If we go and couple Bayesianism with a deterministic universe (before you say anything with the word 'quantum' in it, humour me and recall that this is not physics.stackexchange) we get some interesting results.
Making our assumptions explicit:
- We have a Bayesian agent being part of and observing a deterministic universe.
- The agent has limited computational resources.
Now, the deterministic universe may be one where atoms are newtonian little billiard balls. It may be entirely non-quantum. Let's say it is.
The agent now flips a fair coin. Think about that for a second, what does a fair coin constitute in a deterministic universe? A coin that has a 50/50 probability ratio?
But it is deterministic! With enough computing power you can calculate exactly how the coin will land, purely by simulating a model of a coin being flipped in the same manner.
In a deterministic universe a fair coin would be a disc of metal with uniform density. No force compels it to spend more time with one face down than the other (think about how weighted dice function.)
So the agent flips a fair coin. Yet, the agent is not quite powerful enough. It does not have sharp enough eyes to measure how the coin spins when flipped, it sees but a blur.
And so it says "This coin will land a heads with 50% probability." Lack of information leads to probabilities.
We may look at the phase space of how a coin is thrown. A large multidimensional coordinate system with axes pertaining to direction of throw, force of throw, spin of the coin, speed and direction of wind and so on. A single point in this space corresponds to a single possible coinflip.
If we ask the agent from before to colour in the coordinate system with a greyscale gradient corresponding to the agent's assignment of probability of heads for every given throw, it will most colour it all a uniform shade of grey.
If we the gradually give it more powerful internal computers with which to compute probabilities of heads, it will be able to make more and more discerning colourings. When we finally give it the most powerful internal computer, making it omniscient, it will effectively paint a strange checkerboard.
Fair coins are not made of probabilities, they are made of metal. Probabilities exist only in computational structures. So says the Bayesian.