Ok yes, that’s the definition of a probability measure. But I was talking about the concept of probability, in the world, contrasting with the “objectively defined via frequency in related trials”, which is something people sometimes claim. I misunderstood and thought that was the claim you were making.
Ok.
I would think that, if we have a continuous distribution, then the score should be the probability density of what is observed?
If you say beforehand “I think x will happen”, and I respond “I assign probability 1 that x will not happen”, and then x happens, then I’ve really messed up big time. I’ve messed up to a degree that should never happen.
(And, only countably many events can be described using finite descriptions, and a positive probability could, in principle, be assigned to each, while having the total probability still be 1, so that nothing that can possibly be specified happens while being assigned a probability of 0. Though this isn’t really computable..)
As a more practical thing, if I assign probability 0 to an event which you could describe in a few sentences in under 5 lines (regardless of whether you actually have described it), and it happens, then I’ve really messed up quite terribly, and this should never happen (outside of just, because I made an arithmetic error or something.)
> As a more practical thing, if I assign probability 0 to an event which you could describe in a few sentences in under 5 lines (regardless of whether you actually have described it), and it happens, then I’ve really messed up quite terribly, and this should never happen (outside of just, because I made an arithmetic error or something.)
I'm familiar with the cantor set, and I know it has 0 measure.
Just because you can succinctly describe the cantor set, which has 0 measure, doesn't mean I've messed up.
If I assign a uniform distribution over [0,1] to some number outcome in the world, and an element in the cantor set is the result, then I've messed up.
But, when we measure numbers in the world, we don't measure specific real numbers, as all our measurements have some amount of error.
So, that can't happen.
We can measure that the result is in some interval, and that this interval contains some element of the cantor set, but the probability of what we observed, is not something that I assigned 0 probability to.
Like, heck, every interval will have a rational number in it, and the rational numbers also have measure 0.
"the measured value is in the cantor set" isn't a thing that we can observe to have happened.
("the value, when rounded to the finite amount of precision that our measurement has, is in the cantor set" is something that would have positive probability, under the uniform distribution over the interval, and therefore something I shouldn't assign a probability of 0.)
Ok.
I would think that, if we have a continuous distribution, then the score should be the probability density of what is observed?
If you say beforehand “I think x will happen”, and I respond “I assign probability 1 that x will not happen”, and then x happens, then I’ve really messed up big time. I’ve messed up to a degree that should never happen.
(And, only countably many events can be described using finite descriptions, and a positive probability could, in principle, be assigned to each, while having the total probability still be 1, so that nothing that can possibly be specified happens while being assigned a probability of 0. Though this isn’t really computable..)
As a more practical thing, if I assign probability 0 to an event which you could describe in a few sentences in under 5 lines (regardless of whether you actually have described it), and it happens, then I’ve really messed up quite terribly, and this should never happen (outside of just, because I made an arithmetic error or something.)