but if you use the uniform distribution for x, then you’re not using a uniform distribution for x^2
Well, x^2 isn’t an isometry, so you shouldn’t expect it to leave the prior unchanged.
Let me put it this way: if Omega told you ve had a real number x between 0 and 1, and then ve told you that x^1000 was between 3⁄10 and 4⁄10, you probably should be more surprised than if ve told you that x^1000 was between 0 and 1⁄10. Yes, you could pick your prior to have a uniform distribution for x^1000 rather than for x, but that doesn’t seem a natural choice in general.
I have also been wondering about when it’s appropriate to use a uniform prior.
As an example (I think) of an instance where a uniform prior is not appropriate: NBA stats people calculate points per minute played for all the players. In some cases, bench players have higher points per minutes played than the starters. However, it does not follow that the bench player should be starting (ignoring defensive stats).
This is because bench players tend to enter the game at a time when they will play against the opposing team’s bench. So, presuming that the defensive skills of the opponent’s bench are less than the defensive skills of the opponent’s starters, it’s clear that there is a non-uniform level of defense maintained by the opponent during the game—i.e., bench players should have an easier time scoring than starters. So there is not a simple apples to apples comparison & conclusion based on this stat.
Well, x^2 isn’t an isometry, so you shouldn’t expect it to leave the prior unchanged.
Let me put it this way: if Omega told you ve had a real number x between 0 and 1, and then ve told you that x^1000 was between 3⁄10 and 4⁄10, you probably should be more surprised than if ve told you that x^1000 was between 0 and 1⁄10. Yes, you could pick your prior to have a uniform distribution for x^1000 rather than for x, but that doesn’t seem a natural choice in general.
I have also been wondering about when it’s appropriate to use a uniform prior.
As an example (I think) of an instance where a uniform prior is not appropriate: NBA stats people calculate points per minute played for all the players. In some cases, bench players have higher points per minutes played than the starters. However, it does not follow that the bench player should be starting (ignoring defensive stats).
This is because bench players tend to enter the game at a time when they will play against the opposing team’s bench. So, presuming that the defensive skills of the opponent’s bench are less than the defensive skills of the opponent’s starters, it’s clear that there is a non-uniform level of defense maintained by the opponent during the game—i.e., bench players should have an easier time scoring than starters. So there is not a simple apples to apples comparison & conclusion based on this stat.