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You're simply wrong. What I said was correct. > if there is some finite set and there is a non-zero probability of something existing in it then there must be

by feoren 2y ago

You're simply wrong. What I said was correct. > if there is some finite set and there is a non-zero probability of something existing in it then there must be at least one _possible_ example of how this can occur That is not what GP claimed nor does it really make sense. Yes, if there is a non-zero probability of something existing, then it must be possible. That is what non-zero probability means. That has nothing to do with sets. You've just added the word "possible" to make the claim vague enough to be not provably incorrect. If you have a finite set, and you show that members of that set have a non-zero (but less than 1) probability of having some property, you do not know whether a member of the set actually has that property! Period! Why is this a point of contention? > a frequentist approach would give the probability of something that occured 50 times out of \inf as 0 while other approaches may say a small non-0 probability. No. Nobody would say that 50/infinity is greater than 0. You haven't actually responded to the thing I actually said, nor really to GP's claim. The claim was: >> if something exists with non-zero probability in a finite set there has to be at least one example That is false. I gave an example of how it's false.

moralestapia 2y ago
Agree, @zamadatix seems a bit out if his area of expertise. On finite sets, all the stuff they're saying doesn't make sense completely.
- bubblyworld 2y ago
  I'm an ex-PhD in mathematics. We're talking about two entirely separate things - finite numbers of samples from a probability distribution (what you are talking about) vs the measure of a subset of some finite number of objects (the probabilistic method).
- zamadatix 2y ago
  Just trying to prevent my "false assumption" confirmation bias from growing - did you come to this thread as a result of https://news.ycombinator.com/item?id=41729586 https://news.ycombinator.com/item?id=41729586 or should I be attributing happenstance? Either is cool, no worries!
- deleted 2y ago
  [deleted]
zamadatix 2y ago
> That is not what GP claimed nor does it really make sense. Yes, if there is a non-zero probability of something existing, then it must be possible. That is what non-zero probability means. That has nothing to do with sets. You've just added the word "possible" to make the claim vague enough to be not provably incorrect. I'm not here to assert what GP meant with certainty (and neither should you?). Nor am I going to bother engaging further when you so blatantly end with how your troubles with the vagueness in your understanding of the what the claim meant lie in being unable to call it incorrect. > No. Nobody would say that 50/infinity is greater than 0. Nor did I claim that's how the other approaches come to calculate the probability greater than 0... again, what's the point in me engaging on this if your goal is to slap portions of text together until you find something to disagree with instead of actually engaging with me? A complaint about me not including how the alternative approaches come to a different conclusion would be more than a valid criticism (or even friendly probe, were you actually interested) but you're goal to make each response to anyone here start with "no", "incorrect", or "you're wrong" is apparently too driving to allow that kind of discussion. > You haven't actually responded to the thing I actually said, nor really to GP's claim. The claim was... That is false. I gave an example of how it's false. This, ironically, has got to be the most perfect way I've seen to skip actually responding to anything someone else has said and return to one's original assertions in one swoop. Especially combined with the response opening "You're simply wrong. What I said was correct"... All that aside, it'd be a disservice to myself to say I'm not going to engage as a "last word" situation so I promise to come back to read any response (if one is made) and see what takeaways you have for me. My only ask is it be more than just how you're right/GP is so wrong/I'm wrong - at least something else about how my initial interjection/the GP's original statement (as you understand it from the other side thread now) could have been better, a more rigorous definition (i.e. not just words we've all been saying) of what you've been arguing so I can say "yeah, I did really understand what he was trying to say was so" (even if I disagree that was the ultimate point being made), or the like. Or don't, it's ultimately a service for me - you won't necessarily get anything out of it. Thanks
- feoren 2y ago
  Look, all I can do is respond to the words as actually written. Someone made a claim and I countered it. Now you appear to be simultaneously complaining that I'm responding to exact quoted text (which I "slapped together" by directly quoting?) and also that I'm somehow also not responding to you because after my direct response to the words you wrote, I tried to re-focus the discussion on the original claim? I didn't skip actually responding to you, I did respond to you (50/infinity = 0); you just didn't like my response. My takeaway for you is this: we're talking about mathematical proof techniques, and what you can conclude from the four different situations below. The soundness of a mathematical proof is independent of whether you're a frequentist or not, which is why I brushed off that comment. The particular measure you choose doesn't really matter either. It might matter whether you accept the axiom of choice or not, but I'm the only one who's brought that up so far. It's important because mathematical proofs are very finicky on exact details, and bubblyworld's statement is exactly the kind of pernicious little detail that can trick you into thinking a proof is correct when it's not. That matters! The main free variable here is exactly what "probability" means in terms of how you're "picking" possible objects. Yes, there are some cases where showing that an object exists with "positive probability" (and almost certainly a subset having "positive measure") means that one example must actually exist, but if that's the case, the jump from "positive measure" to "must exist" is actually tautological! That is, your proof doesn't need that step; the act of showing "positive measure" has already proven the existence. The very fact that a mathematical pillar like Erdős was relying on such a step means that it was necessary for his proof. His proofs didn't have random tautological steps in them. We are not talking about the proof technique of "if a subset of X has a thing in it, then X must have a thing in it" -- that's literally tautological. So what does "probability" really mean here? bubblyworld gave a link to examples of what he's talking about (see [1], Example 2), and seemed to misunderstand the proof. It's structured like this: "Let n be very large and consider a random graph G on n vertices ... we show that with positive probability, G satisfies two properties." Let me ask: how are we mathematically certain there exists a graph G that satisfies the two properties? (And we are; the proof is correct.) Which of the 4 situations are we in below? bubblyworld's insistence on finite sets makes it sound like we're in #3: finite set, nonzero probability. And indeed the set of random graphs with n vertices is finite! But that is an important mistake. We are not relying on argument #3 below. Yes, the set of random graphs with n vertices is finite, but we have NOT proven that any specific G has our two properties, and we have NOT proven that any graph with n vertices has those two properties! It's entirely possible that NO graphs with n vertices have the two properties; all we've shown is that every graph in a finite set has some positive probability to have our properties. You might encounter a similar proof, exactly the same up to this point, and that proof might be wrong! If you believe bubblyworld, you could make this mistake. So why does Erdős's proof work? It relies on two things that are both necessary: (1) we're not talking about a finite set at all! We're talking about random subgraphs of size n OR LARGER. We're actually picking from an infinite set here. Our smallest example might be of size 10^10^10^n! We don't know! We just know that if we keep going, eventually we must hit one. And, critically, (2) there is a positive lower bound on the probability, above some n (it's 1/2). That is, if the probability of a random subgraph having those two properties was some small nonzero probability that decreased as n increased, e.g. p ~ 1/n^2, then we still might never find an example! The proof could be incorrect! But it's shown that above some n, the probability is at least 1/2. Thus the whole proof works. Any discussion about frequentism or Lebesgue measure is missing the point: we're talking about whether proofs are correct or not, and that is a very finicky, tricky discussion. bubblyworld's original statement, and his continued refinements of that statement, are incorrect, and believing such things can lead you to accept incorrect proofs. I think that matters. So I corrected it. [1] https://en.wikipedia.org/wiki/Probabilistic_method https://en.wikipedia.org/wiki/Probabilistic_method 1. Infinite set, nonzero probability: we know at least one example must exist. 2. Infinite set, zero probability: not sure! It could be zero, or any subset of smaller cardinality or measure zero. 3. Finite set, nonzero probability: not sure! It could be zero! My coin flip example shows this. 4. Finite set, zero probability: we know no examples can exist.

by feoren 2y ago

moralestapia 2y ago

Agree, @zamadatix seems a bit out if his area of expertise. On finite sets, all the stuff they're saying doesn't make sense completely.

bubblyworld 2y ago
I'm an ex-PhD in mathematics. We're talking about two entirely separate things - finite numbers of samples from a probability distribution (what you are talking about) vs the measure of a subset of some finite number of objects (the probabilistic method).
zamadatix 2y ago
Just trying to prevent my "false assumption" confirmation bias from growing - did you come to this thread as a result of https://news.ycombinator.com/item?id=41729586 https://news.ycombinator.com/item?id=41729586 or should I be attributing happenstance? Either is cool, no worries!
deleted 2y ago
[deleted]

zamadatix 2y ago

> That is not what GP claimed nor does it really make sense. Yes, if there is a non-zero probability of something existing, then it must be possible. That is what non-zero probability means. That has nothing to do with sets. You've just added the word "possible" to make the claim vague enough to be not provably incorrect. I'm not here to assert what GP meant with certainty (and neither should you?). Nor am I going to bother engaging further when you so blatantly end with how your troubles with the vagueness in your understanding of the what the claim meant lie in being unable to call it incorrect. > No. Nobody would say that 50/infinity is greater than 0. Nor did I claim that's how the other approaches come to calculate the probability greater than 0... again, what's the point in me engaging on this if your goal is to slap portions of text together until you find something to disagree with instead of actually engaging with me? A complaint about me not including how the alternative approaches come to a different conclusion would be more than a valid criticism (or even friendly probe, were you actually interested) but you're goal to make each response to anyone here start with "no", "incorrect", or "you're wrong" is apparently too driving to allow that kind of discussion. > You haven't actually responded to the thing I actually said, nor really to GP's claim. The claim was... That is false. I gave an example of how it's false. This, ironically, has got to be the most perfect way I've seen to skip actually responding to anything someone else has said and return to one's original assertions in one swoop. Especially combined with the response opening "You're simply wrong. What I said was correct"... All that aside, it'd be a disservice to myself to say I'm not going to engage as a "last word" situation so I promise to come back to read any response (if one is made) and see what takeaways you have for me. My only ask is it be more than just how you're right/GP is so wrong/I'm wrong - at least something else about how my initial interjection/the GP's original statement (as you understand it from the other side thread now) could have been better, a more rigorous definition (i.e. not just words we've all been saying) of what you've been arguing so I can say "yeah, I did really understand what he was trying to say was so" (even if I disagree that was the ultimate point being made), or the like. Or don't, it's ultimately a service for me - you won't necessarily get anything out of it. Thanks

feoren 2y ago
Look, all I can do is respond to the words as actually written. Someone made a claim and I countered it. Now you appear to be simultaneously complaining that I'm responding to exact quoted text (which I "slapped together" by directly quoting?) and also that I'm somehow also not responding to you because after my direct response to the words you wrote, I tried to re-focus the discussion on the original claim? I didn't skip actually responding to you, I did respond to you (50/infinity = 0); you just didn't like my response. My takeaway for you is this: we're talking about mathematical proof techniques, and what you can conclude from the four different situations below. The soundness of a mathematical proof is independent of whether you're a frequentist or not, which is why I brushed off that comment. The particular measure you choose doesn't really matter either. It might matter whether you accept the axiom of choice or not, but I'm the only one who's brought that up so far. It's important because mathematical proofs are very finicky on exact details, and bubblyworld's statement is exactly the kind of pernicious little detail that can trick you into thinking a proof is correct when it's not. That matters! The main free variable here is exactly what "probability" means in terms of how you're "picking" possible objects. Yes, there are some cases where showing that an object exists with "positive probability" (and almost certainly a subset having "positive measure") means that one example must actually exist, but if that's the case, the jump from "positive measure" to "must exist" is actually tautological! That is, your proof doesn't need that step; the act of showing "positive measure" has already proven the existence. The very fact that a mathematical pillar like Erdős was relying on such a step means that it was necessary for his proof. His proofs didn't have random tautological steps in them. We are not talking about the proof technique of "if a subset of X has a thing in it, then X must have a thing in it" -- that's literally tautological. So what does "probability" really mean here? bubblyworld gave a link to examples of what he's talking about (see [1], Example 2), and seemed to misunderstand the proof. It's structured like this: "Let n be very large and consider a random graph G on n vertices ... we show that with positive probability, G satisfies two properties." Let me ask: how are we mathematically certain there exists a graph G that satisfies the two properties? (And we are; the proof is correct.) Which of the 4 situations are we in below? bubblyworld's insistence on finite sets makes it sound like we're in #3: finite set, nonzero probability. And indeed the set of random graphs with n vertices is finite! But that is an important mistake. We are not relying on argument #3 below. Yes, the set of random graphs with n vertices is finite, but we have NOT proven that any specific G has our two properties, and we have NOT proven that any graph with n vertices has those two properties! It's entirely possible that NO graphs with n vertices have the two properties; all we've shown is that every graph in a finite set has some positive probability to have our properties. You might encounter a similar proof, exactly the same up to this point, and that proof might be wrong! If you believe bubblyworld, you could make this mistake. So why does Erdős's proof work? It relies on two things that are both necessary: (1) we're not talking about a finite set at all! We're talking about random subgraphs of size n OR LARGER. We're actually picking from an infinite set here. Our smallest example might be of size 10^10^10^n! We don't know! We just know that if we keep going, eventually we must hit one. And, critically, (2) there is a positive lower bound on the probability, above some n (it's 1/2). That is, if the probability of a random subgraph having those two properties was some small nonzero probability that decreased as n increased, e.g. p ~ 1/n^2, then we still might never find an example! The proof could be incorrect! But it's shown that above some n, the probability is at least 1/2. Thus the whole proof works. Any discussion about frequentism or Lebesgue measure is missing the point: we're talking about whether proofs are correct or not, and that is a very finicky, tricky discussion. bubblyworld's original statement, and his continued refinements of that statement, are incorrect, and believing such things can lead you to accept incorrect proofs. I think that matters. So I corrected it. [1] https://en.wikipedia.org/wiki/Probabilistic_method https://en.wikipedia.org/wiki/Probabilistic_method 1. Infinite set, nonzero probability: we know at least one example must exist. 2. Infinite set, zero probability: not sure! It could be zero, or any subset of smaller cardinality or measure zero. 3. Finite set, nonzero probability: not sure! It could be zero! My coin flip example shows this. 4. Finite set, zero probability: we know no examples can exist.