What is the Bayesian argument, if one exists, for why quantum dynamics breaks the “probability is in the mind” philosophy?

In my world-view the argument is based on Bell inequalities. Other answers mention them, I will try and give more of an introduction.

First, context. We can reason inside a theory, and we can reason about a theory. The two are completely different and give different intuitions. Anyone talking about "but the complex amplitudes exist" or "we are in one Everett branch" is reasoning inside the theory. The theory, as given in the textbooks, is accepted as true and interpretations built on.

However, both historically and (I think) more generally, we should also reason about theories. This means we need to look at experimental observations, and ask questions like "what is the most reasonable model?".

Many quantum experiments give random-looking results. As you point out, randomness is usually just "in the mind". Reality was deterministic, but we couldn't see everything. The terminology is "local hidden variable". For an experiment where you draw a card from a deck the "local hidden variable" was which card was on top. In a lottery with (assumedly deterministic) pinballs the local hidden variable is some very specific details of the initial momentums and positions of the balls. In other words the local hidden variable is the thing that you don't know, so to you it looks random. Its the seed of your pseudorandom number generator.

Entanglement - It is possible to prepare two (or more) particles in a state, such that measurements of those two particles gives very weird results. What do I mean by "very weird". Well, in a classical setting if Alice and Bob are measuring two separate objects then there are three possible (extremal) situations (1): Their results are completely uncorrelated, for example Alice is rolling a dice in Texas and Bob is rolling a different dice in London. (2) Correlated, for example, Alice is reading an email telling her she got a job she applied for, and Bob is reading an email telling him he failed to get the same job. (4) Signalling (we skipped 3 on purpose, we will get to that). Alice and Bob have phones, and so the data they receive is related to what the other of them is doing. Linear combinations of the above (eg noisy radio messages, correlation that is nor perfect etc) are also possible.

By very weird, I mean that quantum experiments give rise (in the raw experimental data, before any theory is glued on) to a fourth type of relation; (3): Non-locality. Alice and Bob's measurement outcomes (observations) are random, but the correlation between their observation's changes depending on the measurements they both chose to make (inputs). Mathematically its no more complex than the others, but its fiddly to get your head around because its not something seen in everyday life.

An important feature of (3) is that it cannot be used to create signalling (4). However, (3) cannot be created out of any mixture of (1) and (2). (Just like (4) cannot be created by mixing (1) and (2)).  In short, if you have any one of these 4 things, you can use local actions to go "down  hill" to lower numbers but you can't go up.

Anyway, "hidden variables" are shorthand for "(1) and (2)" (randomness and correlation). The "local" means "no signalling" (IE no (3), no radios). The reason we insist on no signalling is because the measurements Alice and Bob do on their particles could be outside one another's light cones (so even a lightspeed signal would not be fast enough to explain the statistics). The "no signalling" condition might sound artificial, but if you allow faster than light signalling then you are (by the standards of relativity) also allowing time travel.

Bell inequality experiments have been done. They measure result (3). (3) cannot be made out of ordinary "ignorance" probabilities (cannot be made from (2)). (3) could be made out of (4) (faster than light signalling), but we don't see the signalling itself, and assuming it exists entails time travel.

So, if we reject signalling, we know that whatever it is that is happening in a Bell inequality experiment it can't be merely apparent randomness due to our ignorance. We also know the individual results collected by Alice and Bob look random (but not the correlations between the results), this backs us into the corner of accepting that the randomness is somehow an intrinsic feature of the world, even the photon didn't "know" if it would go through the polariser until you tried it.

The wiki article on Bell inequalities isn't very good unfortunately.

I dispute the premise. Weights of quantum configurations are not probabilities, they just share some superficial similarities. (They're modeled with complex numbers!) Iirc Eliezer was very clear about this point in the quantum sequence.

Jaynes has argued (jn his book on probability theory) that nearly all progress in physics stopped once quantum mechanics was accepted to be "truly random," and that this anti-scientific attitude has to be abandoned for further progress to be possible. 

From the little I know about QM he was probably wrong - it seems you can't remove the randomness without breaking other stuff we like such as locality. 

Still, I wouldn't be shocked if he (and perhaps you?) were right after all.  

Personally, I can't see any way to rule out the possibility that some things in the universe are "truly random" for all practical purposes - in the sense that, say, a sequence of spin measurements can't be compressed. In order for science to be possible, there have to be some discoverable underlying laws, but Solomonoff induction still works if those laws are probabilistic. If you insist on describing that situation as also not objectively random but only uncertain, fine, but I think it matches our intuitions about objective randomness perfectly and I'm inclined to use that term. This deserves a full post though. 

See Bell's theorem. Basically we know that quantum mechanics is truly random, not just pseudorandom, unless you posit non-locality.

I must admit, I think the "probability is in the mind, not the territory" either vacuously true or possibly false if you think that maps aren't the same things as low-resolution territories.

One example of where probability is in the mind, not the territory is false for non-trivial definitions of maps is Wigner functions, which turn out to be basically equivalent to a wavefunction, where it behaves basically like classical Bayesian probability theory, but with the caveat that for quantum physics, negative probabilities are allowed, and the most important impact here is you can't treat the uncertainty as just ignorance anymore (though they can include our ignorance).

More here:

https://www.lesswrong.com/posts/Y6LhXdGfwsAStMuhr/ackshually-many-worlds-is-wrong#nsebEbJbxqkekTbsK