I am trying to better understand exchangeability. Suppose that I would like to run an experiment. I will pick two people and give each a fair coin and tell them to toss N times in a row. The difference is that to the person A I will tell that the coin is fair, and tell person B that I don't know whether the coin is fair or not. But I assure both of them that there is no monkey business going on.
For person A coin tosses are independent and identically distributed, hence they are exchangeable.
For person B coin tosses are neither independent nor identically distributed but they are exchangeable.
What's strange for me at this point is that for the same physical process of tossing fair coins in a row, one person thinks that the tosses are independent and the other doesn't, while I know that they are all IID.
Does this make independence subjective? Or is my understanding incorrect?
Basically B has something to learn about the process while A is unable to learn anything new. Hence their subjective beliefs affects their assessment of independence.
Should we assess independence based on an idealized true data generating process or based on what we currently know about the process?
Here is a similar rhetoric: "Resonance: From Probability To Epistemology And Back"
It seems it depends on whether you are a Bayesian or a Frequentist. Nevertheless it is quite perplexing that if question was posed as randomly picking a coin from an urn containing one biased and one fair coin (not knowing which one you picked) and then tossing the same coin N times, both the frequentist and the bayesian would probably agree that the tosses are not unconditionally independent, which means they both now assess independence based on what they know about the situation instead of the ground truth that the tosses actually don't affect each other.