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0 and 1 are really extreme probabilities. It's often better to think of a probability of 0.001 being closer to 0.5 than to 0. As odds ratios it's easy to see th

by taejo 8y ago

0 and 1 are really extreme probabilities. It's often better to think of a probability of 0.001 being closer to 0.5 than to 0. As odds ratios it's easy to see that 999:1 is closer to 1:1 than to infinity:1. 0 probability means that even infinitely many pieces of evidence will not change your assessment: no matter how many heads you see restored from cryogenic storage, you will still believe that the probability is 0, and will have to find other, increasingly unlikely, explanations for the evidence. For a more in-depth and slightly more rigorous discussion, see 0 And 1 Are Not Probabilities: https://www.greaterwrong.com/posts/QGkYCwyC7wTDyt3yT/0-and-1-are-not-probabilities https://www.greaterwrong.com/posts/QGkYCwyC7wTDyt3yT/0-and-1...

Retric 8y ago
0.001 is closer to 0 than 0.003 ;) All probability's have error bars, so 0 +/ 10e-20 is still very much 0.
- taejo 8y ago
  Thanks, I can count :) Of course 0.001 is closer to 0 than 0.003, but my point is that linear distance is not a very good way of comparing probabilities-as-degrees-of-certainties: rather than differences of probabilities one should consider the ratios of odds ratios (or differences of log-odds-ratios) because these better represent how we update our certainty in the light of evidence. Frequentist probabilities can have error bars (if I repeat some experiment N times and observe no successes, I estimate the probability as "0 with error bars"), but I'm not sure that Bayesian probabilities-as-degrees-of-certainty naturally have error bars. If you're uncertain of your certainty, isn't that the same as just being less certain? As I say, I'm not sure of the mathematics here.
  - Retric 8y ago
    Bayesian has direct error bars and error bars on your error bars, and error bars on the ... It’s however generally ignored. Edit: I believe I preformed this experiment which updates my prior probability. Anyway, my point is not that you should interpret probability functions in a linear fashion just that non linear interpretations are not nessisarily more accurate. Suppose I am playing a chess I have non zero odds for a 2 win streak, but the odds of winning 1 googleplex games in a row is 0. Not becase it’s some probability ^ a googleplex but rather I am not going to play that many games. Sure, you might say their are some error bars on that zero, and error bars on those error bars etc, but still the direct estimate is still a 0 probability.
    - taejo 8y ago
      > Anyway, my point is not that you should interpret probability functions in a linear fashion just that non linear interpretations are not nessisarily more accurate. Was I supposed to understand that from your previous comment? I didn't, and I don't see how I could have. > their are some error bars on that zero There'd better be, because the true probability is somewhere within those error bars: I'd be very confident (but not certain!) that the true probability is not zero. What if you turn out to have been imagining all your life up till now, a googleplex is much less than you think, and you're actually an immortal chess-playing machine? That's unlikely, but I'm fairly sure the probability is not zero. Zero is a probability (unlike the title of the page I linked to) but it's a very special one, mostly of theoretical interest.
      - Retric 8y ago
        In that case 'I' am not me, so the probability would still be zero as 'I' do not exist. But even more so there are a finite number of possible chess games and it’s less than a googleplex so even if I where in a time loop their are not that many games playable. You are welcome to try and come up with a possibility, but really it's 0 with error bars on something not conceivable. Edit, by which I mean if my understanding if the game is flawed then the estimate is less accurate, but that does not directly improve the odds. Thus that’s an error bar as again it does not directly change the actual probability.
throwawaymath 8y ago
> 0 probability means that even infinitely many pieces of evidence will not change your assessment Well no, this isn’t really what probability 0 means. An event is not impossible if it has a probability of 0. More importantly probability informs you about how many times you should expect an event to occur, not what to do with this information if the event does or does not arise. In the Bayesian view, your probability measures how confident you are about an event occuring. In the frequentist view it measures the expected number of times an event will occur in n trials. But neither of these really mean you’d be impossible to convince :) Probabilities 0 and 1 are legitimate, and correspond to the statements “almost never” and “almost surely” in mathematics [1]. A good theoretical example of probability 0 is picking any given real number x out of an interval a < x < b. The probability of choosing x is 0 but it’s not impossible. A “real world” analogue of this would be flipping a coin infinitely many times and never getting heads. That is not impossible, but it has probability 0. I wouldn’t personally use probability 0 for cryogenics either, but if someone insisted on it the way to interpret that would be to say, “Out of uncountably many trials, a successful resurrection occurs only countably many times.” _____________________ 1. https://en.wikipedia.org/wiki/Almost_surely https://en.wikipedia.org/wiki/Almost_surely
- FeepingCreature 8y ago
  I think there's a legitimate question as to whether every real number is actually physically real. [1] It seems credible that our universe can only actually present us with a finite number of possible sensory experiences, in which case an event with probability 0 actually couldn't ever happen. That said, even if the universe can present you with an infinity of observations you can certainly only consider a finite number of classes of them, leading to the same outcome. [1] https://arxiv.org/abs/math/0411418 https://arxiv.org/abs/math/0411418
  - throwawaymath 8y ago
    Yep, this is why I wouldn't personally use 0 probability to describe anything in the real world. It technically works, it's just the realm of pure measure theory rather than anything applicable to finitely many trials.

by taejo 8y ago

Retric 8y ago

0.001 is closer to 0 than 0.003 ;) All probability's have error bars, so 0 +/ 10e-20 is still very much 0.

taejo 8y ago
Thanks, I can count :) Of course 0.001 is closer to 0 than 0.003, but my point is that linear distance is not a very good way of comparing probabilities-as-degrees-of-certainties: rather than differences of probabilities one should consider the ratios of odds ratios (or differences of log-odds-ratios) because these better represent how we update our certainty in the light of evidence. Frequentist probabilities can have error bars (if I repeat some experiment N times and observe no successes, I estimate the probability as "0 with error bars"), but I'm not sure that Bayesian probabilities-as-degrees-of-certainty naturally have error bars. If you're uncertain of your certainty, isn't that the same as just being less certain? As I say, I'm not sure of the mathematics here.
- Retric 8y ago
  Bayesian has direct error bars and error bars on your error bars, and error bars on the ... It’s however generally ignored. Edit: I believe I preformed this experiment which updates my prior probability. Anyway, my point is not that you should interpret probability functions in a linear fashion just that non linear interpretations are not nessisarily more accurate. Suppose I am playing a chess I have non zero odds for a 2 win streak, but the odds of winning 1 googleplex games in a row is 0. Not becase it’s some probability ^ a googleplex but rather I am not going to play that many games. Sure, you might say their are some error bars on that zero, and error bars on those error bars etc, but still the direct estimate is still a 0 probability.
  - taejo 8y ago
    > Anyway, my point is not that you should interpret probability functions in a linear fashion just that non linear interpretations are not nessisarily more accurate. Was I supposed to understand that from your previous comment? I didn't, and I don't see how I could have. > their are some error bars on that zero There'd better be, because the true probability is somewhere within those error bars: I'd be very confident (but not certain!) that the true probability is not zero. What if you turn out to have been imagining all your life up till now, a googleplex is much less than you think, and you're actually an immortal chess-playing machine? That's unlikely, but I'm fairly sure the probability is not zero. Zero is a probability (unlike the title of the page I linked to) but it's a very special one, mostly of theoretical interest.
    - Retric 8y ago
      In that case 'I' am not me, so the probability would still be zero as 'I' do not exist. But even more so there are a finite number of possible chess games and it’s less than a googleplex so even if I where in a time loop their are not that many games playable. You are welcome to try and come up with a possibility, but really it's 0 with error bars on something not conceivable. Edit, by which I mean if my understanding if the game is flawed then the estimate is less accurate, but that does not directly improve the odds. Thus that’s an error bar as again it does not directly change the actual probability.

throwawaymath 8y ago

> 0 probability means that even infinitely many pieces of evidence will not change your assessment Well no, this isn’t really what probability 0 means. An event is not impossible if it has a probability of 0. More importantly probability informs you about how many times you should expect an event to occur, not what to do with this information if the event does or does not arise. In the Bayesian view, your probability measures how confident you are about an event occuring. In the frequentist view it measures the expected number of times an event will occur in n trials. But neither of these really mean you’d be impossible to convince :) Probabilities 0 and 1 are legitimate, and correspond to the statements “almost never” and “almost surely” in mathematics [1]. A good theoretical example of probability 0 is picking any given real number x out of an interval a < x < b. The probability of choosing x is 0 but it’s not impossible. A “real world” analogue of this would be flipping a coin infinitely many times and never getting heads. That is not impossible, but it has probability 0. I wouldn’t personally use probability 0 for cryogenics either, but if someone insisted on it the way to interpret that would be to say, “Out of uncountably many trials, a successful resurrection occurs only countably many times.” _____________________ 1. https://en.wikipedia.org/wiki/Almost_surely https://en.wikipedia.org/wiki/Almost_surely

FeepingCreature 8y ago
I think there's a legitimate question as to whether every real number is actually physically real. [1] It seems credible that our universe can only actually present us with a finite number of possible sensory experiences, in which case an event with probability 0 actually couldn't ever happen. That said, even if the universe can present you with an infinity of observations you can certainly only consider a finite number of classes of them, leading to the same outcome. [1] https://arxiv.org/abs/math/0411418 https://arxiv.org/abs/math/0411418
- throwawaymath 8y ago
  Yep, this is why I wouldn't personally use 0 probability to describe anything in the real world. It technically works, it's just the realm of pure measure theory rather than anything applicable to finitely many trials.