The distribution JBlack proposes is not over the outcome or bias. It is over possible coin behaviours. If you want an explicit set, you could use this one:
({Coin chosen adversarially} union
{Coin has bias
You seem to be saying that if you define your belief to be a probability distribution over an explicit state space, that then you can’t believe anything that involves information beyond that state space. That seems tautological to the point of uselessness.
Complex beliefs are common. It is nice when we can simplify them to a distribution over state spaces, but if you want to use examples that involve Omega-style self-reference, you are unlikely to be able to simplify them easily. (And if you consider the example of solomonoff induction, then the probability distribution is over all halting Turing machines.)
“Coin chosen adversarially” is an element of a state space. The agent assigns a probability to it being in different possible worlds.
JBlack and I do not seem to have made arguments that depend on the agent’s decision theory. (For all we’ve said, it might be ignoring the world model and acting randomly.) We have only discussed probability distributions that admit representing the Omega situation you brought up.
You would likely want to have your agent able to consider multiple different possible worlds, and evaluate the probability of being in any one of them. The typical highly general formalization of this is Solomonoff induction.