In the particular case I gave, of course frequentists could produce an argument that the mean must be in the given range. But this could not be a statistical argument, it would have to be a deductive logical argument.
The frequentists do have an out here: conditional inference. Obviously, (v2+v1)/2 is sufficient for m, so they don't need any other information for their inference. But it might occur to them to condition on the ancillary statistic v2-v1. In repeated trials where v2-v1 = 0.9, the interval (v1,v2) always contains m.
Edit: As pragmatist mentions below, this is wrong wrong wrong. The minimal sufficient statistic is (v1,v2), although it is true that v2-v1 is ancillary and moreover it is the ancillary complement to the sample mean. That I was working with order statistics (and the uniform distribution!) is a sign that I shouldn't just grope for the sample mean and say good enough.
True, but is there any motivation for the frequentist to condition on the ancillary statistic, besides relying on Bayesian intuitions? My understanding is that the usual mathematical motivation for conditioning on the ancillary statistic is that there is no sufficient statistic of the same dimension as the parameter. That isn't true in this case.
ETA: Wait, that isn't right... I made the same assumption you did, that the sample mean is obviously sufficient for m in this example. But that isn't true! I'm pretty sure in this case the minimal sufficient statis...
I've had a bit of success with getting people to understand Bayesianism at parties and such, and I'm posting this thought experiment that I came up with to see if it can be improved or if an entirely different thought experiment would be grasped more intuitively in that context:
I originally came up with this idea to explain falsifiability which is why I didn't go with say the example in the better article on Bayesianism (i.e. any other number besides a 3 rolled refutes the possibility that the trick die was picked) and having a hypothesis that explains too much contradictory data, so eventually I increase the sides that the die has (like a hypothetical 50-sided die), the different types of die in the jar (100-sided, 6-sided, trick die), and different distributions of die in the jar (90% of the die are 200-sided but a 3 is rolled, etc.). Again, I've been discussing this at parties where alcohol is flowing and cognition is impaired yet people understand it, so I figure if it works there then it can be understood intuitively by many people.