Isn’t your whole argument merely a rediscovery of the mathematical concept of uncertainty? You admit at the outset that the priors for the magically falling pony must be bizzarely specific to render the exact probability of 1/2^100. In contrast, given a fair coin and a hundred coinflips, it’s nothing at all bizzare about the fact that the probability that you’ll get a straight run of heads is exactly 1/2^100. It’s just math. The problem of course, is that the uncertainty as to whether magically falling ponies are possible isn’t a question that is amenable to a quantitative study, given that the event so seldom repeat… Even so, my intuition is exactly opposite to yours: I know how bizzarely unlikely it would be to get a hundred heads in a row, so if that were to happen (under circumstances that ruled out foul play) I would be astonished. If a magically falling pony appeared I would be likewise astonished, but less so since my model of reality is more uncertain to me than my model of abstract math.
I don’t see how your certainty in your model of abstract math is even relevant.
If I flip a coin a hundred times and it comes up heads each time, that does not in any way shake my confidence that the probability of 100 independent binary choices resulting in a specific bitstring is 1/2^100. What it does do is shake my confidence that flipping a coin a hundred times is a physical act that can be reliably modeled as 100 independent binary choices.
That second thing isn’t a statement about abstract mathematics at all, it’s a statement about the physical world… and one I’m significantly less confident of than I am in the continued nonexistence of magically falling ponies.
What sequences of heads and tails would NOT shake your confidence that flipping a coin is reliable enough to model independent binary choices? Any sequence with a prior probability of 1/2^100? 1 tails followed by 99 heads? 49H 1T 50h? Alternating heads and tails? 50heads first, followed by 50 tails? Alternating chains of heads and tails of equal length? Exactly 50 heads and 50 tails, regardless of order?
If you flip that coin 100 times, you will get a sequence with prior probability of roughly 8*10^-31.
Fair enough. How many different sequences can be included in a given ‘called set’? If I said “99 heads out of 100”, then I’m identifying 100 different sequences.
In the end, though, I’m trying to set a ceiling: What’s the most likely prediction I could make which would cause you to reevaluate the math behind odds? So far the lower limit is 1/2^100. Would you accept the call that at least 80% of the coin flips would be heads? My powers of telekinetic manipulation of coin flips are limited, you see, and both exhausting and unreliable.
For example, if I approached you and offered you a bet that you could not predict the flip of a series of coins, and you got it right three times in a row, that wouldn’t particularly shake my confidence that each coin-flip could be modeled as an independent binary choice with equal chances on both sides. OTOH if you approached me and offered me a bet that you could not predict the flip of a series of coins, and you got it right three times in a row, that would indeed shake my confidence.
But if I leave all the real-world stuff out of it, sure, a coin that comes up heads 80% of the time on, say, 100 flips would certainly make me suspicious.
Here’s a bet then- Flip the coin nearest to you 100 times, and report the results. If you get 79 or fewer heads, then I will donate $20US to the cause of your choice (which may be you, personally). If you get 80 or more heads, then consider the possibility that I have the ability to alter the results of coin flips in a way which is unexplained by modern physics.
Or maybe I’m willing to gamble $20US on a very small chance (~half of six standard deviations, if I have the math right) that I can mindscrew you.
It would be a lot more than $50 more impressive (the few times it works) if she said “I bet you a dollar that it’s the three of clubs.”
I was also considering the ‘cider in my ear’ angle. Just because you don’t see any possible way that I could rig the bet, the fact that I proposed it is an indication above baseline that I might have.
If I flip a coin 100 times and get all heads, there are many, many hypotheses that suddenly get a lot more plausible. Perhaps the coin is strongly biased. In fact, a weak prior of slight bias will, post-update, seem much more plausible than the fair coin. Perhaps it’s a double headed coin, and in my inspection I failed to notice that. It seems vanishingly unlikely that I would miss that on repeated inspection… but I’m still more inclined to believe that, than a mathematically fair coin coming up heads every time. Perhaps I’ve unconsciously learned how to predictably flip a coin. Perhaps I’ve failed horribly at counting, and actually only flipped the coin 10 times. Perhaps any of the above are combined with my not noticing a tails or ten appear in the result string.
In other words, exceedingly unlikely events will make me doubt my sanity before they make me start doubting math.
I assumed you did. I just thought it worth explicitly adding to the discussion that considering only some extraordinarily weird ideas when discussing extraordinarily weird events is a form of bias that seems to run rampant in hypotheticals around here. It’s not just the one aspect you mentioned where our confidence should be shaken by such a result.
Isn’t your whole argument merely a rediscovery of the mathematical concept of uncertainty? You admit at the outset that the priors for the magically falling pony must be bizzarely specific to render the exact probability of 1/2^100. In contrast, given a fair coin and a hundred coinflips, it’s nothing at all bizzare about the fact that the probability that you’ll get a straight run of heads is exactly 1/2^100. It’s just math. The problem of course, is that the uncertainty as to whether magically falling ponies are possible isn’t a question that is amenable to a quantitative study, given that the event so seldom repeat… Even so, my intuition is exactly opposite to yours: I know how bizzarely unlikely it would be to get a hundred heads in a row, so if that were to happen (under circumstances that ruled out foul play) I would be astonished. If a magically falling pony appeared I would be likewise astonished, but less so since my model of reality is more uncertain to me than my model of abstract math.
That said, the idea of “metaconfidence” reminds me somewhat of Yvain’s post Confidence levels inside and outside an argument.
Edited to add: And the the concept of falling ponies reminds me of this.
Wait, what?
I don’t see how your certainty in your model of abstract math is even relevant.
If I flip a coin a hundred times and it comes up heads each time, that does not in any way shake my confidence that the probability of 100 independent binary choices resulting in a specific bitstring is 1/2^100. What it does do is shake my confidence that flipping a coin a hundred times is a physical act that can be reliably modeled as 100 independent binary choices.
That second thing isn’t a statement about abstract mathematics at all, it’s a statement about the physical world… and one I’m significantly less confident of than I am in the continued nonexistence of magically falling ponies.
What sequences of heads and tails would NOT shake your confidence that flipping a coin is reliable enough to model independent binary choices? Any sequence with a prior probability of 1/2^100? 1 tails followed by 99 heads? 49H 1T 50h? Alternating heads and tails? 50heads first, followed by 50 tails? Alternating chains of heads and tails of equal length? Exactly 50 heads and 50 tails, regardless of order?
If you flip that coin 100 times, you will get a sequence with prior probability of roughly 8*10^-31.
One that wasn’t specifically identified ahead of time.
Fair enough. How many different sequences can be included in a given ‘called set’? If I said “99 heads out of 100”, then I’m identifying 100 different sequences.
In the end, though, I’m trying to set a ceiling: What’s the most likely prediction I could make which would cause you to reevaluate the math behind odds? So far the lower limit is 1/2^100. Would you accept the call that at least 80% of the coin flips would be heads? My powers of telekinetic manipulation of coin flips are limited, you see, and both exhausting and unreliable.
It depends a lot.
For example, if I approached you and offered you a bet that you could not predict the flip of a series of coins, and you got it right three times in a row, that wouldn’t particularly shake my confidence that each coin-flip could be modeled as an independent binary choice with equal chances on both sides.
OTOH if you approached me and offered me a bet that you could not predict the flip of a series of coins, and you got it right three times in a row, that would indeed shake my confidence.
But if I leave all the real-world stuff out of it, sure, a coin that comes up heads 80% of the time on, say, 100 flips would certainly make me suspicious.
Here’s a bet then- Flip the coin nearest to you 100 times, and report the results. If you get 79 or fewer heads, then I will donate $20US to the cause of your choice (which may be you, personally). If you get 80 or more heads, then consider the possibility that I have the ability to alter the results of coin flips in a way which is unexplained by modern physics.
Or maybe I’m willing to gamble $20US on a very small chance (~half of six standard deviations, if I have the math right) that I can mindscrew you.
I used to date a girl who had a favorite card trick: she would hand you a deck of cards, ask you to pick one, and say “is it the three of clubs?”
Her theory was that she’d be wrong most of the time, but when she was right it would be really impressive.
It would be a lot more than $50 more impressive (the few times it works) if she said “I bet you a dollar that it’s the three of clubs.”
I was also considering the ‘cider in my ear’ angle. Just because you don’t see any possible way that I could rig the bet, the fact that I proposed it is an indication above baseline that I might have.
See also.
I instantly thought about that, too.
If I flip a coin 100 times and get all heads, there are many, many hypotheses that suddenly get a lot more plausible. Perhaps the coin is strongly biased. In fact, a weak prior of slight bias will, post-update, seem much more plausible than the fair coin. Perhaps it’s a double headed coin, and in my inspection I failed to notice that. It seems vanishingly unlikely that I would miss that on repeated inspection… but I’m still more inclined to believe that, than a mathematically fair coin coming up heads every time. Perhaps I’ve unconsciously learned how to predictably flip a coin. Perhaps I’ve failed horribly at counting, and actually only flipped the coin 10 times. Perhaps any of the above are combined with my not noticing a tails or ten appear in the result string.
In other words, exceedingly unlikely events will make me doubt my sanity before they make me start doubting math.
Agreed with all of that. (Not sure if you think we disagree on any of it.)
I assumed you did. I just thought it worth explicitly adding to the discussion that considering only some extraordinarily weird ideas when discussing extraordinarily weird events is a form of bias that seems to run rampant in hypotheticals around here. It’s not just the one aspect you mentioned where our confidence should be shaken by such a result.
Ah, gotcha… sure, agreed on all counts.