Heh, that's a cheeky example. To explain why it's cheeky, I have to briefly run through it, which I'll do here (using Jaynes's symbols so whoever clicked through and has pages 22-24 open can directly compare my summary with Jaynes's exposition).

Call N the sample size and θ the minimum possible widget lifetime (what bill calls X). Jaynes first builds a frequentist confidence interval around θ by defining the unbiased estimator θ∗, which is the observations' mean minus one. (Subtracting one accounts for the sample mean being >θ.) θ∗'s probability distribution turns out to be y^(N-1) exp(-Ny), where y = θ∗ - θ + 1. Note that y is essentially a measure of how far our estimator θ∗ is from the true θ, so Jaynes now has a pdf for that. Jaynes integrates that pdf to get y's cdf, which he calls F(y). He then makes the 90% CI by computing [y1, y2] such that F(y2) - F(y1) = 0.9. That gives [0.1736, 1.8259]. Substituting in N and θ∗ for the sample and a little algebra (to get a CI corresponding to θ∗ rather than y) gives his θ CI of [12.1471, 13.8264].

For the Bayesian CI, Jaynes takes a constant prior, then jumps straight to the posterior being N exp(N(θ - x1)), where x1's the smallest lifetime in the sample (12 in this case). He then comes up with the smallest interval that encompasses 90% of the posterior probability, and it turns out to be [11.23, 12].

Jaynes rightly observes that the Bayesian CI accords with common sense, and the frequentist CI does not. This comparison is what feels cheeky to me.

Why? Because Jaynes has used different estimators for the two methods [edit: I had previously written here that Jaynes implicitly used different estimators, but this is actually false; when he discusses the example subsequently (see p. 25 of the PDF) he fleshes out this point in terms of sufficient v. non-sufficient statistics.]. For the Bayesian CI, Jaynes effectively uses the minimum lifetime as his estimator for θ (by defining the likelihood to be solely a function of the smallest observation, instead of all of them), but for the frequentist CI, he explicitly uses the mean lifetime minus 1. If Jaynes-as-frequentist had happened to use the maximum likelihood estimator -- which turns out to be the minimum lifetime here -- instead of an arbitrary unbiased estimator he would've gotten precisely the same result as Jaynes-as-Bayesian.

So it seems to me that the exercise just demonstrates that Bayesianism-done-slyly outperformed frequentism-done-mindlessly. I can imagine that if I had tried to do the same exercise from scratch, I would have ended up faux-proving the reverse: that the Bayesian CI was dumber than the frequentist's. I would've just picked up a boring, old-fashioned, not especially Bayesian reference book to look up the MLE, and used its sampling distribution to get my frequentist CI: that would've given me the common sense CI [11.23, 12]. Then I'd construct the Bayesian CI by mechanically defining the likelihood as the product of the individual observations' likelihoods. That last step, I am pretty sure but cannot immediately prove, would give me a crappy Bayesian CI like [12.1471, 13.8264], if not that very interval.

Ultimately, at least in this case, I reckon your choice of estimator is far more important than whether you have a portrait of Bayes or Neyman on your wall.

[Edited to replace my asterisks with ∗ so I don't mess up the formatting.]