

An exact sum over all 21 counts at each of 199 true proportions. The equal-tailed interval's worst case is 89.64%, the shortest interval's is 85.54%, and the interval computed on the log-odds scale is at 90.46%. The shortest interval falls below 90% at 28 of the 199 proportions against the equal-tailed interval's 4.

On the proportion scale this posterior's mode and median are 0.406 of its own spread apart; in log-odds they are 0.168 apart. A nearly symmetric density has almost the same shortest and equal-tailed interval, which is why the two agree here and not on the other scale.

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.

Summed over all 21 counts at each of 97 true proportions. Transforming an interval's endpoints leaves its coverage exactly where it was; computing a fresh standard error on the new scale does not, and the log-odds version is the worst of the four at 27.2%.

The posterior median of the odds is the odds of the posterior median, exactly, because a median is a quantile. The posterior mean of the odds is not the odds of the posterior mean — it is larger at every count — and on 1 of the 21 counts it is infinite, because the integral diverges.

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.

A prior worth 35 observations, centred 0.75 away from the truth. Its 95% interval covers 0.0% at 10 observations and does not reach 90% until 17409 — 497 times the prior's own weight, because what must shrink is the displacement in standard errors and that falls only as one over the square root of the sample size.

The prior predictive distribution of the count, in closed form from the Beta–binomial. It expects 17.0 and gives the observed 2 a probability of 1.55e-8; a count at least this extreme has probability 1.68e-8. The interval built from this prior carries none of that.

The posterior mean is displaced from the truth by w(c − p)/(w + n) and the interval round it has a width of order the square root of p(1 − p) over n, so the displacement measured in standard errors grows like the square root of n over (w + n). That rises and then falls, and it peaks at n = 35 against a prior worth 35 — the sample size at which the data is worth exactly what the prior is worth. The peak here is 7.40 standard errors.