Laplace’s rule of succession is for this problem. Laplace defines orthodox bayesianism, not the heretic (infidel?) you quote. Why shouldn’t I believe that sometimes objective probabilities make sense, even if not all probabilities are objective? In any event, I can choose it as a model of the coin. I have a prior over biases and update it every time I see a flip of the coin. Ahead of time, my prediction for each flip is the same, but after a few flips, my new prediction is different from my old prediction. Of course, I could have other models, like that Alice always flips heads and Bob always flips tails, but if I’m flipping and I know I’m not cheating, then a constant bias seems like a pretty good model to me.
Laplace’s rule of succession is for this problem. Laplace defines orthodox bayesianism, not the heretic (infidel?) you quote. Why shouldn’t I believe that sometimes objective probabilities make sense, even if not all probabilities are objective? In any event, I can choose it as a model of the coin. I have a prior over biases and update it every time I see a flip of the coin. Ahead of time, my prediction for each flip is the same, but after a few flips, my new prediction is different from my old prediction. Of course, I could have other models, like that Alice always flips heads and Bob always flips tails, but if I’m flipping and I know I’m not cheating, then a constant bias seems like a pretty good model to me.
Laplace’s rule of succession implicitly relies on a particular choice of prior over possible biases, so I don’t see how this answers the question.