Other failure modes could be to fail to have properties of probabilty distributions. Negative numbers, imaginary amounts? Not an unknown probability distribuiton because its not a probabilty distribution[...]
Not every probability distribution has to result in real numbers. A distribution that gets me complex numbers or letters from the set { A, B, C } is still a distribution. And while some things may be vastly easier to describe when using quasiprobability distributions (involving “negative probabilities”), that is a choice of the specific approach to modeling, not a necessity. You still get observations for which you can form at least an empirical distribution. A full model using only “real probabilities” may be much more complex, but it’s not fundamentally impossible.
If your random pull is aleph-null does it have any sensible distributions to be drawn from? Dealing with other “exotic” numbers systematically might get tricksy.
Use a mixture distribution, combining e.g. a discrete part (specific huge numbers outside of the reals) with a continuous part (describing observable real numbers).
Questions involving infidesimals like “Given a uniformly random number between 0 and 1 is it twice more likely it to be in the interval from epsilon to 3*epsilon than in the interval from 3*epsilon to 4*epsilon”[...]
That doesn’t typecheck. Epsilon isn’t a number, it’s a symbol for a limit process (“for epsilon approaching zero, …”) If you actually treat it as the limit process, then that question makes sense and you can just compute the answer. If you treat it as an infinitesimal number and throw it into the reals without extending the rest of your theoretical building blocks, then of course you can’t get meaningful answers.
I mean negative or imaginary probablities. Quasipropability distributions fail to be probability distributions. If I have a “random apple” and somebody ask what proprtion of it might be “pear” then that will be 0 as pears are not apples. If I meant to ask “random fruit” then pears would be relevant. While you get some analysis and there is hope to get to a probability distribution analysis, you would need a entirely depenable way to produce the reformulation. Just because some vechicles are amphibous doesn’t mean you can take a boat and drive it on land (because some boats are also cars).
{0,1,2,3,4,5,6...|}=omega is an exact surreal number and 1/omega = epsilon is an exact surreal number. Yes, the base approach is to make your terms clear and if there remain ambiguity in the core part of the question it is going to critically confuse you. I didn’t provide enough clues to glue in what I was talking about. It is kind of telling that the “default frame” will push into all spaces not specifically specified to be against it even if it is pushing square peg throught a round hole.
One could easily think that which such a “easy” construction “uniform between 0 and 1″ seems like easy to understand. I am trying to highlight a situation where the thign is so basic it seems it would be reasonable to trancend particular formalizations. “getting a propability” and “throwing into the reals” can be slightly different operations when you would need to throw it into others than reals to make your calculation work.
Here specifically you can dance it around if you cast small against small into real numbers or bigs against bigs into real numbers. But when you would need to respect the things compared to belong to different archimedean fields things break down. For casting into a single archimedean field everything that fails to be a finite length line will get rounded to nearest real precision of 0 and then all zeroes are equal failing to distinguish single points from infinitely short lines.
Not every probability distribution has to result in real numbers. A distribution that gets me complex numbers or letters from the set { A, B, C } is still a distribution. And while some things may be vastly easier to describe when using quasiprobability distributions (involving “negative probabilities”), that is a choice of the specific approach to modeling, not a necessity. You still get observations for which you can form at least an empirical distribution. A full model using only “real probabilities” may be much more complex, but it’s not fundamentally impossible.
Use a mixture distribution, combining e.g. a discrete part (specific huge numbers outside of the reals) with a continuous part (describing observable real numbers).
That doesn’t typecheck. Epsilon isn’t a number, it’s a symbol for a limit process (“for epsilon approaching zero, …”) If you actually treat it as the limit process, then that question makes sense and you can just compute the answer. If you treat it as an infinitesimal number and throw it into the reals without extending the rest of your theoretical building blocks, then of course you can’t get meaningful answers.
I mean negative or imaginary probablities. Quasipropability distributions fail to be probability distributions. If I have a “random apple” and somebody ask what proprtion of it might be “pear” then that will be 0 as pears are not apples. If I meant to ask “random fruit” then pears would be relevant. While you get some analysis and there is hope to get to a probability distribution analysis, you would need a entirely depenable way to produce the reformulation. Just because some vechicles are amphibous doesn’t mean you can take a boat and drive it on land (because some boats are also cars).
{0,1,2,3,4,5,6...|}=omega is an exact surreal number and 1/omega = epsilon is an exact surreal number. Yes, the base approach is to make your terms clear and if there remain ambiguity in the core part of the question it is going to critically confuse you. I didn’t provide enough clues to glue in what I was talking about. It is kind of telling that the “default frame” will push into all spaces not specifically specified to be against it even if it is pushing square peg throught a round hole.
One could easily think that which such a “easy” construction “uniform between 0 and 1″ seems like easy to understand. I am trying to highlight a situation where the thign is so basic it seems it would be reasonable to trancend particular formalizations. “getting a propability” and “throwing into the reals” can be slightly different operations when you would need to throw it into others than reals to make your calculation work.
Here specifically you can dance it around if you cast small against small into real numbers or bigs against bigs into real numbers. But when you would need to respect the things compared to belong to different archimedean fields things break down. For casting into a single archimedean field everything that fails to be a finite length line will get rounded to nearest real precision of 0 and then all zeroes are equal failing to distinguish single points from infinitely short lines.