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.
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.
> 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.
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.
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.