Subjective Bayesian's care for real world validation and classical statistician's worry about CI related paradoxes for a given data set? I was thinking about CI and subjective Bayesian and I have following two questions:


*

*If a subjective (not objective) Bayesian would care if her predictions don't do well in the real world.

*A classical statistician would not care if her confidence statement is (obviously) wrong for a given data set (as in Welch's Paradox, where conditioning is on ancillary statistics leads to the resolution of pathological behavior).
I think my answer for 1. is YES and 2. is NO. 
But I don't know if I am thinking along the right lines. Can I have some more insights?

UPDATE
Welch's example: This example works for any $n$, but we will take $n=2$ for simplicity. $X_1, X_2 \sim U(\theta - 1/2, \theta +1/2)$ (iid), $\theta \in R$. This implies $X_1 - \theta \sim U(-1/2, 1/2)$ (iid). $(X_1 + X_2) /2 - \theta$ (note that this is NOT a statistic) has a distribution independent of $\theta$. We can choose c > 0 s.t. $Prob_{\theta} [-c \le (X_1 + X_2) /2 - \theta \le c] = 1- \alpha (~0.99)$, implying $((X_1 + X_2) /2 - c, (X_1 + X_2) /2 + c)$ is the 99% CI of $\theta$. The interpretation of this CI is: if we sample repeatedly, we will get different $(X_1 + X_2) /2$ and (at least) 99% times it will contain true \theta. But for a particular set of $X_1, X_2$, we can't say if the CI contains $\theta$. Now, consider the following data: $X_1 =0$ and $X_2=1$, as $|X_1 - X_2|=1$, we know FOR SURE (Prob =1) that the interval $(X_1, X_2)$ contains theta (one possible criticism, $P{|X_1 - X_2|=1} = 0$, but we can handle it mathematically and I won't discuss it). (Better details are in Pratt, 1961; Lehman, Chap 10, 2nd Edition, Prob 27, 28; Kiefer, 1977; Berger and Wolpert, 1988)
Thanks,
S.
 A: For the second question, I believe the answer is "Yes". I will quote Andrew Gelman here, "..in general there is no coverage guarantee because frequency properties depend on nuisance
parameters which can only be ignored in some special cases of pivotal test statistics". 
You can take a look at the following paper for some really nice discussion of relative merits and demerits of these procedures. 
http://www.stat.columbia.edu/~gelman/research/published/badbayesresponsemain.pdf
V.S.
A: For many reasons you're right about 1.  I certainly wouldn't heed the advice of someone who did not care about whether it is any good!
Number 2, as you have expressed it, does not characterize good practice.  If there are possible datasets where a CI (or any decision procedure, for that matter) is clearly wrong, then that procedure is inadmissible by definition (because you can replace it by one that is at least as good and sometimes better, no matter what).  Yes, inadmissible procedures are sometimes used in practice, but it can be argued that such procedures actually are admissible when we include the cost of performing the procedure itself within the loss function.  (In other words, a quick and dirty method that works ok can be considered superior to one that requires extensive time and effort to calculate and works only a little better.)  But in this hypothetical case, the "obviously" clause indicates it takes no effort to recognize some wrong intervals and replace them with better ones.  Therefore, although I do think your reasoning is good, we should conclude that a thoughtful concerned "classical" statistician would indeed care: the answer should be YES to both questions.
