Right, that’s essentially what I mean. You’re of course right that this doesn’t let you get around the existence of nonmeasurepreserving automorphisms. I guess what I’m saying is that, if you’re trying to find a prior on [0,1], you should try to think about what system of finite measurements this is idealizing, and see if you can apply a symmetry argument to those bits. Which isn’t always the case! You can only apply the principle of indifference if you’re actually indifferent. But if it’s the case that X∈[0,1] is generated in a way where “there’s no reason to suspect that any bit should be 0 rather than 1, or that there should be correlations between the bits”, then it’s of course not the case that √X has this same property. But of course you still need to look at the actual situation to see what symmetry exists or doesn’t exist.
Haar measures are a high-powered way of doing this, I was just thinking about taking the iterated product measure of the uniform probability measure on {0,1} (justified by symmetry considerations). You can of course find maps from {0,1}∞ to all sorts of spaces but it seems harder to transport symmetry considerations along these maps.
Right, that’s essentially what I mean. You’re of course right that this doesn’t let you get around the existence of nonmeasurepreserving automorphisms. I guess what I’m saying is that, if you’re trying to find a prior on [0,1], you should try to think about what system of finite measurements this is idealizing, and see if you can apply a symmetry argument to those bits. Which isn’t always the case! You can only apply the principle of indifference if you’re actually indifferent. But if it’s the case that X∈[0,1] is generated in a way where “there’s no reason to suspect that any bit should be 0 rather than 1, or that there should be correlations between the bits”, then it’s of course not the case that √X has this same property. But of course you still need to look at the actual situation to see what symmetry exists or doesn’t exist.
Haar measures are a high-powered way of doing this, I was just thinking about taking the iterated product measure of the uniform probability measure on {0,1} (justified by symmetry considerations). You can of course find maps from {0,1}∞ to all sorts of spaces but it seems harder to transport symmetry considerations along these maps.