But if we start with a prior over all possible six-sided dice and do Bayesian updating, we get a different answer that diverges from fairness more and more as the number of throws goes to infinity!

In this example, what information are we Bayesian updating on?

In the large N limit, and only the information that the mean is exactly 3.5, the obvious conclusion is that one is in a thought experiment, because that's an absurd thing to choose to measure and an adversary has chosen the result to make us regret the choice.

More generally, one should revisit the hypothesis that the rolls of the die are independent. Yes, rolling only 1 and 6 is more likely to get a mean of 3.5 than rolling all six numbers, but still quite unlikely. Model checking!

But if we start with a prior over all possible six-sided dies and do Bayesian updating, we get a different answer that diverges from fairness more and more as the number of throws goes to infinity!

I'm nearly positive that the linked paper (and in particular, the above-quoted conclusion) is just wrong. Many years ago I checked the calculations carefully and found that the results come from an unavailable computer program, so it's definitely possible that the results were just due to a bug. Meanwhile, my paper copy of PT:LOS contains a section which purports to show that Bayesian updating and maximum entropy give the same answer in the large-sample limit. I checked the math there too, and it seemed sound.

I might be able to offer more than my unsupported assertions when I get home from work.

We throw it many times (say, a billion) and obtain an average value of 3.5. Using that information alone

So my prior state of knowledge about the die is entirely characterized by N=10^9 and m=3.5, with no knowledge of the shape of the distribution? It's not obvious to me how you're supposed to turn that, plus your background knowledge about what sort of object a die is, into a prior distribution; even one that maximizes entropy. The linked article mentions a "constraint rule" which seems to be an additional thing.

This sort of thing is rather thoroughly covered by Jaynes in PT:TLOS as I recall, and could make a good exercise for the Book Club when we come to the relevant chapters. In particular section 10.3 "How to cheat at coin and die tossing" contains the following caveat:

The results of tossing a die many times do not tell us any definite number char- acteristic only of the die. They tell us also something about how the die was tossed. If you toss ‘loaded’ dice in different ways, you can easily alter the relative frequencies of the faces. With only slightly more difficulty, you can still do this if your dice are perfectly ‘honest’.

And later:

The problems in which intuition compels us most strongly to a uniform probability assignment are not the ones in which we merely apply a principle of ‘equal distribution of ignorance’. Thus, to explain the assignment of equal probabilities to heads and tails on the grounds that we ‘saw no reason why either face should be more likely than the other’, fails utterly to do justice to the reasoning involved. The point is that we have not merely ‘equal ignorance’. We also have positive knowledge of the symmetry of the problem; and introspection will show that when this positive knowledge is lacking, so also is our intuitive compulsion toward a uniform distribution.

EDIT: I am an eejit. Dangit, need to remember to stop and think before posting.

Umm, not quite. The die being biased towards 2 and 5 gives the same probability of 3.5 as the die being 3,4 biased.

As does 1,6 bias.

So, given these three possibilities, an equal distribution is once again shown to be correct. By picking one of the three, and ignoring the other two, you can (accidentally) trick some people, but you cannot trick probability.

This is before even looking at the maths, and/or asking about the precision to which the mean is given (ie. is it 2 SF, 13 SF, 1 billion sf? Rounded to the nearest .5?)

EDIT: this appears to be incorrect, sorry.