5 ms·

It might be of some general interest to note that there exist such objects - distributions over distributions, or metaprobabilities. They have practical use - f

It might be of some general interest to note that there exist such objects - distributions over distributions, or metaprobabilities. They have practical use - for example, if you are playing a game with uncertainty, you might observe something in the next step that changes your belief, or distribution over world state, b = P(s). If you have some expected distribution of next observations then you can talk about the probability of having some belief in the next state. Roughly P(b'|b) = P(P(s')|P(s)). You can collapse this into simple probabilities if you just care about your new observations, P(obs'|b) = \sum_s P(obs'|s)P(s|b). I could imagine some frameworks where confidence intervals in this way would be useful - ex. I have 3 fairly different world models that I think are equally likely, each has a P(election), what's the P(election|world) and get confidence intervals across world models rather than just summing over them to P(election). But I agree that for most common approaches that simple probabilities are most useful and clear.

OscarCunningham 8y ago
Something similar certainly is useful in the case of flipping a coin. Suppose we have two coins, one of which we know is fair and one of which we know is biased but we don't know which way. Then our knowledge about the bias of the first coin would be described by a distribution with a sharp peak around 0.5, but the distribution for the second coin would have two lumps either side of 0.5. Then the probability of heads on the next flip would be 0.5 for either coin. But if someone asked us to bet on the probability of heads on the next two flips, then the second coin would have a higher probability than the first. The different distributions for the two coins don't make a difference for one flip, but do for multiple flips. But there's no meaningful way to repeat an election, so I don't think similar distributions or confidence intervals are useful in that case. EDIT: We can imagine rerolling an election, but since we can't actually do it we don't have to bet on it so the information wouldn't be very useful to us.

OscarCunningham 8y ago

Something similar certainly is useful in the case of flipping a coin. Suppose we have two coins, one of which we know is fair and one of which we know is biased but we don't know which way. Then our knowledge about the bias of the first coin would be described by a distribution with a sharp peak around 0.5, but the distribution for the second coin would have two lumps either side of 0.5. Then the probability of heads on the next flip would be 0.5 for either coin. But if someone asked us to bet on the probability of heads on the next two flips, then the second coin would have a higher probability than the first. The different distributions for the two coins don't make a difference for one flip, but do for multiple flips. But there's no meaningful way to repeat an election, so I don't think similar distributions or confidence intervals are useful in that case. EDIT: We can imagine rerolling an election, but since we can't actually do it we don't have to bet on it so the information wouldn't be very useful to us.