Comment author:Epictetus
23 December 2014 05:20:04AM
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If I roll a die, then one of the events that can happen will happen. That's just saying that if S is my sample space, then P(S) = 1. Similarly, P(~S) = 0, which is just saying that impossible things won't happen. The former statement is an axiom in the standard mathematical treatments of the subject. These statements may be trivial, but I distrust any mathematics that can't handle trivial cases.

Rejecting 1 as a probability would be catastrophic when you're dealing with discrete spaces. If you're the sort to reject infinity, then it would follow that all probability spaces are discrete. At that point probability loses its rigor. Preference for odds or log odds just means that you have to live with using the extended reals with special conventions for the infinities.

Comment author:ike
23 December 2014 05:38:11AM
1 point
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You can reject infinity without being able to enumerate every possibility. Your sample space will never practically contain all the possibilities. (How many times has something you never thought of happened?) There are 2^(however many bits of input come into my brain) possibilities for me to observe for any period of time, and I can never think about all of them. Any explicit sample space is going to miss possibilities. S is not well-defined.

I think the point of the post was that 1 shouldn't be used for practical cases.

Comment author:Epictetus
23 December 2014 11:37:45AM
0 points
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Real life is complex enough that there is merit to the philosophical position that one should refrain from assigning probabilities of 0 or 1 to nontrivial events. Categorically denying that any event can have probability 0 or 1 is an extreme position (which, applied to itself, would really mean that a given event would have a high probability of not occurring with probability 0 or 1).

From the purely mathematical standpoint, removing 0 and 1 from the set of possible probabilities breaks the current foundations of the theory. The existence of a sample space containing all possibilities does not depend on whether we humans can comprehend them all. If the sample space of all possibilities exists and P(S) < 1, then a lot of theorems break down. That's where you live with idealizations like absolute certainty (or almost certainty in the infinite case) or else find something other than probability to use to model the real world.

Comment author:ike
23 December 2014 02:46:27PM
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0 points
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In theory, if you could list every possible observation you could make, that will have a 1 probability. It would take infinite time, because the following class of outcomes:

my brain bandwidth is increased to X bits, and X random bits are my next input

has an infinite cardinality. I could get into how Godel means you can't even in principle describe all possible outcomes in a finite amount of space, even by referencing classes like I did, but I'll leave that up to you.

There was a suggested fix to your problem in the post, why isn't that good enough for you?

If you made a magical symbol to stand for "all possibilities I haven't considered", then you could marginalize over the events including this magical symbol, and arrive at a magical symbol "T" that stands for infinite certainty.

Sounds like he agrees that S has probability 1.

Note: I agree that the way he "proves" the claim is not very good. He basically tries to switch your intuition by switching the wording of the question. Not too rigorous.

Comment author:Epictetus
23 December 2014 06:40:53PM
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When I say that the possibilities can be listed in principle, what I mean is that there some set S that contains them and make no reference to any practical problems with describing or storing its elements. Like the points and lines of geometry, it's a Platonic idealization.

There was a suggested fix to your problem in the post, why isn't that good enough for you?

Because talk of magical symbols is a good sign that the passage was meant to ridicule the use of infinity. The very next paragraph seeks to expunge such "magical symbols" from probability theory.

Comment author:ike
23 December 2014 06:48:04PM
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If he has a rigorous way to ground probability theory without 0 and 1, I'm fine with it. He seemed to be saying that he wishes there was such a way, but until someone develops one, he's stuck with magical symbols. He acknowledges all your problems in the end of the post.

## Comments (128)

OldIf I roll a die, then one of the events that can happen will happen. That's just saying that if S is my sample space, then P(S) = 1. Similarly, P(~S) = 0, which is just saying that impossible things won't happen. The former statement is an axiom in the standard mathematical treatments of the subject. These statements may be trivial, but I distrust any mathematics that can't handle trivial cases.

Rejecting 1 as a probability would be catastrophic when you're dealing with discrete spaces. If you're the sort to reject infinity, then it would follow that all probability spaces are discrete. At that point probability loses its rigor. Preference for odds or log odds just means that you have to live with using the extended reals with special conventions for the infinities.

You can reject infinity without being able to enumerate every possibility. Your sample space will never practically contain all the possibilities. (How many times has something you never thought of happened?) There are 2^(however many bits of input come into my brain) possibilities for me to observe for any period of time, and I can never think about all of them. Any explicit sample space is going to miss possibilities. S is not well-defined.

I think the point of the post was that 1 shouldn't be used for practical cases.

Real life is complex enough that there is merit to the philosophical position that one should refrain from assigning probabilities of 0 or 1 to nontrivial events. Categorically denying that any event can have probability 0 or 1 is an extreme position (which, applied to itself, would really mean that a given event would have a high probability of not occurring with probability 0 or 1).

From the purely mathematical standpoint, removing 0 and 1 from the set of possible probabilities breaks the current foundations of the theory. The existence of a sample space containing all possibilities does not depend on whether we humans can comprehend them all. If the sample space of all possibilities exists and P(S) < 1, then a lot of theorems break down. That's where you live with idealizations like absolute certainty (or almost certainty in the infinite case) or else find something other than probability to use to model the real world.

*0 points [-]In theory, if you could list every possible observation you could make, that will have a 1 probability. It would take infinite time, because the following class of outcomes:

has an infinite cardinality. I could get into how Godel means you can't even in principle describe all possible outcomes in a finite amount of space, even by referencing classes like I did, but I'll leave that up to you.

There was a suggested fix to your problem in the post, why isn't that good enough for you?

Sounds like he agrees that S has probability 1.

Note: I agree that the way he "proves" the claim is not very good. He basically tries to switch your intuition by switching the wording of the question. Not too rigorous.

When I say that the possibilities can be listed in principle, what I mean is that there some set S that contains them and make no reference to any practical problems with describing or storing its elements. Like the points and lines of geometry, it's a Platonic idealization.

Because talk of magical symbols is a good sign that the passage was meant to ridicule the use of infinity. The very next paragraph seeks to expunge such "magical symbols" from probability theory.

*0 points [-]If he has a rigorous way to ground probability theory without 0 and 1, I'm fine with it. He seemed to be saying that he wishes there was such a way, but until someone develops one, he's stuck with magical symbols. He acknowledges all your problems in the end of the post.