OK. But if you yourself state that you “certainly know”—certainly—that p is fixed, then you have already accounted for that particular item of knowledge.
If you do not, in fact, “certainly know” the probability of p—as could easily be the case if you picked up a coin in a mafia-run casino or whatever—then your prior should be 0.5 but you should also be prepared to update that value according to Bayes’ Theorem.
I see that you are gesturing towards assigning also the probability that the coin is a fair coin (or generally such a coin that has a p of a certain value). That is also amenable to Bayes’ Theorem in a normal way. Your prior might be based on how common biased coins are amongst the general population of coins, or somewhat of a rough guess based on how many you think you might find in a mafia-run casino. But by all means, your prior will become increasingly irrelevant the more times you flip the coin. So, I don’t think you need to be too concerned about how nebulous that prior and its origins are!
Knowing (or assuming) that the value of p does not change between experiments is a different kind of knowledge than knowing the value of p.
OK. But if you yourself state that you “certainly know”—certainly—that p is fixed, then you have already accounted for that particular item of knowledge.
If you do not, in fact, “certainly know” the probability of p—as could easily be the case if you picked up a coin in a mafia-run casino or whatever—then your prior should be 0.5 but you should also be prepared to update that value according to Bayes’ Theorem.
I see that you are gesturing towards assigning also the probability that the coin is a fair coin (or generally such a coin that has a p of a certain value). That is also amenable to Bayes’ Theorem in a normal way. Your prior might be based on how common biased coins are amongst the general population of coins, or somewhat of a rough guess based on how many you think you might find in a mafia-run casino. But by all means, your prior will become increasingly irrelevant the more times you flip the coin. So, I don’t think you need to be too concerned about how nebulous that prior and its origins are!