Dutch books also seems to be a popular way of showing what works and what doesn’t, so here’s a simple Dutch argument against improper priors: I have two real numbers: x and y. Suppose they have a uniform distribution. I offer you a bet at 2:1 odds that x has a higher magnitude. They’re equally likely to be higher, so you take it. I then show you the value of x. I offer you a new bet at 2:1 odds that y has a higher magnitude. You know y almost definitely has a higher magnitude than that, so you take it again. No matter what happens, I win.
To fix this example, replace “real” with “positive real” and make the bets 2:4 and 100:1.
Still, an example that comes from using improper priors as probability distributions, which they are explicitly not, doesn’t seem like a strong argument. Better to show that they can’t come up in any interesting situations—this may be impossible, though.
Why did you say 2:4 instead of 1:2? Do you mean 2:1?
Just to emphasize that the victim should have more money riding on the first bet if they are to consistently lose money.
If they’re not probability distributions, what are they?
Since they’re invalid probability distributions but can be updated into a probability distribution given some evidence, you might think of these as representing states where you have some knowledge, but not enough to assign consistent probabilities. For example, if all you know is that X is a member of some infinite set, you cannot assign consistent probabilities, but you still have some knowledge, which might be represented as a uniform function.
To fix this example, replace “real” with “positive real” and make the bets 2:4 and 100:1.
Still, an example that comes from using improper priors as probability distributions, which they are explicitly not, doesn’t seem like a strong argument. Better to show that they can’t come up in any interesting situations—this may be impossible, though.
I used “real” because with positive reals, you’re more likely to use a logarithmic prior.
Oops. Why did you say 2:4 instead of 1:2? Do you mean 2:1?
If they’re not probability distributions, what are they?
Just to emphasize that the victim should have more money riding on the first bet if they are to consistently lose money.
Since they’re invalid probability distributions but can be updated into a probability distribution given some evidence, you might think of these as representing states where you have some knowledge, but not enough to assign consistent probabilities. For example, if all you know is that X is a member of some infinite set, you cannot assign consistent probabilities, but you still have some knowledge, which might be represented as a uniform function.