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Vladimir_Nesov comments on Bead Jar Guesses - Less Wrong
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I do not know how related this is to your comment, but it made me think of another response to the Dutch book objection. (Am I using that term correctly?)
If Omega asks me about a red bead I can say 100%. If he then asks about a blue bead I can adjust my original estimate so that both red and blue are the same at 50/50. Every question asked is adding more information. If Omega asks about green beads all three answers get shifted to 1/3.
This translates into an example with numbered balls just fine. The more colors or numbers Omega asks about decreases the expected probability that any particular one of them will come out of the jar simply because the known space of colors and numbers is growing. Until Omega acknowledges that there could be a bead of that color or number there is no particular reason to assume that such a bead exists.
If the example was rewritten to simply say any type of object could be in the jar, this still makes sense. If Omega asks about a red bead, we say 100%. If Omega asks about a blue chair, both become 50%. The restriction of colors and numbers is our assumed knowledge and has nothing to do with the problem at hand. We can meta-game all we want, but it has nothing to do with what could be in the jar.
The state of the initial problem is this:
After the second question:
I suppose it makes some sense to include an "other" category, but there is no knowledge of anything other than red and green beads. The question of probability implies that another may exist, but is that enough to assign it a probability?
Don't think of probability as being mutable, as getting updated. Instead, consider a fixed comprehensive state space, that has a place on it for every possible future behavior, including the possible questions asked, possible pieces of evidence presented, possible actions you make. Assign a fixed probability measure to this state space.
Now, when you do observe something, this is information, an event, a subset on the global state space. This event selects an area on it, and encompasses some of the probability mass. The statements, or beliefs (such as "the ball #2 will be red"), that you update on this info, are probabilistic variables. A probabilistic variable is a function that maps the state space on a simpler domain, for example a binary discrete probabilistic variable is basically an event, a subset of the state space (that is, in some states, the ball #2 is indeed defined to be red, these states belong to the event of ball #2 being red).
Your info about the world retains only the part of the state space, and within that part of the state space, some portion of the probability mass goes to the event defining your statement, and some portion remains outside of it. The "updating" only happens when you focus on this info, as opposed to the whole state space.
If that picture is clear, you can try to step back to consider what kind of probability measure you'd assign to your state space, when its structure already encodes all possible future observations. If you are indifferent to a model, the assignment is going to be some kind of division into equal parts, according to the structure of state space.
IAWYC, but as pedagogy it's about on the level of "How should you imagine a 7-dimensional torus? Just imagine an n-dimensional torus and let n go to 7."
Eliezer's post on priors explains the same idea more accessibly.
EDIT: Sorry, I didn't notice you already linked it below.