Christian apologist William Lane Craig claims the skeptical slogan “extraordinary claims require extraordinary evidence” is contradicted by probability theory, because it actually wouldn’t take all that much evidence to convince us that, for example, “the numbers chosen in last night’s lottery were 4, 2, 9, 7, 8 and 3.” The correct response to this argument is to say that the prior probability of a miracle occurring is orders of magnitude smaller than mere one in a million odds.
I’m not sure that response works. Flip a fair coin two hundred times, tell me the results, then show me the video and I’ll almost certainly believe you. But if the results were H^200, I won’t; I’ll assume you were wrong or lying about the coin being fair, or something.
H^200 isn’t any less likely than any other sequence of two hundred coin flips, but it’s still one of the most extraordinary. Extraordinariness just doesn’t feel like it’s a mere question of prior probability.
H^200 isn’t any less likely under the assumption that the coin is fair, and the person reporting the coin is honest. But! H^200—being a particularly simple sequence—is massively more likely than most other sequences under the alternative assumption that the reporter is a liar, or that the coin is biased.
So being told that the outcome was H^200 is at least a lot of evidence that there’s something funny going on, for that reason.
This has nothing to do with simplicity. Any other apriori selected sequence, such as first 200 binary digits of pi, would be just as unlikely. It seems like it is related to simplicity because “non-simple” sequences are usually described in an aggregate way, such as “100 heads and 100 tails” and in fact include a lot of individual sequences, resulting in an aggregate probability much higher than 1/2^200.
This has nothing to do with simplicity. Any other apriori selected sequence, such as first 200 binary digits of pi, would be just as unlikely.
Yes, under the hypothesis that the coin is fair and has been flipped fairly all sequences are equally unlikely. But under the hypothesis that someone is lying to us or has been messing with the coin simple sequences are more likely. So (via Bayes) if we hear of a simple sequence we will think it’s more likely to have be artificially created than if we hear of a complicated one.
Well, what’s most interestingly improbable here is the prediction of a 200-coin sequence, not the sequence itself.
I suspect what’s going on with such “extraordinary” sequences is a kind of hindsight bias… the sequence seems so simple and easy to understand that, upon revealing it, we feel like “we knew it all along.” That is, we feel like we could have predicted it… and since such a prediction is extraordinarily unlikely, we feel like something extraordinarily unlikely just happened.
You seem to be suggesting that if I actually flipped a coin 200 times, the actual result would be just as extraordinary as the hypothetical result H^200, having an equal prior probability. I’m not sure why you’d be suggesting that, so maybe we have crossed wires?
For one thing, it might be that an extraordinary event is one which causes us to make large updates to the probabilities assigned to various hypotheses. H^200 makes hypotheses like “double headed coin” go from “barely worth considering” to “really quite plausible”, so is extraordinary. (I think this idea has a bunch of fiddly little bits that I’m not particularly interested in hammering out. But the idea that the extraordinariness of an event is purely a function of its prior probability, just seems plain wrong.)
You seem to be suggesting that if I actually flipped a coin 200 times, the actual result would be just as extraordinary as the hypothetical result H^200, having an equal prior probability.
I agree that I seemed to suggest that; I indeed disagree that some arbitrary (expected to come with attached justifications, but those justifications are not present) sequence would be just as “extraordinary”. This is where simplicity comes in—the only reason a sequence of 200 bits would be interesting to humans would be if it were simple—if it had some special property that allowed us to describe it without listing out the results of all 200 flips. Most sequences of 200 flips won’t have this property, which makes the sequences that do extraordinary. So I’d consider T^200, (HT)^100, and (TH)^100 extraordinary sequences, but not 001001011101111001101011001111100101001110001011101000100011100
000111000101110001011100011010101001000011011001011011010110101
011100011000000101011001000100011000010100001100110000110010101
However, if I were to take out a coin and flip it now, and get those results, I could say “that sequence I posted on Less Wrong”, and thus it would be extraordinary.
So I agree that extraordinariness has nothing to do with the prior probability of a particular sequence of flips, but rather the fact that such a sequence of flips belongs to a privileged reference class (sequences of flips you can easily describe without listing all 200 flips), and getting a sequence from that reference class is an event with a low prior probability. The combination of being in that particular reference class and the fact that such an event (being in that reference class, not the individual sequence itself) is unlikely together might provide a sense of extraordinariness.
I’ve suggested elsewhere that this sense of extraordinariness when faced with a result like H^200 is the result of a kind of hindsight bias.
Roughly speaking, the idea is that certain notions seem simple to our brains… easier to access and understand and express and so forth. When such an notion is suggested to us, we frequently come to believe that “we knew it all along.”
A H^200 string of coin-flips is just such a notion; it seems simpler than a (HHTTTHHTHHT)^20 string, for example. So when faced with H^200 we have a stronger sense of having predicted it, or at least that we would have been able to predict it if we’d thought to.
But, of course, predicting the result of 200 coin flips is extremely unlikely, and we know that. So when faced with H^200 we have a much stronger sense of having experienced something extremely unlikely (aka extraordinary) than when faced with a more “random-seeming” string.
Getting 200 heads only in coinflipping is just as likely as any other result. However, it is of incredibly low entropy—you should not expect to see a pattern of that sort (less bits to describe than listing the results). It’s also impossibly unlikely as a result from a fair coin, compared to as a result of fraud or an unfair coin.
Isn’t it also the case that you are, in that case, receiving extraordinary evidence? If people were as unreliable about lottery numbers as they are about religion you would in fact remain pretty skeptical about the actual number.
I’m not sure that response works. Flip a fair coin two hundred times, tell me the results, then show me the video and I’ll almost certainly believe you. But if the results were H^200, I won’t; I’ll assume you were wrong or lying about the coin being fair, or something.
H^200 isn’t any less likely than any other sequence of two hundred coin flips, but it’s still one of the most extraordinary. Extraordinariness just doesn’t feel like it’s a mere question of prior probability.
H^200 isn’t any less likely under the assumption that the coin is fair, and the person reporting the coin is honest. But! H^200—being a particularly simple sequence—is massively more likely than most other sequences under the alternative assumption that the reporter is a liar, or that the coin is biased.
So being told that the outcome was H^200 is at least a lot of evidence that there’s something funny going on, for that reason.
This has nothing to do with simplicity. Any other apriori selected sequence, such as first 200 binary digits of pi, would be just as unlikely. It seems like it is related to simplicity because “non-simple” sequences are usually described in an aggregate way, such as “100 heads and 100 tails” and in fact include a lot of individual sequences, resulting in an aggregate probability much higher than 1/2^200.
Yes, under the hypothesis that the coin is fair and has been flipped fairly all sequences are equally unlikely. But under the hypothesis that someone is lying to us or has been messing with the coin simple sequences are more likely. So (via Bayes) if we hear of a simple sequence we will think it’s more likely to have be artificially created than if we hear of a complicated one.
Well, what’s most interestingly improbable here is the prediction of a 200-coin sequence, not the sequence itself.
I suspect what’s going on with such “extraordinary” sequences is a kind of hindsight bias… the sequence seems so simple and easy to understand that, upon revealing it, we feel like “we knew it all along.” That is, we feel like we could have predicted it… and since such a prediction is extraordinarily unlikely, we feel like something extraordinarily unlikely just happened.
And, that, my friend, is how an algorithm feels from inside. What else could extraordinariness possibly be? It might also help to read “Probability Is Subjectively Objective”.
You seem to be suggesting that if I actually flipped a coin 200 times, the actual result would be just as extraordinary as the hypothetical result H^200, having an equal prior probability. I’m not sure why you’d be suggesting that, so maybe we have crossed wires?
For one thing, it might be that an extraordinary event is one which causes us to make large updates to the probabilities assigned to various hypotheses. H^200 makes hypotheses like “double headed coin” go from “barely worth considering” to “really quite plausible”, so is extraordinary. (I think this idea has a bunch of fiddly little bits that I’m not particularly interested in hammering out. But the idea that the extraordinariness of an event is purely a function of its prior probability, just seems plain wrong.)
I agree that I seemed to suggest that; I indeed disagree that some arbitrary (expected to come with attached justifications, but those justifications are not present) sequence would be just as “extraordinary”. This is where simplicity comes in—the only reason a sequence of 200 bits would be interesting to humans would be if it were simple—if it had some special property that allowed us to describe it without listing out the results of all 200 flips. Most sequences of 200 flips won’t have this property, which makes the sequences that do extraordinary. So I’d consider T^200, (HT)^100, and (TH)^100 extraordinary sequences, but not 001001011101111001101011001111100101001110001011101000100011100 000111000101110001011100011010101001000011011001011011010110101 011100011000000101011001000100011000010100001100110000110010101
However, if I were to take out a coin and flip it now, and get those results, I could say “that sequence I posted on Less Wrong”, and thus it would be extraordinary.
So I agree that extraordinariness has nothing to do with the prior probability of a particular sequence of flips, but rather the fact that such a sequence of flips belongs to a privileged reference class (sequences of flips you can easily describe without listing all 200 flips), and getting a sequence from that reference class is an event with a low prior probability. The combination of being in that particular reference class and the fact that such an event (being in that reference class, not the individual sequence itself) is unlikely together might provide a sense of extraordinariness.
I’ve suggested elsewhere that this sense of extraordinariness when faced with a result like H^200 is the result of a kind of hindsight bias.
Roughly speaking, the idea is that certain notions seem simple to our brains… easier to access and understand and express and so forth. When such an notion is suggested to us, we frequently come to believe that “we knew it all along.”
A H^200 string of coin-flips is just such a notion; it seems simpler than a (HHTTTHHTHHT)^20 string, for example. So when faced with H^200 we have a stronger sense of having predicted it, or at least that we would have been able to predict it if we’d thought to.
But, of course, predicting the result of 200 coin flips is extremely unlikely, and we know that. So when faced with H^200 we have a much stronger sense of having experienced something extremely unlikely (aka extraordinary) than when faced with a more “random-seeming” string.
Getting 200 heads only in coinflipping is just as likely as any other result. However, it is of incredibly low entropy—you should not expect to see a pattern of that sort (less bits to describe than listing the results). It’s also impossibly unlikely as a result from a fair coin, compared to as a result of fraud or an unfair coin.
Isn’t it also the case that you are, in that case, receiving extraordinary evidence? If people were as unreliable about lottery numbers as they are about religion you would in fact remain pretty skeptical about the actual number.