7 ms·

Yes, with no prior knowledge the mathematically correct estimate is: time left = time so far But as you note prior knowledge will enable a better guess.

matsemann 1y ago
Yeah, the Copernican Principle. > I visited the Berlin Wall. People at the time wondered how long the Wall might last. Was it a temporary aberration, or a permanent fixture of modern Europe? Standing at the Wall in 1969, I made the following argument, using the Copernican principle. I said, Well, there’s nothing special about the timing of my visit. I’m just travelling—you know, Europe on five dollars a day—and I’m observing the Wall because it happens to be here. My visit is random in time. So if I divide the Wall’s total history, from the beginning to the end, into four quarters, and I’m located randomly somewhere in there, there’s a fifty-percent chance that I’m in the middle two quarters—that means, not in the first quarter and not in the fourth quarter. > Let’s suppose that I’m at the beginning of that middle fifty percent. In that case, one-quarter of the Wall’s ultimate history has passed, and there are three-quarters left in the future. In that case, the future’s three times as long as the past. On the other hand, if I’m at the other end, then three-quarters have happened already, and there’s one-quarter left in the future. In that case, the future is one-third as long as the past. https://www.newyorker.com/magazine/1999/07/12/how-to-predict-everything https://www.newyorker.com/magazine/1999/07/12/how-to-predict...
- hshdhdhehd 1y ago
  Is this a weird Monty hall thing where the person next to you didnt visit the wall randomly (maybe they decided to visit on some anniversary of the wall) so for them the expected lifetime of the wall is different?
- tsimionescu 1y ago
  This thought process suggests something very wrong. The guess "it will last again as long as it has lasted so far" doesn't give any real insight. The wall was actually as likely to end five months from when they visited it, as it was to end 500 years from then. What this "time-wise Copernican principle" gives you is a guarantee that, if you apply this logic every time you have no other knowledge and have to guess, you will get the least mean error over all of your guesses. For some events, you'll guess that they'll end in 5 minutes, and they actually end 50 years later. For others, you'll guess they'll take another 50 years and they actually end 5 minutes later. Add these two up, and overall you get 0 - you won't have either a bias to overestimating, nor to underestimating. But this doesn't actually give you any insight into how long the event will actually last. For a single event, with no other knowledge, the probability that it will after 1 minute is equal to the probability that it will end after the same duration that it lasted so far, and it is equal to the probability that it will end after a billion years. There is nothing at all that you can say about the probability of an event ending from pure mathematics like this - you need event-specific knowledge to draw any conclusions. So while this Copernican principle sounds very deep and insightful, it is actually just a pretty trite mathematical observation.
  - jwarden 1y ago
    But you will never guess that the latest tik-tok craze will last another 50 years, and you'll never guess that Saturday Night Live (which premiered in 1075) will end 5-minutes from now. Your guesses are thus more likely to be accurate than if you ignored the information about how long something has lasted so far.
    - delichon 1y ago
      > Saturday Night Live (which premiered in 1075) They probably had a great skit about the revolt of the Earls against William the Conquerer.
    - tsimionescu 1y ago
      Sure, but the opposite also applies. If in 1969 you guessed that the wall would last another 20 years, then in 1989, you'll guess that the wall of Berlin will last another 40 years - when in fact it was about to fall. And in 1949, when the wall was a few months old, you'll guess that it will last for a few months at most. So no, you're not very likely to be right at all. Now sure, if you guess "50 years" for every event, your average error rate will be even worse, across all possible events. But it is absolutely not true that it's more likely that SNL will last for another 50 years as it is that it will last for another 10 years. They are all exactly as likely, given the information we have today.
      - post-it 1y ago
        However, > Well, there’s nothing special about the timing of my visit. I’m just travelling—you know, Europe on five dollars a day—and I’m observing the Wall because it happens to be here. It's relatively unlikely that you'd visit the Berlin Wall shortly after it's erected or shortly before it falls, and quite likely that you'd visit it somewhere in the middle.
        tsimionescu 1y ago
        No, it's exactly as likely that I'll visit it at any one time in its lifetime. Sure, if we divide its lifetime into 4 quadrants, its more likely I'm in quadrant 2-3 than in either of 1 or 4. But this is slight of hand: it's still exactly as likely that I'm in quadrant 2-3 than in quadrant (1 or 4) - or, in other words, it's as likely I'm at one of the ends of the lifetime as it is that I am in the middle.
tsimionescu 1y ago
Note that this is equivalent to saying "there's no way to know". This guess doesn't give any insight, it's just the function that happens to minimize the total expected error for an unknowable duration. Edit: I should add that, more specifically, this is a property of the uniform distribution, it applies to any event for which EndsAfter(t) is uniformly distributed over all t > 0.
- froobius 1y ago
  I'm not sure about that. Is it not sometimes useful for decision making, when you don't have any insight as to how long a thing will be? It's better than just saying "I don't know".
  - tsimionescu 1y ago
    Not really, unless you care about something like "when I look back at my career, I don't want to have had a bias to underestimating nor overestimating outages". That's all this logic gives you: for every time you underestimate a crisis, you'll be equally likely to overestimate a different crisis. I don't think this is in any way actually useful. Also, the worse thing you can get from this logic is to think that it is actually most likely that the future duration equals the past duration. This is very much false, and it can mislead you if you think it's true. In fact, with no other insight, all future durations are equally likely for any particular event. The better thing to do is to get some even-specific knowledge, rather than trying to reason from a priori logic. That will easily beat this method of estimation.
    - froobius 1y ago
      You've added some useful context, but I think you're downplaying it's use. It's non-obvious, and in many cases better than just saying "we don't know". For example, if some company's server has been down for an hour, and you don't know anything more, it would be reasonable to say to your boss: "I'll look into it, but without knowing more about it, stastically we have a 50% chance of it being back up in an hour". > The better thing to do is to get some even-specific knowledge, rather than trying to reason from a priori logic True, and all the posts above have acknowledged this.

matsemann 1y ago

Yeah, the Copernican Principle. > I visited the Berlin Wall. People at the time wondered how long the Wall might last. Was it a temporary aberration, or a permanent fixture of modern Europe? Standing at the Wall in 1969, I made the following argument, using the Copernican principle. I said, Well, there’s nothing special about the timing of my visit. I’m just travelling—you know, Europe on five dollars a day—and I’m observing the Wall because it happens to be here. My visit is random in time. So if I divide the Wall’s total history, from the beginning to the end, into four quarters, and I’m located randomly somewhere in there, there’s a fifty-percent chance that I’m in the middle two quarters—that means, not in the first quarter and not in the fourth quarter. > Let’s suppose that I’m at the beginning of that middle fifty percent. In that case, one-quarter of the Wall’s ultimate history has passed, and there are three-quarters left in the future. In that case, the future’s three times as long as the past. On the other hand, if I’m at the other end, then three-quarters have happened already, and there’s one-quarter left in the future. In that case, the future is one-third as long as the past. https://www.newyorker.com/magazine/1999/07/12/how-to-predict-everything https://www.newyorker.com/magazine/1999/07/12/how-to-predict...

hshdhdhehd 1y ago
Is this a weird Monty hall thing where the person next to you didnt visit the wall randomly (maybe they decided to visit on some anniversary of the wall) so for them the expected lifetime of the wall is different?
tsimionescu 1y ago
This thought process suggests something very wrong. The guess "it will last again as long as it has lasted so far" doesn't give any real insight. The wall was actually as likely to end five months from when they visited it, as it was to end 500 years from then. What this "time-wise Copernican principle" gives you is a guarantee that, if you apply this logic every time you have no other knowledge and have to guess, you will get the least mean error over all of your guesses. For some events, you'll guess that they'll end in 5 minutes, and they actually end 50 years later. For others, you'll guess they'll take another 50 years and they actually end 5 minutes later. Add these two up, and overall you get 0 - you won't have either a bias to overestimating, nor to underestimating. But this doesn't actually give you any insight into how long the event will actually last. For a single event, with no other knowledge, the probability that it will after 1 minute is equal to the probability that it will end after the same duration that it lasted so far, and it is equal to the probability that it will end after a billion years. There is nothing at all that you can say about the probability of an event ending from pure mathematics like this - you need event-specific knowledge to draw any conclusions. So while this Copernican principle sounds very deep and insightful, it is actually just a pretty trite mathematical observation.
- jwarden 1y ago
  But you will never guess that the latest tik-tok craze will last another 50 years, and you'll never guess that Saturday Night Live (which premiered in 1075) will end 5-minutes from now. Your guesses are thus more likely to be accurate than if you ignored the information about how long something has lasted so far.
  - delichon 1y ago
    > Saturday Night Live (which premiered in 1075) They probably had a great skit about the revolt of the Earls against William the Conquerer.
  - tsimionescu 1y ago
    Sure, but the opposite also applies. If in 1969 you guessed that the wall would last another 20 years, then in 1989, you'll guess that the wall of Berlin will last another 40 years - when in fact it was about to fall. And in 1949, when the wall was a few months old, you'll guess that it will last for a few months at most. So no, you're not very likely to be right at all. Now sure, if you guess "50 years" for every event, your average error rate will be even worse, across all possible events. But it is absolutely not true that it's more likely that SNL will last for another 50 years as it is that it will last for another 10 years. They are all exactly as likely, given the information we have today.
    - post-it 1y ago
      However, > Well, there’s nothing special about the timing of my visit. I’m just travelling—you know, Europe on five dollars a day—and I’m observing the Wall because it happens to be here. It's relatively unlikely that you'd visit the Berlin Wall shortly after it's erected or shortly before it falls, and quite likely that you'd visit it somewhere in the middle.
      - tsimionescu 1y ago
        No, it's exactly as likely that I'll visit it at any one time in its lifetime. Sure, if we divide its lifetime into 4 quadrants, its more likely I'm in quadrant 2-3 than in either of 1 or 4. But this is slight of hand: it's still exactly as likely that I'm in quadrant 2-3 than in quadrant (1 or 4) - or, in other words, it's as likely I'm at one of the ends of the lifetime as it is that I am in the middle.

tsimionescu 1y ago

Note that this is equivalent to saying "there's no way to know". This guess doesn't give any insight, it's just the function that happens to minimize the total expected error for an unknowable duration. Edit: I should add that, more specifically, this is a property of the uniform distribution, it applies to any event for which EndsAfter(t) is uniformly distributed over all t > 0.

froobius 1y ago
I'm not sure about that. Is it not sometimes useful for decision making, when you don't have any insight as to how long a thing will be? It's better than just saying "I don't know".
- tsimionescu 1y ago
  Not really, unless you care about something like "when I look back at my career, I don't want to have had a bias to underestimating nor overestimating outages". That's all this logic gives you: for every time you underestimate a crisis, you'll be equally likely to overestimate a different crisis. I don't think this is in any way actually useful. Also, the worse thing you can get from this logic is to think that it is actually most likely that the future duration equals the past duration. This is very much false, and it can mislead you if you think it's true. In fact, with no other insight, all future durations are equally likely for any particular event. The better thing to do is to get some even-specific knowledge, rather than trying to reason from a priori logic. That will easily beat this method of estimation.
  - froobius 1y ago
    You've added some useful context, but I think you're downplaying it's use. It's non-obvious, and in many cases better than just saying "we don't know". For example, if some company's server has been down for an hour, and you don't know anything more, it would be reasonable to say to your boss: "I'll look into it, but without knowing more about it, stastically we have a 50% chance of it being back up in an hour". > The better thing to do is to get some even-specific knowledge, rather than trying to reason from a priori logic True, and all the posts above have acknowledged this.