“But iterated expectations, all with the same conditioning, is superfluous. That’s why I took care not to put any conditioning into my expectations.”
Fair enough. My point is that the de Finetti theorem provides a way to think sensibly about having a probability of a probability, particularly in a Bayesian framework.
Let me give a toy example to demonstrate why the concept is not superfluous, as you assert. Compare two situations:
(a) I toss a coin that I know to be as symmetrical in construction as possible.
(b) A magician friend of mine, who I know has access to double-headed and double-tailed coins, tosses a coin. I have no idea about the provenance of the coin she is using.
My epistemic probability for the outcome of the toss, in both cases, is 0.5, from symmetry arguments. (Not physical symmetry, epistemic symmetry—that is, symmetry of the available pre-toss information to an interchange of heads and tails.) My epistemic “probability of the probability” of the toss is different in the two cases. In case (a) it is nearly a delta function at 0.5, the sharpness of distribution being a function of my knowledge of the state of the art in symmetrical coin minting. In case (b), it is a mixture of distributions encoding the possible types of coins my friend might have chosen.
“But iterated expectations, all with the same conditioning, is superfluous. That’s why I took care not to put any conditioning into my expectations.”
Fair enough. My point is that the de Finetti theorem provides a way to think sensibly about having a probability of a probability, particularly in a Bayesian framework.
Let me give a toy example to demonstrate why the concept is not superfluous, as you assert. Compare two situations:
(a) I toss a coin that I know to be as symmetrical in construction as possible.
(b) A magician friend of mine, who I know has access to double-headed and double-tailed coins, tosses a coin. I have no idea about the provenance of the coin she is using.
My epistemic probability for the outcome of the toss, in both cases, is 0.5, from symmetry arguments. (Not physical symmetry, epistemic symmetry—that is, symmetry of the available pre-toss information to an interchange of heads and tails.) My epistemic “probability of the probability” of the toss is different in the two cases. In case (a) it is nearly a delta function at 0.5, the sharpness of distribution being a function of my knowledge of the state of the art in symmetrical coin minting. In case (b), it is a mixture of distributions encoding the possible types of coins my friend might have chosen.
And this could make a real difference, if you are shown the product of 5 tosses (they were all heads) and then asked to bet on the following result.