The sleeping beauty problem (click) has been discussed in 3 specialized TRF blog posts with the "sleeping beauty" in the title (the first, oldest texts start at the bottom of the search page) and in several vaguely related texts.
Now, economist Brad DeLong – whom I know primarily as the co-author of a bunch of famous economics papers written primarily by Larry Summers – posted another defense of the incorrect result,
Sleeping Bae [sic] Again: Lobos [sic] Motl April Fools Day Comes Early Here [sic] on the Internet Blogging [sic]He has previously written about the problem in November 2014.
Almost everything has been written (on both sides) but I will try to be at least somewhat innovative. First, let us look at the simple sociological stuff.
One should notice how sloppy he is and how many "assorted errors" an economist claiming to understand the probability calculus is doing. His November 2014 was titled "Wednesday Cognitive Science Blogging: What Are the Odds Princeton's David Lewis Understands Probability Properly?": DeLong wasn't able to to find out that the famous Princeton philosopher David Lewis (who wrote a paper about the problem) died in 2001 – and he can no longer defend himself directly, only indirectly through your humble correspondent.
Now, in March 2015, he wasn't able to learn my name. It is Luboš, not Lobos. Speakers of Slavic languages wouldn't make this mistake. Lobos may sound OK to Spanish ears (it means "wolves"; in Chile, they must apparently believe that wolves are close relatives to seals which are known as "lobos del mar") but Luboš is visibly related to the much kinder Slavic (but not Czech) word for "[to] love".
Dyslexia is just one of DeLong's problems. He also claims that I celebrated the April Fools Day almost a "month" earlier. However, he clearly failed to notice that the "offensive" text of mine that someone sent him was written in July 2014 so I was actually 9 months, and not 1 month, ahead of time!
DeLong also seems to be obsessed with affiliations. In November 2014, he asked "What Are the Odds Princeton's David Lewis Understands Probability Properly?" and apparently used late David Lewis' affiliation with Princeton as an argument that Lewis must have been wrong. He did something similar with your humble correspondent:
Reading it, I feel obliged to calculate the probability that there is something very wrong with a Physics Department that would offer him a job...Maybe it's a good idea to start with a simpler exercise before this complex one – what about learning how to calculate the probability that a coin landed heads, Bradford?
Good news – I escaped from all these departments many years ago. But my former affiliation with Harvard etc. (where David Lewis got his PhD) cannot calculate the probabilities from the sleeping beauty problem, and if my or Lewis' affiliations with Princeton and Harvard had to be used as circumstantial evidence, they would increase, and not decrease, the probability that we are right – yes, because Princeton and Harvard are substantially better than other places, including UC Berkeley ;-), so there are reasons to think that a guy chosen to Harvard faculty from 66 candidates may be right even if a loud activist from UC Berkeley wants to believe otherwise.
I hope that all the principles of good manners and aristocratic diplomacy were obeyed in my subtle way to convey the information that DeLong is an overrated mediocre pseudointellectual.
Reminding you of the problem
The sleeping beauty (whose ID won't be capitalized, sorry) is hired for an experiment they will do with her, and told about all the details. They toss a fair coin on Sunday and make her sleep. If the coin lands heads, she will be woken up (and interviewed...) once – sometimes on Monday. If it lands tails, she will be woken twice – on Monday and on Tuesday, and her memory about the Monday awakening will be erased before the Tuesday awakening (a fact she is told in advance, too).
Now, when she is woken up, she should calculate the probability that the coin landed "heads" on Sunday.
The correct answer: 50%
The correct answer is, of course, 1/2. It is a fair coin so on Sunday, the sleeping beauty knows that the odds for heads (=one awakening) are 50%. This knowledge about an event in the past may only change if she is provided with any evidence that discriminates the possible states of the coin, that makes one more likely than the other.
However, the only thing that she learns when she wakes up is that "the number of her awakenings is at least one". The problem is that this is completely unsurprising – both hypotheses, "heads" and "tails", predict that she will be woken at least once, with 100% certainty. So she is getting no new information – it's just like hearing that 1+1=2. Everyone knows that in advance so one gets zero information when we hear it.
For this reason, the odds for "the coin is showing heads" and "the coin is showing tails" stay at 50%. No new information means no change of the probabilities of the hypotheses. You may express this lack of change by equations of Bayes' theorem where \(P(B|A)/P(B)=1/1=1\) simply cancels. If one has to determine the probabilities of the heads-Monday, tails-Monday, and tails-Tuesday combinations (possible arrangements of the state of the coin and the day when she is woken up), the two tails possibilities have to share the 50%, so if Monday and Tuesday are equally likely for tails, the most sensible arrangement is 25% for tails-Monday and 25% for tails-Tuesday.
The probabilities are 50-25-25 for the three arrangements which also allows you to say that the probability is 50/(50+25) = 2/3 that it's "heads" if she's told that it's Monday, 25/(50+25) = 1/3 for "tails" if she's told it's Monday, and 25/25 = 1 = 100% for "tails" if she is told that it is Tuesday.
The most widespread wrong answer: 33.3%
The most popular wrong answer is that the probability of "heads" drops to 1/3 when she's woken up because heads-Monday, tails-Monday, and tails-Tuesday are three indistinguishable (by her), and therefore equally likely, arrangements of the coin state and the day.
Also, if the experiment is repeated many times, 1/3 of the awakenings simply will be "heads" ("heads-Monday") ones which, the "thirders" believe, implies that the probability is equal to this fraction.
However, both of these two arguments in favor of the answer 1/3 are wrong.
The argument that the probability should be 1/3 because "there are three equally good arrangements and 100% must be equally divided between them" is wrong because if there are three options, it simply doesn't mean that they are equally likely. One would need something like the ergodic theorem (clearly not applicable here because different coin_state-time combinations can't be compared and thermalized); or some \(\ZZ_3\) or \(S_3\) symmetry to establish that the three probabilities are equal.
However, no such thermalization and no such symmetry exists here which is why there is absolutely no reason to expect that the three probabilities are equally likely. And indeed, a different, correct argument may be employed to show that they are not equal.
The other argument, based on the "counting of awakenings" over many weeks, is wrong because the ratio cannot be interpreted as the probability due to the sampling bias. If a lie (or unlikely proposition) is repeated 97 times, it doesn't become more likely.
Imagine that the Republican and Democratic presidents are equally likely a priori – which is not so unrealistic. If a Republican president is elected, someone at Fox News says "good news". If a Democratic president is elected, millions of members of moveon.org say "good news". Clearly, the probability that the president is a Republican can't be obtained as the fraction of sentences "good news" that are said by Republican voters over the century – because the Democratic celebrations have been overrepresented and everyone knows so.
The probability remains at 50%. Incidentally, the people who believe that there is a 97% probability of a dangerous climate change are victims of the same Goebbelsian fallacy. If such a claim is repeated 97 times (or in 97 papers) and the opposite claim just 3 times (or in 3 papers), and if you know that it's sensible to expect the first answer to be published much more often (because it produces grants for the authors), you know that the fraction of the papers simply can't be interpreted as the probability of a dangerous climate change.
This global warming example was used just to be specific. Even the people who believe that there is a threat of global warming should be able to agree with the general idea.
All those things have been said many times. But I think that the "bookmaker" argument has been under-discussed by our side, the "halfers". Another popular wording for the basic frequentist argument of the "thirders" is the following:
The answer "the probability of heads is 50%" cannot be correct because this probability would imply that one wins 2 times the price of the lottery ticket and if the sleeping beauty made a 50% bet on "heads" every time she is woken up, she would be losing money because only 1/3 of the awakenings would be "heads" so she would only win on 1/3 of her lottery tickets but she would need to win 1/2 of the tickets for her investment to return.Why isn't this argument right? Because the bookmaker isn't honest, and that's why the numbers calculated from this experimental setup cannot be interpreted as probabilities.
Why do I say that he is not honest?
He is not honest exactly because the sleeping beauty's memory is being erased. This fact implies nothing else than that when the beauty's guess turns out to be wrong (because the coin landed tails), the bookmaker forces the beauty to pay twice on the same match (because the coin state is the same on Monday and on Tuesday), to buy two losing tickets instead of one. She can't see that she's being robbed exactly because her memory was erased but in these "rules of the game", she is clearly being robbed by a dishonest bookmaker and she would know that if her memory were not erased.
Things are hopefully clear but let's reiterate the situation in different words, anyway.
Imagine that her memory is not being erased. In that case, she knows whether she was woken up before or not. So she knows whether it's Monday or Tuesday. If it is Tuesday, the coin must have landed tails (0% heads, 100% tails) because there is no Tuesday interview if it lands heads. It she remembers it is Monday, the probability of "heads" and "tails" are 50% (copied from the fair coin on Sunday) because what is going to happen or not happen later, on Tuesday, clearly can't have an impact on these Monday probabilities.
Note that these four numbers, 0%, 100%, 50%, 50% add up to 200% and not 100% because they are conditional probabilities and the union of all these four options covers "all alternatives that may occur on two days", but only the probabilities of mutually exclusive options on the same day have a reason to add up to 100%.
So these are the probabilities without the memory erasure. The role of the bookmaker would be obvious: the fair lottery tickets would have to offer prices that depend on the day. On Monday, both bets – heads or tails – would have to win you twice what you paid. On Tuesday, there is no uncertainty so it's a complete ripoff to sell heads tickets on Tuesday.
But if she is made forgotten about the Monday interview and sold another ticket on Tuesday even though the "result of the Monday match" is already known, then the bookmaker isn't acting in the honest way that is needed to identify the numbers in this thought experiment with the probabilities.
The problem is the asymmetric information: on Tuesday, the bookmaker knows that it is Tuesday, and because they're going to wake up the beauty anyway, he inevitably knows that the coin landed tails. That's why it's dishonest for him to offer the same odds on Tuesday as he did on Monday. It's only fair to sell lottery tickets (for a nonzero price) that have a (nonzero) chance to be winning. But on Tuesday, the "heads" lottery tickets are guaranteed to be losing, so selling them (for a nonzero price) as something that quantifies the uncertainty that existed on Sunday and Monday is a ripoff.
Of course that if a clerk knows that thousands of lottery tickets can't be winning, it's financially beneficial for him to sell them for the usual price – instead of opening them himself, for example. By this asymmetric knowledge, he is robbing the consumers.
Many people are employing the analogous tricks as the dishonest bookmaker above. For example, to return to the climate change discussion, when the weather is warm and/or agreeing with some global warming predictions, you read many newspapers articles about it, and if the weather is cold and/or refutes the global warming predictions, you read almost nothing (or the relationship to the climate isn't mentioned). Because of this selective "rate of publication", it is obvious that the ratio of newspaper articles can't be interpreted as the odds. The journalists are deliberately introducing the bias and the bias is why the texts they offer can't be considered representative of the underlying probabilities and it's a trivial mistake to mindlessly interpret these ratios as probabilities.
Similarly, the number of schools that are worse than Princeton and Harvard is greater than their number – i.e. greater than two – but that doesn't necessarily mean that the beliefs that are widespread at the subpar schools such as the Marxist den at UC Berkeley are more likely to be correct than the views believed and conclusions reached at Princeton, Harvard, and in Pilsen.
Finally, I must mention that thankfully, fewer than 1/3 DeLong's readers actually support his wrong answer 1/3.
An off-topic comment, only related because the two economists know each other.
Ireland's GDP up 5% in a year
The Celtic Tiger we knew between 1995 and 2000 is back. In 2014, the GDP grew by 4.8%, clearly the fastest rate in Europe, and the GNP by 5.2%. Austerity apparently didn't kill them. They're the role models (not only!) for countries like Greece but this fact is unpopular because there are too many lazy, Syriza-like folks in Greece and many other countries. Not to mention folks like Paul Krugman, the head of Syriza's fifth column in the United States who has been bashing Ireland for years but now he is bound to be silent because he is a dishonest ideologue who is working to make the Western civilization rot away 7 days a week and the real-world data aren't convenient for him.