> 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.

Sorry, I don’t follow. Let’s say I want to compute a 50% confidence interval for the unknown mean of a Gaussian distribution. I sample two numbers from the distribution, get 9 and 7, compute the interval according to the trivial procedure and get (-∞, ∞). Does the interval (-∞, ∞) have a 50% probability of containing the mean of that Gaussian distribution? I would think it’s closer to 100%.

If what you are saying is “it’s meaningless to talk about the probability of that specific interval containing the unknown-but-fixed parameter” then that’s the purely frequentist view and then you also agree that it’s meaningless to say that [-2.31, 10.31] has a 90% chance of containing the location parameter of the Cauchy distribution that happened to yield the samples 3 and 5. Incidentally, what asymmetry are you referring to in the Cauchy example?