If you think there is a 50% chance that your credences will say go from 10% to 30%+. Then you believe that with a 50% probability, you live in a “30%+ world.” But then you live in at least a 50% * 30%+ = 15%+ world rather than a 10% world, as you originally thought.
“Good forecasts should be a martingale” is another (more general) way to say the same thing, in case the alternative phrasing is helpful for other people.
I imagine a proof (by contradiction) would work something like this:
Suppose you place > 1/x probability on your credences moving by a factor of x. Then the expectation of your future beliefs is > prior * x* 1/x = prior, so your credence will increase. With our remaining probability mass, can we anticipate some evidence in the other direction, such that our beliefs still satisfy conservation of expected evidence? The lowest our credence can go is 0, but even if we place our remaining < 1 − 1/x probability on 0, we would still find future beliefs > prior * x * 1/x + 0 * [remaining probability] = prior. So we would necessarily violate conservation of expected evidence, and we conclude that Joe’s rule holds.
Note that all of these comments apply, symmetrically, to people nearly certain of doom. 99.99%? OK, so less than 1% than you ever drop to 99% or lower?
But I don’t think this proof works for beliefs decreasing (because we don’t have the lower bound of 0). Consider this counterexample:
prior = 10%
probability of decreasing to 5% (factor of 2) = 60% > 1⁄2 —> violates the rule
So conservation of expected evidence doesn’t seem to imply Joe’s rule in this direction? (Maybe it holds once you introduce some restrictions on your prior, like in his 99.99% example, where you can’t place the remaining probability mass any higher than 1, so the rule still bites.)
This asymmetry seems weird?? Would love for someone to clear this up.
If you think there is a 50% chance that your credences will say go from 10% to 30%+. Then you believe that with a 50% probability, you live in a “30%+ world.” But then you live in at least a 50% * 30%+ = 15%+ world rather than a 10% world, as you originally thought.
“Good forecasts should be a martingale” is another (more general) way to say the same thing, in case the alternative phrasing is helpful for other people.
I imagine a proof (by contradiction) would work something like this:
Suppose you place > 1/x probability on your credences moving by a factor of x. Then the expectation of your future beliefs is > prior * x * 1/x = prior, so your credence will increase. With our remaining probability mass, can we anticipate some evidence in the other direction, such that our beliefs still satisfy conservation of expected evidence? The lowest our credence can go is 0, but even if we place our remaining < 1 − 1/x probability on 0, we would still find future beliefs > prior * x * 1/x + 0 * [remaining probability] = prior. So we would necessarily violate conservation of expected evidence, and we conclude that Joe’s rule holds.
But I don’t think this proof works for beliefs decreasing (because we don’t have the lower bound of 0). Consider this counterexample:
prior = 10%
probability of decreasing to 5% (factor of 2) = 60% > 1⁄2 —> violates the rule
probability of increasing to 17.5% = 40%
Then, expectation of future beliefs = 5% * 60% + 17.5% * 40% = 10%
So conservation of expected evidence doesn’t seem to imply Joe’s rule in this direction? (Maybe it holds once you introduce some restrictions on your prior, like in his 99.99% example, where you can’t place the remaining probability mass any higher than 1, so the rule still bites.)
This asymmetry seems weird?? Would love for someone to clear this up.