I know this is an old thread, and I'm not totally sure how this affects the debate here, but for what it's worth I think applying principle of indifference-type reasoning here implies that the appropriate uninformative prior is an exponential distribution.
I apply the principle of indifference (or maybe of invariance, following Jaynes (1968)) as follows: If I wake up tomorrow knowing absolutely nothing about the world and am asked about the probability of 10 days into the future containing the most important time in history conditional on it being in the future, I should give the same answer as if I were to be woken up 100 years from now and were asked about the day 100 years and 10 days from now. I would need some further information (e.g. about the state of the world, of human society, etc.) to say why one would be more probable than the other, and here I'm looking for a prior from a state of total ignorance.
This invariance can be generalized as: Pr(X>t+k|X>t) = Pr(X>t'+k|X>t') for all k, t, t'. This happens to be the memoryless property, and the exponential distribution is the only continuous distribution that has this property. Thus if we think that our priors from a state of total ignorance should satisfy this requirement, our prior needs to be an exponential distribution. I imagine there are other ways of characterizing similar indifference requirements that imply memorylessness.
This is not to say our current beliefs should follow this distribution: we have additional information about the world, and we should update on this information. It’s also possible that the principle of indifference might be applied in a different way to give a different uninformative prior as in the Bertrand paradox.
(The Jaynes paper: https://bayes.wustl.edu/etj/articles/prior.pdf)
I know this is an old thread, and I'm not totally sure how this affects the debate here, but for what it's worth I think applying principle of indifference-type reasoning here implies that the appropriate uninformative prior is an exponential distribution.
I apply the principle of indifference (or maybe of invariance, following Jaynes (1968)) as follows: If I wake up tomorrow knowing absolutely nothing about the world and am asked about the probability of 10 days into the future containing the most important time in history conditional on it being in the future, I should give the same answer as if I were to be woken up 100 years from now and were asked about the day 100 years and 10 days from now. I would need some further information (e.g. about the state of the world, of human society, etc.) to say why one would be more probable than the other, and here I'm looking for a prior from a state of total ignorance.
This invariance can be generalized as: Pr(X>t+k|X>t) = Pr(X>t'+k|X>t') for all k, t, t'. This happens to be the memoryless property, and the exponential distribution is the only continuous distribution that has this property. Thus if we think that our priors from a state of total ignorance should satisfy this requirement, our prior needs to be an exponential distribution. I imagine there are other ways of characterizing similar indifference requirements that imply memorylessness.
This is not to say our current beliefs should follow this distribution: we have additional information about the world, and we should update on this information. It’s also possible that the principle of indifference might be applied in a different way to give a different uninformative prior as in the Bertrand paradox.
(The Jaynes paper: https://bayes.wustl.edu/etj/articles/prior.pdf)