Fundamental or not I think my point still stands that “the prior is infinite so the whole thing’s wrong” isn’t quite enough of an argument, since you still seem to conclude that improper priors can be used if used carefully enough. A more satisfying argument would be to demonstrate that the 9⁄10 case can’t be made without incorrect use of an improper prior. Though I guess it’s still showing where the problem most likely is which is helpful.
As far as being part of the foundations goes, I was just going by the fact that it’s in Jaynes, but you clearly know a lot more about this topic than I do. I would be interested to know your answer to the following questions though: “Can a state of ignorance be described without the use of improper priors (or something mathematically equivalent)?”, and “Can Bayesian probability be used as the foundation of rational thought without describing states of ignorance?”.
On the Doomsday argument, I would only take the Dice Room as a metaphor not a proof of anything, but it does help me realise a couple of things. One is that the setup you describe of a potentially endlessly exponentially growing population is not a reasonable model of reality (irrespective of the parameters themselves). The growth has to stop, or at least converge, at some point, even without a catastrophe.
It’s interesting that the answer changes if he rolls the dice first. I think ultimately the different answers to the Dice Room correspond to different ways of handling the infinite population correctly—i.e. taking limits of finite populations. For any finite population there needs to be an answer to “what does he do if he doesn’t roll snake-eyes in time?” and different choices, for all that you might expect them to disappear in the limit, lead to different answers.
If the dice having already being rolled is the best analogy for the Doomsday argument then it’s making quite particular statements about causality and free will.
Thanks, interesting reading.
Fundamental or not I think my point still stands that “the prior is infinite so the whole thing’s wrong” isn’t quite enough of an argument, since you still seem to conclude that improper priors can be used if used carefully enough. A more satisfying argument would be to demonstrate that the 9⁄10 case can’t be made without incorrect use of an improper prior. Though I guess it’s still showing where the problem most likely is which is helpful.
As far as being part of the foundations goes, I was just going by the fact that it’s in Jaynes, but you clearly know a lot more about this topic than I do. I would be interested to know your answer to the following questions though: “Can a state of ignorance be described without the use of improper priors (or something mathematically equivalent)?”, and “Can Bayesian probability be used as the foundation of rational thought without describing states of ignorance?”.
On the Doomsday argument, I would only take the Dice Room as a metaphor not a proof of anything, but it does help me realise a couple of things. One is that the setup you describe of a potentially endlessly exponentially growing population is not a reasonable model of reality (irrespective of the parameters themselves). The growth has to stop, or at least converge, at some point, even without a catastrophe.
It’s interesting that the answer changes if he rolls the dice first. I think ultimately the different answers to the Dice Room correspond to different ways of handling the infinite population correctly—i.e. taking limits of finite populations. For any finite population there needs to be an answer to “what does he do if he doesn’t roll snake-eyes in time?” and different choices, for all that you might expect them to disappear in the limit, lead to different answers.
If the dice having already being rolled is the best analogy for the Doomsday argument then it’s making quite particular statements about causality and free will.