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The reason why Bob should be much more skeptical when Alice says "I just got HHHHHHHHHHHHHHHHHHHH" than when she says "I just got HTHHTHHTTHTTHTHHHH" is that there are specific other highish-probability hypotheses that explain Alice's first claim, and there aren't for her second. (Unless, e.g., it turns out that Alice had previously made a bet with someone else that she would get HTHHTHHTTHTTHTHHHH, at which point we should suddenly get more skeptical again.)

Bob's perfectly within his rights to be skeptical, of course, and if the number of coin flips is large enough then even a perfectly honest Alice is quite likely to have made at least one error. But he isn't entitled to say, e.g., that Pr(Alice actually got HTHHTHHTTHTTHTHHHH | Alice said she got HTHHTHHTTHTTHTHHHH) = Pr(Alice actually got HTHHTHHTTHTTHTHHHH) = 2^-20 because Alice's testimony provides non-negligible evidence, because empirically when people report things they have no particular reason to get wrong they're quite often right.

(But, again: if Bob learns that Alice had a specific reason to want it thought she got that exact sequence of flips, he should get more skeptical again.)

So, now suppose Alice says "I just won the lottery" and Amanda says "I just saw a ghost". What should Bob's probability estimates be in the two cases?

Empirically, so far as I can tell, a good fraction of people who claim to have won the lottery actually did so. Of course people sometimes lie, but you have to weigh "most people don't win the lottery on any given occasion" against "most people don't falsely claim to have won the lottery on any given occasion". I guess Bob's posterior Pr(Alice won the lottery) should be somewhere in the vicinity of 1/2. Enough to be decently convinced by a modest amount of further evidence, unless some other hypothesis -- e.g., Alice is trying to scam him somehow, or she's being seriously hoaxed -- gets enough evidence to be taken seriously (e.g., Alice, having allegedly won the lottery, asks Bob for a loan to be repaid with exorbitant interest).

On the other hand, there are lots and lots of tales of ghosts and (at best) very few well verified ones. It looks as if many people who claim to have seen ghosts probably haven't. Further, there are reasons to think it very unlikely that there are ghosts at all (e.g., it seems clear that human thinking is done by human brains, and by definition a ghost's brain is no longer functioning) and those reasons seem quite robust -- they aren't, e.g., dependent on details of our current theories of quantum physics or evolutionary biology. So we should set Pr(ghosts are real) extremely small, and Pr(Amanda reports a ghost | Amanda hasn't really seen a ghost) not terribly small, which means Pr(Amanda has seen a ghost | Amanda reports a ghost) is still small.

Bob's last comparison (claims of seeing ghosts, against actual wins of big lottery prizes) is of course nonsensical, and as long as one of it's of the form "more claims of ghosts than X" it actually goes the wrong way for his purposes. What he wants is more actual sightings of ghosts and fewer claims of ghosts.

My response is here, a post on my blog from last August.

Basically when Bob sees Alice present the particular sequence, he is seeing something extremely improbable, namely that she would present that individual sequence. So he is seeing extremely improbable evidence which strongly favors the hypothesis that something extremely improbable occurred. He should update on that evidence by concluding that it probably did occur.

Regarding the lottery issue, we have the same situation. If you play the lottery, see the numbers announced, and go, "I just won the lottery!" you are indeed probably wrong. Look again. In most cases you will see that the numbers don't quite match. In the few cases where they do match, you are seeing extremely improbable evidence that you won the lottery, namely that your numbers match after repeated comparisons.

I don't see the paradox. P(Alice saw this sequence) is low, and P(Alice presented this sequence) is low, but P(Alice saw this sequence | Alice presented this sequence) is high, so Bob has no reason to be incredulous.

Suppose Alice and Bob are the same person. Alice tosses a coin a large number of times and records the results.

Should she disbelieve what she reads?

I would personally argue that, even given any particular non-fatal objection to the core of this article, there is something interesting to be found here, if one is charitable. I recommend Chapter 2, Section 4 of Nick Bostrom's Anthropic Bias: Observation Selection Effects in Science and Philosophy, and the citations therein, for further reading. There also might be more recent work on this problem that I'm unaware of. We might refer to this as defining the distinction between surprising and unsurprising improbable events.

It also seems noteworthy that user:cousin_it has done precisely what Bostrom does in his book: sidesteps the issue by focusing on determining the conditional probability implicit in the problem. (In Bostrom's case, however, it is P(There exists an ensemble of universes | An observer observes a fine-tuned universe)). Perhaps this concentration on conditional probabilities is a reduction of the problem, but it does not seem to cause the confusion to evaporate in a puff of smoke, as we should probably expect.

It is true that sometimes humans systematically happen upon wrong-headed ideas, but it may also be the case that user:CronoDAS and others have converged upon a substantial problem.

I have seen this argument on LessWrong before, and don't think the other explanations are as clear as they can be. They are correct though, so my apologies if this just clutters up the thread.

The Bayesian way of looking at this is clear: the prior probability of any particular sequence is 1/2^[large number]. Alice sees this sequence and reports it to Bob. Presumably Alice intends on telling Bob the truth about what she saw, so let's say that there's a 90% chance that she will not make a mistake during the reporting. The other 10% will cover all cases ranging from misremembering/misreading a flip to outright lying. The point is that if Alice is lying, this 10% has to be divided up between the other 2^[large number]-1 other possible sequences - if Alice is going to lie, any particular sequence is very unlikely to be presented by her as the true sequence, since there are a lot of ways for her to lie. So, assuming that Alice was intending to speak the truth, her giving that sequence is very strong (in my example 9*(2^[large number]-1):1) evidence that that particular sequence was indeed the true one over any specific other sequence - 'coincidentally' precisely strong enough to turn the posterior belief of Bob that that sequence is correct to 90%.

A fun side remark is that the above also clearly shows why Bob should be more skeptical when Alice presents sequences like HHHHHHHHHH or HTHTHTHTHTHT - if Alice were planning on lying these are exactly the sequences that she might pick with a greater than uniform probabilty out of all the sequences that were not thrown, and therefore each possible actual sequence contributes a higher-than-average amount of probability that Alice would present one of these special sequences, so the fact that Alice informs Bob of such a sequence is weaker evidence for this particular sequence over any other one than it would be in the regular case, and Bob ends up with a lower posterior that the sequence is actually correct.

An analogous question that I encountered recently when buying a powerball lottery ticket just for the heck of it (also because its jackpot was $1.5 billion and the expected value of buying a ticket was actually approaching a positive net reward) :

I was in a rush to get somewhere when I was buying the ticket, so I thought, "instead of trying to pick meaningful numbers, why not just pick something like 1-1-1-1-1-1? Why would that drawing be strictly more improbable than any other random permutations of 6 numbers from 1 to 60, such as 5-23-23-16-37-2? But then the store clerk told me that I could just let the computer pick the numbers on my ticket, so I said "OK."

Picking 1-1-1-1-1-1 SEEMS like you are screwing yourself over and requiring an even more improbable outcome to take place in order to win...but are you REALLY? I don't see how....

I'm sure if 1-1-1-1-1-1 were actually drawn, there would be investigations about whether that drawing was rigged. And if I won with ANY ticket (such as 5-23-23-16-37-2), I would start to wonder whether I was living in a simulation centered around my life experience. But aren't these intuitions going astray? Aren't the probabilities all the same?

It seems to me that either Alice is lying or she is telling the truth. The actual amount of possible lies at her disposal is pretty irrelevant to the question of whether she is lying or not.

For any n coin flips p(sequence) = 1/2^n right?

for 100 coin flips, p(sequence) as a result is 1/2^100 = 7.8886091e-31

you have observed an event that could have gone 2^100 different ways, and found one version of the result. just because you have done something with a specific probability doesn't mean it's a low probability.

The probability of getting a sequence is (pretty much) 1 (given that flipping 100 coins in a thought experiment is pretty safe)

The probability of getting that sequence again is quite low.

Let's divide possible sequences into two broad classes: Distinguished, and undistinguished. Distinguished sequences are those which, for example, are predicted in advance of the coin flips; they have a property which sets them apart from the undistinguished sequences. Undistinguished sequences are all sequences which are isomorphic with respect to the rest of the universe.

All heads is a naturally distinguished sequence; all tails, likewise. Repeating patterns in general. Likewise simple ASCII encodings of binary messages ("This is God, quit flipping that coin").

Once you notice that every sequence can be distinguished somehow (some more readily or naturally than others; for any given input, there's a function that will turn that input into any given output), and that all sequences belong to the "distinguished" set, the odds of getting an interesting-by-some-criteria sequence become 1. Therefore, the mere fact of an interesting quality of a sequence isn't, in itself, interesting, but rather the class of qualities that interesting quality belongs to.

All heads is equivalent to all tails. HTHTHT, repeated as necessary, is equivalent to THTHTH, is lesser interestingness than HHHHHH and TTTTTT. HHTHHTHHT, TTHTTHTTH, HTHHTHHTH, THTTHTTHT, HHHHHHHHH, TTTTTTTTT.

You'll notice that all-heads belongs to multiple sets of interesting qualities; repeated single digit, repeated digit pairs, repeated digit triplets.

There's probably some mathematical language that could be used to describe the "interestingness" of a sequence (I imagine the complexity of the property counts against it?), but I am simply ignorant of it. Depending on the interestingness of the actual sequence Alice shows to Bob, the odds of something of that level of interest occurring should be computable, and may or may not be more likely than some kind of deception or bizarre behavior on the part of the universe.

The probability of getting some head/tails sequence is near 1 (cuz it could land on it's edge). The probability of predicting said sequence beforehand is extremely low.

The probability of someone winning the lottery is X, where X = the % of the possible ticket combinations sold. The probability of you winning the lottery with a particular set of numbers is extremely low.

As far as we can tell, and with the exception of the Old Testament heros, the probability of someone living to be 500 years old is much lower than winning most lotteries or predicting a certain high number of coin flips, though I suppose a smart ass could devise some exceptions to either. We'd have to better define "vampire" to arrive on a probability for that bit.

A house being haunted by real ghosts is actually extremely probable, depending on the neighborhood.

two ways to approach this, depending on which direction Bob takes the argument.

1) Alice seems to be accepting Bob's implication that probability exists in reality, rather than only in the minds and models we use to predict it

A fair bit of recent discussion can be found in this thread, and in the sequences in Probability is in the Mind.

I'd summarize as "the probability of something that happened is 1". There's no variance in whether or not it will occur. If you like, you can add uncertainty about whether you know it happened, but a lot of things approach 1 pretty quickly.

The probability of future experiences will be 1 or 0 at some point, but for now, different agents assign likelihoods based on their priors. The probably that the next sequence of flips will be exactly this is, in the world, 0 or 1 - what will be will be. The probability that an agent can use to predict this experience is 1/2^n. That probability expectation changes as evidence is added, such as the evidence of seeing some or all of the flips.

alternate way of showing this: ask Bob to show his update after each of the N flips. The prior is indeed 1/2^n, but each observation makes it twice as likely.

2) Bob's just wrong if his priors give a significantly higher chance to supernatural physics violations than to winning the lottery. This is probably some form of scope insensitivity. For both ghosts and lottery-winning, I do assign a much higher chance that I'm being tricked than that it actually happened, but if the question comes up, I can gather evidence to change those ratios.

Bob is lumping all events of low probability in the same category, without distinguishing between "not too likely, but still could happen someday" and "ridiculously unlikely, why are you even considering it" events.

Nothing is wrong with this picture -- it's just Bob trolling Alice :-)

People need this

https://en.wikipedia.org/wiki/Axiom

in order to avoid "paralysis" and go further.