[equark]

I don't think this is a worthwhile distinction, even if it's historically accurate. Both Bayesian and Frequentists focus on point estimation and distributions. EAP and MAP are just as Bayesian and the full posterior distribution. And the sampling distribution is just as important to Frequentists as the posterior distribution is to a Bayesian.

The key difference is whether inference is based on the sampling distribution or the posterior distribution.

[ot]

Very interesting, can you expand? How do frequentists use the sampling distribution other than the classical MLE etc?

[equark]

The sampling distribution is not the likelihood. It's the fundamental basis of all frequentist inference. Amazingly, even though this is typically taught in introductory statistics virtually no students actually digest its importance. You literally cannot understand frequentist statistics without understanding the idea of a sampling distribution.

The sampling distribution is the distribution of your statistic (MLE estimate, mean, EAP, MAP, or whatever you want) under repeated sampling from the population distribution. Frequentism is an evaluation procedure, which can be applied to any estimator whether it be Bayesian or something like MLE. Frequentists are interested in whether this distribution has "good" properties. Supposedly good properties include things like unbiasedness, consistency, minimum variance, etc. Inference is typically expressed as a function of this distribution (confidence intervals) or by comparing the sampling distribution under some restriction (the null hypothesis) to the actual value of the statistic in the observed sample.

Given that you can't typically sample from the population distribution, the practical question becomes how do you approximate the sampling distribution. Typically this is done by appealing to a central limit theorem. Bootstrapping provides another intuitive approximation.

There are all sorts of problem with this approach to statistics despite its success.