the only sensible prior may be determined by those symmetries, and that in such cases you can save some mental effort by just talking about the symmetries
Okay, but that's really just treating other views of probability as convenient fictions, which isn't quite what I was hoping for.
I do not believe that the subjective interpretation of probability is a good match for the application of probability to the analysis of events that have already occurred. Conduct an experiment, observe (say) a normal distribution in some variable. That we subjectively expected a normal distribution prior to the run of the experiment, or that we now expect it in future runs, is all well and good, but it is not our expectation that accounts for the actual appearance of the normal distribution itself. If anything accounts for the actual appearance of a normal distribution, it is some property of the experimental setup, rather than an expectation of ours.
I am not denying that you can go ahead and talk about our degree of belief and the unique rational way to update that degree of belief. I think that's a perfectly legitimate topic. What I find unconvincing is the idea is that that's all that we can profitably apply the probability calculus to. I did google "bayesian quantum" and didn't find anything that answered my concern, though I did find assertions of the very position that I have a problem with. I don't deny that you can approach quantum mechanics from a bayesian perspective - obviously your beliefs can be informed by quantum mechanics and you can have beliefs about the results of experiments and so on; I simply think it leaves something out, because at the end we have not only our own belief about the result of the next quantum experiment to arrive at rationally based on our priors in combination with experiments we have observed, but also actual observed frequencies of past quantum experiments, and these are out there, they are not subjective, and the mathematical theory of probability strongly recommends itself as a tool in the analysis of the observed frequencies.
Followup to: Illusion of Transparency: Why No One Understands You, Expecting Short Inferential Distances
A few years ago, an eminent scientist once told me how he'd written an explanation of his field aimed at a much lower technical level than usual. He had thought it would be useful to academics outside the field, or even reporters. This ended up being one of his most popular papers within his field, cited more often than anything else he'd written.
The lesson was not that his fellow scientists were stupid, but that we tend to enormously underestimate the effort required to properly explain things.
He told me this, because I'd just told him about my experience publishing "An Intuitive Explanation of Bayesian Reasoning". This is still one of my most popular, most blogged, and most appreciated works today. I regularly get fan mail from formerly confused undergraduates taking statistics classes, and journalists, and professors from outside fields. In short, I successfully hit the audience the eminent scientist had thought he was aiming for.
I'd thought I was aiming for elementary school.
Today, when I look back at the Intuitive Explanation, it seems pretty silly as an attempt on grade school:
(Then again, I get a roughly equal number of complaints that the Intuitive Explanation is too long and drawn-out, as that it is too short. The current version does seem to be "just right" for a fair number of people.)
Explainers shoot way, way higher than they think they're aiming, thanks to the illusion of transparency and self-anchoring. We miss the mark by several major grades of expertise. Aiming for outside academics gets you an article that will be popular among specialists in your field. Aiming at grade school (admittedly, naively so) will hit undergraduates. This is not because your audience is more stupid than you think, but because your words are far less helpful than you think. You're way way overshooting the target. Aim several major gradations lower, and you may hit your mark.
PS: I know and do confess that I need to work on taking my own advice.
Addendum: With his gracious permission: The eminent scientist was Ralph Merkle.