I have to run but I'll quickly respond

You're saying for certain models it is hard to obtain a sampling distribution but easy to obtain a posterior. I agree. But this is a bit like saying, sometimes when it's hard to come by a banana, it is easy to come by a pickup truck. We want the sampling distribution for a reason, namely because it allows us to determine whether our procedure has good properties under repeated sampling. Either the posterior gives us that, in which case the sampling distribution is just as easy to obtain, or it doesn't, in which case Bayes is giving us something irrelevant.

Again, distinguish classical and frequentist. "There is no guarantee that estimators derived from frequentist procedures actually have good frequentist properties" makes no sense. A frequentist procedure is a procedure with good frequentist properties. A Bayesian procedure is a procedure with good Bayesian properties. The question is when we have a Bayesian procedure which is not a frequentist procedure, and a frequentist procedure which is not a Bayesian procedure, and we have to choose. Which would you choose, per these definitions?

Coverage-free 95% posterior intervals may be pathological, but they're relevant to the philosophical question. If we see such intervals as problematic, it means we care about coverage. On the other hand if someone has a true 95% confidence interval with close to 0 posterior probability, do we see that as a problem? If we see the one as a problem but not the other, that tells us something about what we want from our statistical procedures.

Btw most people call me a Bayesian. I'm an applied statistician, and most of what I do is employ Bayesian methods. But I do so because I think they often get me good frequentist properties. If I do a simulation and find I'm wrong, I start looking elsewhere. And almost all of my applied stat colleagues would do the same. Gotta go though, peace out

(1) I can fit any model by making up parameter values. What makes the Bayesian fit better than mine?

I'm talking about the ability to estimate the model. With complex models that have hierarchal structure/varying dimension of parameter space/etc it is very hard to actually produce sampling distribtions, which often leads frequentists to fit simplistic models that make stronger assumptions. In general, this is less of an issue for Bayesians, because obtaining the posterior distribtuion of parameters in complex models is much easier.

Short story is, Bayes is better if you don't care about the frequentists properties of your statistical procedures. If you do, then you have to use methods with, well, good frequentist properties--namely, frequentist procedures. Sometimes Bayesian procedures have good frequentist properties, but when they don't (e.g., when a 95% posterior interval has close to 0 coverage), what is their rationale?

There is no guarantee that estimators derived from frequentist procedures actually have good frequentist properties, and Bayesian estimators often do better. The classic example is inference for Binomial proportions, where the credible derived under the Jeffrey's prior has better coverage properties than most frequentist intervals. Yes, everyone should be worrying about the performance of estimators under repeated sampling, but that itself is no reason to use frequentist procedures. Frequentism isnt just a case of worrying about repeated sampling/coverage/etc (which many Bayesians also care about), its about commitment to a whole host of dubious baggage like the idea that estimators/intervals should be minimax, and so on.

Examples where 95% posterior intervals have close to zero coverage when a sensible non-informative prior has been used tend to be pathological and dont arise often. Frequentist confidence intervals can also break down in pathological situations too, that isnt really ...See full post