It's no longer enough to have an April Fools Day.

It's no longer enough to have an April Fools Week.

It looks like we need an April Fool's Month. I do know that the question is not whether to laugh or cry, but rather of how much to do of both.

A correspondent--I presume who wishes me ill--informs me--probably because I mentioned @seanmcarroll in a tweet earlier today--that Lubos Motl, often wrong but never in doubt, has been opining on "Sleeping Bae", which he analyzes about as well as a fish would analyze Lorentz invariance.

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...

Let's review the state of play:

"Sleeping Bae" goes to sleep Sunday night. At some point a fair coin is flipped. If it is heads, they will awaken him only on Monday, after which he will sleep until Wednesday. If it is tails, they will awaken him on both Monday and Tuesday--but after awakening him on Monday they will give him a drug that will disrupt his short term memory, so that when awakened on Tuesday he will not remember his Monday experience. When awakened and asked "what is your confidence that the coin flip winds up heads?", what answer should he give? If he is then told "it is Monday; what is your confidence that the coin flip winds up heads?", what answer should he give?

"Thirders" say that he should answer "1/3" when awakened, and "1/2" after being told it is Monday. If you run the experiment over and over again, 1/3 of the correct answers to the first question are "heads", and 1/2 of the correct answers to the second question are heads. You make money if you bet on heads as the answer to the first question at better than 2-1 odds, and if you bet on heads as the answer to the second questions at better than 1-1 odds. Sensible probabilities should converge in the long run to sample averages, and should guide you to betting decisions that do not lose money.

"Double-Halfers" say that he should answer "the odds are 1/2 that the coin came up heads and it is Monday, and 1/2 that it is heads and I will or have answered this question twice" when awakened, and "1/2" after being told it is Monday--for the definition of a fair coin is that its probability of coming heads is 1/2. Never mind that betting on heads as your answer to the first question will lose you money, and that only 1/3 of the occasions on which the first question is asked will the right answer be "heads".

"Lewis-Halfers" say he should answer "1/2" when awakened, and "2/3" after being told it is Monday. Why? Because they do not understand how probabilities work, or how to properly apply the Rule of the Reverend Thomas Bayes.

In short: Thirders are correct, Double Halfers are somewhat confused, Lewis-Halfers are simply are wrong.. ([Click here if you want the detailed argument first.][#detailed])

Which of those do you think Lubos Motl is?

Let's raise the curtain!

**:
The Reference Frame: The sleeping beauty problem: "Sean Carroll's blog posts are getting stupider and stupider....**

...A sleeping forgetful lady in it. Miss Czechia 2012 Tereza Fajksová in Paris. How many legs does the sleeping beauty in a Škoda car have? I find the obsession of similar (Czech) women with getting the suntan excessive.... Carroll--along with some stupid philosophers--gives a completely wrong answer to a very trivial problem. What is it?

On Sunday, a blonde is told that they would flip a coin once (and never again).... They will play with her.... The games will include no sex.... What should the babe say? Well, it's an intelligent blonde who was previously told that the probability for the coin to come up heads was 50%. So she answers that she would assign the probability 50% that the coin came up heads.... A smart blonde. Unlike Sean "Dumber Than a Blonde" Carroll whose answer is 33.3%. No kidding! ;-).... This is a puzzle I would correctly solve when I was 7 years old. Why is it so hard for Sean Carroll who is an adult?

The probability that a coin comes up heads is: Pheads=Nheads/Nflips, i.e. the ratio of coin flips (in repetitions of the same situation) that come up tails and the number of all flips. It's clearly 50% in the problem we were explained above. We were told that it is so. That's how the problem was defined. To answer correctly, we just need to repeat one sentence we were explicitly told!

It is very obvious where his retarded childish mistake comes from.... The probability that a coin comes up tails has nothing whatsoever to do with some number of some interviews.... The behavior of a coin doesn't depend on what someone does with a woman's brain. The ratio that leads to the "result" P=1/3 isn't even a "probability" because the "units" that are being added in the numerator as well as the denominator aren't even "outcomes in mutually different repetitions of the history".... The probability that a randomly chosen interview occurred after "tails" may be 33.3% but that's an a priori different question and a possibly different probability than the question about the probability that the coin came up tails...

Incidentally, the probability that a random interview occurred after "heads" is still equal to 50%! The result "tails" may lead to more words and interviews but its probability is still 50%....

For Sean Carroll, however, it's too difficult. What a moron.... He isn't able to solve this trivial problem for the schoolkids.... I guess that many of the "physicists" who talk about the "many worlds" would do the same mistakes as Wagner and Carroll. It's insane that Carroll--and arguably many others--are pretending to do cutting-edge science by presenting completely wrong solutions to some problems that intelligent enough schoolkids can solve correctly....

Most of the papers about the problem also discuss the value of P

_{m}(HEADS), the credence that the coin is "heads" right after she is told on Monday that it is Monday--and let's assume that the rules now include the comment that she is always told it is Monday on Monday evening. The original "thirder" Adam Elga would say that P_{m}(HEADS)=1/2.... However, the correct answer is obviously P_{m}(HEADS)=2/3 exactly because in this case, we are incorporating new information (about "Monday") by Bayesian inference....I've concluded that the elementary mistakes that lead some people to say that the correct answer to the Sleeping Beauty Problem is P=1/3 is the primary cause of these people's totally invalid claims about the arrow of time as well as the anthropic principle as well as their irrational fear of the Boltzmann Brains....

The people who want to defend P(HEADS)=1/3 like to imagine that one throws some marbles or fruits into a jar whenever she is woken up.... After a long time, the number of heads-fruits in the jar will exceed the number of tails-fruits by a factor of two which they interpret by saying that the probability that she wakes up after "heads" is twice as high. They completely miss the obvious problem, the sampling bias. The fruits are more likely to be thrown to the jar if they're tails-fruits.... It's a bias, a mistake, and it has to be removed if you want to determine the actual probabilities of the underlying phenomenon--the coin, in this case. She is asked about the probability of a statement about the coin, not about fruits, so if she finds fruits helpful to answer a question about the coin, she must do it correctly....

Let me write the full table of the conditional probabilities.... Monday heads 1/2, Monday tails 1/2, Tuesday tails 1/2... These are the correct probabilities after you normalize them.... The probability of "heads" is P=1/2 while the two wakeup days share the remaining P=1/2 reserved for the "tails".... The hypothesis "tails Monday" (just like "tail Tuesday") predicts that there is a wakeup both on Monday and Tuesday--these two wakeups are logically connected, not mutually exclusive.... Both "Monday tails" and "Tuesday tails" hypotheses share the possibility that she wakes up and t=PWI=Monday or Tuesday, so these predicted conditional probabilities are 50%.

Let me get to Gitmo.... With the probability 1/5,000, i.e. approximately once in 100 years, the Sleeping Beauty is accused by CIA of having drunk the Siberian Crown, a Russian beer, with Agent Mulder. The CIA Heads Office therefore storms the building and transfers the woman to Guantánamo Bay. They don't wake her on Monday, they don't wake her on Tuesday, but they wake her up on Wednesday. 50,000 times, approximately once per second.... A vast majority of the wakeup incidents will be wakeups by the CIA....

Imagine that you are in the position of the Sleeping Beauty and you will wake up sometime in the week. Do you really believe that it follows that you have probably been captured by the CIA?... If you are not a conspiracy nut and a whackadoodle... the fact that you are woken up 50,000 times can't really change this fact. These 50,000 wakeups just share a predetermined, reserved, "small piece of the probability pie".... You can't inflate the likelihood of a (conspiracy) theory that looks extremely unlikely by decorating the (conspiracy) theory with lots of big numbers what happens when it happens.... This "fallacy of the Thirders" is an important driver behind people's beliefs in the Boltzmann Brains, claims that statistical physics doesn't explain the arrow of time, and even global warming.

The Boltzmann Brains (or at least their worshipers) think that just by inventing quadrillions of people who live "somewhere" and who are similar to you implies that it becomes likely that you are one of them. But it doesn't because there is no real "democracy" or "equivalence" between you and random elements of that ensemble. Equal probabilities are only justifiable if there exists an argument, e.g. symmetry argument. But we're qualitatively different--at least by our environment--from the Boltzmann Brains so there is no reason why the two theories about our location should be equally likely....

For a positive example of the symmetry, in the Sleeping Beauty Problem, there is a symmetry ℤ2 between the two sides of the coin, and that's why these two sides are equally likely. Similarly, in the "heads" case, there is a symmetry between Monday and Tuesday, which is why these two days are equally likely assuming "heads". When combined, the probabilities are 50%, 25%, 25% for the three arrangements. Just because the three combinations look similar, the "thirders" assume a non-existing ℤ3 or S3 symmetry between "Monday heads", "Tuesday heads", and "Monday tails" to claim that the probabilities are 1/3. But there is no symmetry which is why there is no reason to think that the probabilities are the same....

I added the "global warming" believers.... Some people think that they can make the ludicrous global warming fears "more important" by inventing a "bigger impact" of the global warming catastrophe.... People whose thinking is overwhelmed by possible events that occur less than once in a trillion of years are nuts. The other reason why I added the "global warming" believers is that many of them love sampling bias.

If you ask Michael Mann and dozens of his "peers" about "global warming" many times, you may find out that 97% of the answers support the "global warming" fears. But that doesn't mean that these fears are justified. It doesn't even mean that most of the competent people share the fears (which would still not be sufficient to conclude that the fears are justified). You have been just asking people in a biased sample. Even if you count people who write into some standardized climate science journals, they are still a heavily biased sample--and they aren't really among the most competent people on the planet to discuss such matters rationally or scientifically....

So the "thirders' fallacy" may be the pseudointellectual Urquell of many kinds of conspiracy theories by various types of whackadoodles who believe that the arguments that "something is extremely unlikely" may be "beaten" (and perhaps, the probabilities themselves may be rewritten and inflated) by inventing far-reaching fantasies "how important it would be if the unlikely thing were true" or simply by repeating (or forcing you to observe) one picked answer many times....

The belief in P=1/3 is equivalent to a major flaw in thinking that turns many people into conspiracy theorists: they think that even if some scenario is very unlikely (e.g. that the woman is transferred to the facility in Cuba), the low value of the probability may be "beaten" and made irrelevant by inventing "huge implications" such as a very long sequence of torture and interviews....

The Bayesian inference implies Ptails=1/2 if you do it right even if you include the "hypotheses about your state" among the competing hypotheses. "Doing it right" involves realizing that "Monday tails" and "Tuesday tails" aren't really mutually exclusive possibilities.... When she wakes up, she is getting no nontrivial information that would reduce or increase the odds of "heads" or "tails"--because both hypotheses imply that she would be woken up at least once, and that's the only thing she is observing. Because she is learning nothing new, her subjective probabilities of "heads" and "tails" have to be the same as they were on Sunday, namely 1/2 and 1/2....

This assumption of "mutual exclusiveness" of the evolutions--in our case, the mutual exclusiveness of "Monday tails", "Monday heads", "Tuesday tails"--is clearly wrong.... What's the probability that "Sunday" evolves to "Monday tails" or "Tuesday tails"?... Both terms... are equal to 1/2 because we know that the coin must end up "heads". You see that the sum of these two probabilities--and the thirders had to use the sum because they incorrectly assumed "Monday heads" and "Tuesday heads" to be mutually exclusive--is already 2×1/2=1, so it should mean that there is a certainty that one gets heads. For the evolution to "Monday tails" (in one step), there is only one term that is again equal to 1/2. That's twice smaller than the sum P=1 we got for "heads" which is why thirders ultimately incorrectly assign the probability Ptails=1/3 to the "tails" and the remaining Pheads=2/3 to the "heads".

However, you see that the intermediate results giving this answer really involve the assumption that the evolution from "Sunday" to "any tails" is equal to one--although there's surely no certainty that you get "tails". This is obviously a manifestation of the fact that their probabilities do not sum to one. Their sum exceeds one. I have mentioned that the thirders' invalid assumption of the mutual exclusiveness of different "days" is mathematically expressed by the wrong equation.... We are not allowed to sum over all values of n, the number of steps, because this sum yields the "total probability" for all options to be greater than one, as we have repeatedly said. Instead, the right ways to calculate n-blind probabilities of different states is to compute a weighted average of the probabilities assuming a certain value of n. The weights aren't determined in general but if there is no difference between the "Monday tails" and "Tuesday tails" interviews, we may choose the weights to be equal to each other i.e. equal to 1/2....

You may see that the probabilities of A "or" B are sometimes the sums and sometimes the (weighted) averages. This is very analogous to (and in fact, a special case of) the point made in many blog posts about the second law of thermodynamics: probabilities for the evolution of "one ensemble" to "another ensemble" are summed over final microstates but averaged over the initial microstates. Because n, the number of steps in a Markov chain that have led to the present state, is a part of the information about the past, it follows the rules of the past so we must average over different values of n and not sum! More generally, thirders are the people who would always like to sum probabilities and interpret the sum as a probability--while not caring whether the sum exceeds one.... It's an error leading (or at least helping to lead) various people to say lots of dumb things about "problems with the second law of thermodynamics", "many worlds of quantum mechanics", "anthropic principle", or--in the case of the most unhinged whackadoodles – "Boltzmann Brains"...

**The Detailed Context:**

**: Sleeping Beauty: Reply to Elga: "Researchers at the Experimental Philosophy Laboratory... Sleeping Beauty...**

...On Sunday evening they will put her to sleep [P

_{s}]. On Monday they will awaken her briefly. At first they will not tell her what day it is [P], but later they will tell her that it is Monday [P_{m}]. Then they will subject her to memory erasure. Perhaps they will again awaken her briefly on Tuesday... depend[ing] on the toss of a fair coin: if heads they will awaken her only on Monday, if tails they will awaken her on Tuesday as well.... Let P be her credence function just after she is awakened on Monday [uncertain whether it is Monday or Tuesday]. Let P_{m}be her credence function just after she’s told that it’s Monday. Let P_{s}be her credence function just before she’s put to sleep on Sunday. [And let P_{t}be her credence function if awakened on Tuesday after being told that it is Tuesday]...

There is a well-known principle which says that credences about future chance events should equal the known chances.... I reply that the principle requires a proviso.... Imagine that there is a prophet whose extraordinary record of success forces us to take seriously the hypothesis that he is getting news from the future by means of some sort of backward causation. Seldom does the prophet tell us outright what will happen, but often he advises us what our credences about the outcome should be, and sometimes his advice disagrees with what we would get by setting our credences equal to the known chances. What should we do? If the prophet’s success record is good enough, I say we should take the prophet’s advice and disregard the known chances.

Now when Beauty is told during her Monday awakening that it’s Monday, or equivalently not-T2, she is getting evidence--centred evidence--about the future: namely that she is not now in it. That’s new evidence: before she was told that it was Monday, she did not yet have it. To be sure, she is not getting this new evidence from a prophet or by way of backward causation, but neither is she getting it just by setting her credences equal to the known chances. The news is relevant to HEADS, since it raises her credence in it by 1/6; see my (L7). Elga agrees; see his (E6). Therefore the proviso applies, and we cannot rely on it that P+(HEADS) = chance(HEADS) and P+(TAILS) = chance(TAILS).

_{m}(HEADS)=1/2, that P

_{t}(HEADS)=0, and that if she wants to win money her credence, her belief, about P(HEADS) should be such as to lead her to bet like a thirder--and thus act as if P(HEADS)=1/3--is completely consistent, intelligible, and sane.

Less intelligible than the Lewis-Halfers is Robert Walters's Double-Halfer position:

**: The sleeping beauty problem: how some philosphers and physicists calculate probability: "The three probabilities discussed above are...**

(i) the probability p1 of going from [Sunday to Monday HEADS]... (ii) the probability [p3 of going from [Sunday to Monday TAILS]... and (iii) the probability p4 of going from [Sunday to Monday TAILS] in two steps. Each of these three values is clearly 1/2. What however should the beauty calculate for the probability p3,4 of being either in 3 [on Monday] or 4 [on Tuesday]? The philosophers/physicists are claiming that p3,4=2p1. I claim p3,4=p1...

What Walters says (I think: he is confused) is that Sleeping Beauty should not say P(HEADS) = 1/2 or P(HEADS) = 1/3 but, rather:

The probability that the coin came up heads is 1/2. It is either heads, Monday, and this is my only awakening; or it is tails, either Monday or Tuesday, and this is one of my two awakenings...

And then, if being told it is Monday, Double-Halfer Sleeping Beauty gives the perfectly correct P_{m}(HEADS)=1/2.

And then, if being told it is Tuesday, Double-Halfer--Sleeping Beauty gives the perfectly correct P_{t}(HEADS)=0.

But when pressed--while still ignorant of the day of the week--with the question: "You don't know whether it's heads or tails, you don't know whether it's Monday or Tuesday, but you do know that it isn't Tuesday-Heads. What are your odds, respectively, that it is Monday-Heads, Monday-Tails, and Tuesday-Tails?" all Double-Halfer Sleeping Beauty can do is filibuster:

Even though the probability that the coin is tails is 1/2, and even though I don't know whether it is Monday or Tuesday, that does not mean that the probability it is Monday-tails is 1/4 and the probability that it is Tuesday-tails is 1/4.

Double-Halfer Sleeping Beauty gives those answers because her goal is not to *give the right answer to the question* or *win money by betting* but rather to *be right about the state of the world*. In the Monday world, because it is pre-coin flip, the answer she should give *if she wants to be right about the state of the world* to P(HEADS) is 1/2. In the Tuesday world, because she is not awakened if the coin is heads, the answer she should give *if she wants to be right* to P(Heads) is 0. Double-Halfer Sleeping Beauty is right about the probabilities *all the time* on Monday, and wrong on Tuesday. Thirder Sleeping Beauty is wrong about the probabilities all the time: on Monday they are not 1/3 but 1/2, and on Tuesday they are not 1/3 but 0). But Double-Halfer Sleeping Beauty loses if she bets on her probabities.

By contrast, the Thirder gives coherent and consistent answers: precisely because the day is uncertain and the coin is correlated with the day, the probability the coin was or will be heads is 1/3; the probability that it is Tuesday (and tails) is 1/3; and the probability that it is Monday-Tails is 1/3.

The problem with Walters--and here I suspect the problem is that he did not read Elga's paper sufficiently caerfully--is that Sleeping Beauty is not asked "what is the probability": she is not asked about a parameter describing the world. In fact, the word "probability" does not appear in Elga's initial paper. The words "belief" and "credence" do. The phrase "decision theory" does. She is asked not to nail the value of a parameter describing the world, but rather to form rational beliefs in order to guide her actions.

It matters.

Double-Halfers get confused because they want to reserve "probability" for cases in which you are unsure which possible world you are in. They do not think "probability" applies to cases in which you are unsure where you are in one particular possible world. You can perhaps restrict the meaning of "probability" to use it in such a way. But you cannot reserve "belief" or "credence" in that way.

Whenever you are awakened, the probability that the coin falls heads and you are being awakened once is one-half; and the probability that the coin falls tails, that you are being awakened twice, and that this is one of those two awakenings is one-half. That is why the Double-Halfers are confused. But are they definitively wrong? Not quite.

Are they right? No. For Elga asks:

When

youare awakened...

that is, at that specific moment...

...to what degree ought you believe...

that is, not what are the objective probabilities of various possible worlds that might have happened, but what does it make sense for you to subjectively think are the odds...

...that the outcome of the coin toss is Heads?

And the answer to this question is crystal clear.

When you are awakened you do not know the date. It might be Monday, in which case P_{m}(HEADS)=1/2. It might be Tuesday, in which case P_{t}(HEADS)=0. Since experimental subjects are awakened twice as often on Monday as on Tuesday Tuesday, it is twice as likely that you are in a Monday as in a Tuesday Tuesday. So even though it is a fair coin, for Sleeping Beauty P(HEADS) = (2/3)P_{m}(HEADS) + (1/3)P_{t}(HEADS) = 1/3.

_{m}(HEADS)=2/3--is:Let us be very clear where the Lewis-Halfers have gotten themselves here on P

_{m}:(5), (6), and (7) are the only things relevant to the coin's probabilities (that the coin is fair, and about to be flipped) and to the future environment (that her memory is about to be disrupted, and if it is tails she will be awakened on Tuesday).

(1) through (4) have absolutely nothing to do--neither as cause, nor as effect--with how the coin flip comes out.How the hell could anybody ever think that (5), (6), and (7) together imply not P

_{m}(HEADS) = 1/2 but P_{m}(HEADS) = 2/3?Lewis-Halfers provide no answer, but only incoherent word-salad: