There are two ways to look at this at a higher level. The first is that the algebra doesn’t really apply in the first place, because this is a domain error: 0 and 1 aren’t probabilities, in the same way that the string “hello” and the color blue aren’t.
The second way to look at it is that when we say P(H)=1.0 and P(E|H)=0, what we really meant was that P(H)=1.0−ϵ1 and P(E|H)=0+ϵ2; that is, they aren’t precisely one and zero, but they differ from one and zero by an unspecified, very small amount. (Infinitesimals are like infinities; ϵ is arbitrarily-close-to-zero in the same sense that an infinity is arbitrarily-large). Under this interpretation, we don’t have a contradiction, but we do have an underspecified problem, since we need the ratio ϵ1ϵ2 and haven’t specified it.
Thanks for the answer! i was somewhat amused to see that it ends up being a zero divided by zero.
Does the ratio between 1epsilon over 2epsilon being undefined means that it’s arbitrarily close to half (since 1 over two is half, but that wouldn’t be exactly it)? or means that we get the same problem i specified in the question, where it could be anything from (almost) 0 to (almost) 1 and we have no idea what exactly?
The latter; it could be anything, and by saying the probabilities were 1.0 and 0.0, the original problem description left out the information that would determine it.
and also, when you use epsilons, does it mean you get out of the “dogma” of 100%? or you still can’t update down from it?
And what i did in my post may just be another example of why you don’t put an actual 1.0 in your prior, cause then even if you get evidence of the same strength in the other direction, that would demand that you divide zero by zero. right?
Using epsilons can in principle allow you to update. However, the situation seems slightly worse than jimrandomh describes. It looks like you need P(E|h), or the probability if H is false, in order to get a precise answer. Also, the missing info that jim mentioned is already enough in principle to let the final answer be any probability whatsoever.
If we use log odds (the framework in which we could literally start with “infinite certainty”) then the answer could be anywhere on the real number line. We have infinite (or at least unbounded) confusion until we make our assumptions more precise.
If you do out the algebra, you get that P(H|E) involves dividing zero by zero:
There are two ways to look at this at a higher level. The first is that the algebra doesn’t really apply in the first place, because this is a domain error: 0 and 1 aren’t probabilities, in the same way that the string “hello” and the color blue aren’t.
The second way to look at it is that when we say P(H)=1.0 and P(E|H)=0, what we really meant was that P(H)=1.0−ϵ1 and P(E|H)=0+ϵ2; that is, they aren’t precisely one and zero, but they differ from one and zero by an unspecified, very small amount. (Infinitesimals are like infinities; ϵ is arbitrarily-close-to-zero in the same sense that an infinity is arbitrarily-large). Under this interpretation, we don’t have a contradiction, but we do have an underspecified problem, since we need the ratio ϵ1ϵ2 and haven’t specified it.
Thanks for the answer! i was somewhat amused to see that it ends up being a zero divided by zero.
Does the ratio between 1epsilon over 2epsilon being undefined means that it’s arbitrarily close to half (since 1 over two is half, but that wouldn’t be exactly it)? or means that we get the same problem i specified in the question, where it could be anything from (almost) 0 to (almost) 1 and we have no idea what exactly?
The latter; it could be anything, and by saying the probabilities were 1.0 and 0.0, the original problem description left out the information that would determine it.
I see. so -
If P(H) = 1.0 - ϵ1
And P(E|H) = 0 + ϵ2
Then it equals “infinite confusion”.
Am i correct?
and also, when you use epsilons, does it mean you get out of the “dogma” of 100%? or you still can’t update down from it?
And what i did in my post may just be another example of why you don’t put an actual 1.0 in your prior, cause then even if you get evidence of the same strength in the other direction, that would demand that you divide zero by zero. right?
Using epsilons can in principle allow you to update. However, the situation seems slightly worse than jimrandomh describes. It looks like you need P(E|h), or the probability if H is false, in order to get a precise answer. Also, the missing info that jim mentioned is already enough in principle to let the final answer be any probability whatsoever.
If we use log odds (the framework in which we could literally start with “infinite certainty”) then the answer could be anywhere on the real number line. We have infinite (or at least unbounded) confusion until we make our assumptions more precise.