Comment by spider-mario
4 hours ago
It’s the same principle. The “trivial interval” from the first link (the one you called pedantry), which is applicable to a Gaussian and tweakable to have different coverage than 50%, clearly shows that being an n% confidence interval, on its own, doesn’t guarantee the ability to directly translate that n% into post-data inferences. Likewise, in the truncated exponential example, when you compute that confidence interval of [12.1471, 13.8264], you know that the probability that it contains θ is 0%. In the Cauchy example, when you compute [-2.31, 10.31] as your 90% CI from the two samples 3 and 5, you can then compute that it actually has a >99% chance of containing θ (table I).
The trivial example demonstrates that once you pick your confidence interval you don't truly have odds anymore. But if we're judging whether we have the right odds number, what matters is if a different number is correct. And the only coherent way I can think of to interpret "odds" says the odds are 50%. The trivial example survives that challenge.
The examples that actually ruin the number so far have been asymmetrical. Is there a way to do it with a gaussian? Especially if you're trying to make a reasonable internal?