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Julia Galef and the Sleeping Beauty Problem

Julia Galef is a statistician and the president plus co-founder of the Center for Applied Rationality in San Francisco, an entity that fights against biases as well as paranormal phenomena. Two months ago, she posted her most popular YouTube video so far (over 60,000 views), one about the Sleeping Beauty Problem.

The 4-minute video seems very comprehensible, concise, and powered by some beautiful cartoons. She presents both major answers to the Sleeping Beauty Problem that has earned about 5 previous TRF blog posts.

Recall that in this controversial exercise, the sleeping beauty is put to sleep on Sunday and a coin is tossed. If it lands "heads", she is woken once – on Monday. If it lands "tails", she is woken twice – on Monday and on Tuesday – and after the Monday awakening, she's made to forget about it. So the three possible awakenings in the two possible histories look "indistinguishable" from her viewpoint.

The question is what probability she should assign to "the coin landed heads" when she's woken up. The correct answer is \(P=1/2\) because that's the probability for the fair coin. That was the probability of "heads" on Sunday and because she learned nothing new when she woke up – she was guaranteed to wake up at least once, regardless of the coin, and having woken up at least once is the only data she observes – the update to her probabilities and beliefs is trivial (the change of the probabilities is zero) and the "heads" vs "tails" are still 50:50.

The most popular wrong answer is \(P=1/3\). There are three possible combinations of the coin and the day of the week, "Monday-heads", "Monday-tails", and "Tuesday-tails". They are equally likely (???) so each of them has \(P=1/3\).

She presents the two possibilities as equally good – exactly in the way you expect from a very good and articulate female communicator and a diplomat. At the end, I think it's obvious from her words that she prefers the wrong answer \(P=1/3\).

The two answers look "equally good" but there is a clear difference. Those who say \(P=1/2\), the "halfers", may say what's wrong with the reasoning of the "thirders" who say \(P=1/3\). In particular, it is not true that the three possible coin-day arrangements are equally likely. Already on Sunday, all "heads" possibilities are being assigned just \(P=1/2\) which they may share and the "tails" possibilities are assigned the other \(P=1/2\). On the other hand, the "thirders" can't meaningfully describe any bug in the obviously correct reasoning of the "halfers". No new evidence means no change of the beliefs.

There exist various ways to describe the fallacy committed by the "thirders". For example, they like to run the experiment for many weeks and note that \(1/3\) of the awakenings in this period were "Monday-tails" awakenings. But that fraction cannot be interpreted as the probability of a property of the coin (and nothing else). It's a fraction of measurements that would have produced "heads". The probability that a measurement chosen from a multi-week sample yields "heads" is \(1/3\). But note that to get \(1/3\), one has to talk about the probability of something whose description refers to the measurement or awakening, not just to the coin itself! The problem is that the awakenings or "measurements" are designed so that you make twice as many measurements if the coin is showing "tails" than if it is showing "heads". So the counting of these measurements is biased – distorted by the so-called sampling bias. The fractions of the measurements can't be directly interpreted as the information about the coin only. And the question in the Sleeping Beauty Problem was unambiguously about the probability of the coin state only!

If you repeat one of the two possible answers twice, it just doesn't make it more likely! If you organize two celebrations after a Democratic Party candidate wins the presidency and only one celebration if the Republican wins, it doesn't mean that the Democrats become twice as likely to win!

With no justification, the "thirders" claim that certain possibilities that are not equally likely as others are equally likely. And they directly interpret the fractions in certain biased samples as probabilities which is clearly a fallacy, too. (It is pretty much the same fallacy as to seriously believe Göbbels' claim that "a lie repeated 100 times becomes the truth".) The same holds for their arguments involving bets. They simply promise a bigger total amount of money in the case of "tails" (because she gets the money twice) and incorrectly interpret this higher amount of money as a higher probability of the "tails" option.

If you're promised a trillion of dollars in the lottery if your guess is right and if it is 1-2-3-4-5-6, it may be a good idea to make a bet on 1-2-3-4-5-6. But the reason why it is a good idea is that you try to maximize the profit and the profit has a higher value for 1-2-3-4-5-6. The reason is not that 1-2-3-4-5-6 became more likely to be the result! The probability has not increased. It's not the probability \(p_i\) but the other factor in \[

\langle {\rm Profit} \rangle = \sum_i p_i \cdot {\rm Profit}_i,

\] namely \({\rm Profit}_i\), that became larger for \(i=(1,2,3,4,5,6)\).

Strangely enough, she also quotes the generalization with thousands of awakenings as a modification of the thought experiment that makes \(P=1/2\) look "more absurd". That's bizarre because I have used exactly this extreme version of the thought experiment to show that the \(P=1/3\) reasoning becomes more absurd!

By the way, her monologue also correctly says that the thought experiment has important relationships to the anthropic principle. And indeed, the \(P=1/3\) result arises from the same incorrect reasoning that is often used by the people using the so-called anthropic principle. The "thirders" are a similar group of people who also commit the "typicality fallacy" in the context of the anthropic reasoning.

While the video sounds "fair" at the sociological level, it is "fair" because it gives the same room to the correct answer and the wrong one (and perhaps, the wrong answer is a bit oversold in comparison with the correct one). But she sells another wrong idea – not directly related to the "fight between the two answers" – in the comments under the video.

Julia Galef: I can see that a common response is, "Philosophers don't understand math, it's clearly 1/2."

Sigh. Perhaps I should have clarified that mathematicians are divided on this puzzle, too, not just philosophers. For example, here are several mathematicians and physicists arguing the 1/3 position:

1. Nick Wedd is an International Math Olympiad winner, and a thirder:
2. Jeffrey Rosenthal, winner of the "Nobel prize of statistics", is a thirder:
3. Another International Math Olympiad winner, Tanya Khovanova, is a thirder:
4. Physicist Sean Carroll is a thirder (but agrees it's controversial):
Sorry but this is an ad hominem argument that in no way disproves the "common response". It is true that the result \(P=1/3\) is tightly linked to someone's being a philosopher.

The claim that this is a problem dividing mathematicians is highly misleading especially because there obviously doesn't exist any "truly rigorous mathematical formulation" of the problem that could be controversial.

As long as one defines the notion of probabilities so that they are fully well-defined and applicable to problems like this one, it's obviously possible to settle the question what the probability is. The technical portion of the problem is extremely simple. As long as the rules of the game are clearly stated, the correct solution is clear, too!

The real controversy is about "how we should actually think", what we should believe is likely or not likely. A necessary condition for a controversy to exist is that the probability of "heads" is neither 0% nor 100%; it is strictly in between. But pretty much by definition, mathematicians as experts stay silent in all such intermediate situations. A statement has either been proven, or disproven, or neither of those two things. And we are dealing with the third option, indeed. Mathematicians may be biased in the situations where the proof doesn't exist in either way but they realize that such a bias or asymmetry is their personal belief and not something they may tightly link to their expertise. Mathematics is the ultimate rigorous business where emotions don't play a role when disputes are being settled.

It's great she may enumerate three winners of math-related prizes and one third-class crackpot cosmologist as examples of people who prefer the wrong answer but whatever the question is, one may find some influential people who give wrong answers. This very fact doesn't prove anything and one should avoid this kind of "evidence".

Even if we talked about the sociological questions only, I am absolutely confident that the percentage of the wrong answer \(P=1/3\) among philosophers is much higher than the percentage of \(P=1/3\) among technically skilled mathematicians so the very promotion of \(P=1/3\) as "at least one of the two equally plausible answers" (if not something better!) is a part of the movement promoting the misconceptions and fallacies of a group of people who are not as good as experts as the more relevant experts, namely mathematicians and natural scientists who actually need to solve similar (and much more difficult) problems all the time.

Incidentally, in her URLs, she's overselling the "thirder" responses just like the thirders in general are overselling the "tails" answer to the question about the coin! ;-) You may see that her first URL shows the dominance of the "halfers" relatively to the thirders approximately in the "expected" 3:2 ratio. ;-)

The Rosenthal text is an isolated text by a "thirder" but the remaining two pages contain lots of responses that are often from "peers" of the thirder authors and those peers usually argue in favor of \(P=1/2\). So I think that these sociological arguments in favor of answers should better be avoided but if someone is trying to build on them, he or she should do it honestly. Neither the technical arguments nor the sociology among non-philosophers favors \(P=1/3\).

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