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Sleeping beauty: attempts for peace between thirders and halfers are inconsistent

At the end of March, the Quanta Magazine, by its article

Why Sleeping Beauty Is Lost in Time (by Pradeep Mutalik),
has joined those who argue that both widespread answers to the Sleeping Beauty problem, \(P=1/2\) and \(P=1/3\), are correct and they answer two different questions. The halfers (such as your humble correspondent) do care about the "background" of events while the thirders (which includes fans of the radical anthropic principle and Boltzmann Brains) don't.

Such a reconciliation could sound "nice" except for one problem. The "framework" that implies \(P=1/3\) just can't be turned into a logically consistent version of the probability calculus.

Just to remind you. In the problem, a coin is tossed on Sunday. If it lands Heads or Tails, respectively, the Sleeping Beauty is awaken once (Monday) or twice (Monday, Tuesday), respectively, for an interview. During the interview, she's asked about her subjective probability that the coin is just showing heads.

You may remember that heads is "1 awakening" while tails is "2 awakenings" by internalizing Czechia's coat of arms. The double-tailed lion has one head and two tails. ;-)

The correct answer \(P=1/2\) is the solution to the original puzzle while \(P=1/3\) may be said to be an answer to a slightly different question:
If you were asked about the most popular incorrect answer, what would you answer? Or: If you believed that a lie repeated 100 times becomes the truth and you don't need to worry about this sampling bias, what probability would you assign? If you believe that two answers should be assumed to be equally likely even though there's no valid argument for such a conclusion, what would you answer? What is the most convincing silly answer if you assume that probabilities don't have to add up to one? Or that probabilities of statements about "now" may be influenced by events in the future? Or that probabilities may suddenly jump even though nothing changes about the things you know? Or anything similarly stupid?
And so on. In all cases, there is something wrong about the reasoning that makes people believe that \(P=1/3\).

If you read Mutalik's essay, you will find the actual #1 reason that makes him think that both answers must be "correct" in some framework or interpretation:
It isn't possible that so many people are wrong for such a long time.
However, this reasoning is incorrect. As we can observe in numerous other situations, the human stupidity is very widespread, it can make many people very stubborn, and it is often highly persistent, too.

Even though I don't believe that there exists any logically consistent "version of probability calculus" that would justify the answer \(P=1/3\), I do agree with Mutalik's description of a key difference between the thinking of the halfers and thirders. Halfers (like myself) always care about the "background" of an event, e.g. the event of the coin toss. Thirders try to "count" heads if not Boltzmann Brains regardless of their evolution.

The reason why the halfers' attitude is pretty much necessary for any logically consistent treatment of probababilities was implicitly described in a previous blog post Sleeping beauty thirders' rudimentary Markov chain error partly dedicated to an argument by fellow halfer Bob Walters.

A key point is that whenever we say that "some property \(A\) holds for/in the region of spacetime we just inhabit", this probability \(P(A)\) must always be identified with the probability of the evolution\[

P(BB\to A)

\] from the "Big Bang" (which is my somewhat arbitrary name for the "last event we can be 100% certain about") to the present. The evolution operator also depends on the information about how much time the Universe evolves and where the region where \(A\) is evaluated sits.

But the implicit assumption that there's some "universal initial state" is necessary because it's the fact that\[

P(BB) = 1

\] that is the source of the rule that the sum of the probabilities of all the mutually exclusive outcomes must always be 100%. This basic rule needed for the consistency of probabilities always arises because some "certainty about a moment in the past" evolves into an "uncertainty" at a later time. The probabilities are always slices of a "pie" that had to be baked before we could look at the pieces.

The laws of Nature – including e.g. the law that a fair coin lands heads-or-tails at 50-50 odds – are the primary laws that produce well-defined numbers, and those are always probabilities of an earlier state's evolution to a later state. On the other hand, all of our "reconstructions" of the past are examples of a "reverse engineering" that requires Bayesian inference. And there can't be any additional "simply yet quantitatively exact" laws that could allow you to assign probabilities of an earlier state from the knowledge of a later state.

The thirders generally ignore the logical arrow of time, in one way or another, which "allows" them to end up with all the "wonderful" answers such as \(P=1/3\) or the claim that we should be afraid of being Boltzmann Brains.

In the correct, halfers' probability calculus, it's obvious why \(P=1/2\). There was the Big Bang whose probability was 100%. Afterwards, a coin was tossed and the 100% pie was divided to two slices, heads (50%) and tails (50%). When the Sleeping Beauty is woken up, she doesn't learn any new information – it was guaranteed that she would be awakened at least once, and that's the only thing that she observes – so these 50%-50% slices remain. At most, she may ask what is the day assuming that the coin is showing tails. She may split the slice 50% for tails to two smaller slices, in the symmetric case 25% for tails-Monday and 25% for tails-Tuesday.

But only a 50% slice was "reserved" for tails when the coin was tossed and she can't "inflate" the size of this slice proportionally to the number of awakenings because the awakenings occurred after the coin toss. For her, on Monday or Tuesday, to change the 50-to-50 division of the pie that took place on Sunday would mean to change her past. But one simply cannot change the past! So whatever details about the experiment (the day that the Sleeping Beauty experiences) are asked after the coin toss simply has to work with the pieces of the pie that were previously reserved for "tails".

One may compute more complicated probabilities that are ratios whose numerators as well as denominators are smaller-than-100-percent pieces of the original pie, 100%. But one can never assume that the pie grows bigger than 100%. Thirders constantly commit this fallacy because they basically assume that they have 300% of a pie to work with if there are three possible (coin_state, day) combinations. You just can't ever start with a 300% total probability. One may estimate probabilities by a measurement, in the frequentist way, as \(N/N_{\rm total}\), but these are always just measurements of something that still require the knowledge of the laws of Nature to deduce something interesting – and the laws of Nature always have 100% as the denominator for probabilities.

Moreover, if you want to determine a probability as \(P=N/N_{\rm total}\) in the frequentist way, you must actually measure some \(N,N_{\rm total}\). The Sleeping Beauty doesn't measure any \(N,N_{\rm total}\), so she simply can't use any "frequentist" formula for the probability. It's that simple! Thirders are "imagining" some über-observer who picks some random ensemble in a whole spacetime and defines "probabilities" as some relative sizes of subsets of this set. But that's never the meaning of a probability. \(N/N_{\rm total}\) is only interpreted as the "approximately measured probability", in the frequentist way, if one has a control over the initial conditions, repeats them \(N_{\rm total}\) times, and counts how many times, \(N\), a property holds. The Sleeping Beauty doesn't have a control over the initial conditions of the experiment – she can't prepare it \(N_{\rm total}\) times (e.g. because her memory is erased) – so she simply cannot use the frequentist formula for the probability. Instead, she has to use Bayesian inference and assume the laws of Nature (the fairness of the coin).

Whenever someone else has the control over the "preparation of the experiment", you simply can't interpret the ratio \(N/N_{\rm total}\) as the probability. Readers of newspapers in North Korea can't interpret the fraction of articles saying that "North Korea is the most prosperous country in the world" as the probability that North Korea is the most prosperous country in the world. They may have read it 100,000 times in the newspapers but they were not preparing the ensemble of \(N_{\rm total}\) experiments by themselves which makes it illegitimate to assume that the ratios are representative of the actual probabilities.

If we take Matulik's explanation of thirders' thinking seriously and if we add some old language of your humble correspondent, thirders typically imagine that all copies or events involving the Sleeping Beauty (or Boltzmann Brains) in the whole spacetime define the "ensemble", the denominator in the fractions that quantify the probabilities, but it's simply never possible.

If all three awakening events in the whole spacetime (or all lives of the Boltzmann Brains in the spacetime) defined the denominator, and 100% would be the sum of their probabilities, it would include the assumption that all these statements "I am in the spacetime locus" are mutually exclusive with each other and they have an extra predetermined metaphysical law (e.g. the claim that they're equally likely) that determines what their probabilities actually are.

But that's impossible in Nature because it's the laws of physics that actually dictate the right relationships between all these probabilities. So there simply cannot be another law on top of them that would impose additional relationships between the probabilities. You may either believe that there exist laws of Nature, or you may believe that all awakenings (or all lives of the Boltzmann Brains) are equally likely, but you just can't believe both because the union is logically contradictory.

Let me offer you a random new problem to make the point.

Cloning babes

Imagine that on Monday, there is one babe. Every evening, every babe's memory from the day is erased and at midnight, she's cloned into 2 copies. So there are 2 babes on Tuesday, 4 babes on Wednesday, ... and 64 babes on Sunday. When a babe is awakened, she's asked what is the probability that it's Monday. On the following week, the babes are allowed to survive (although I originally had more drastic plans) but they're assured that the week-long experiment is over.

I have eliminated all randomness, coins, and dice.

It seems obvious to me that the thirders will answer \(P=1/127\). In total, there are 1+2+...+64 arrangements "which babe, which day", and only the first one one of them lives on Monday. For an anthropic, Boltzmann-Brain-loving thirder, all these options are equally likely, because they define the "experiments" and the number of elements in this set determines the denominator from which they calculate their "probabilities".

Halfers know that this "equal probability" assumption is inconsistent with any laws of physics.

When a babe is created by splitting her "mother" by cloning, the slice of her probability pie is being divided to halves, too. If the babe has no other information or bias concerning the "current date", it's actually right for her to assume that the probability of Monday is \(P=1/7\). The probability that you're a particular babe who lives on Thursday is 50% of the probability that you're "her cloning mother" who lives on Wednesday. The probability pie for the babe's spacetime identity has to be cut from the pre-given slices.

The difference between the halfers' (now, seventhers') correct answer and the thirders' (now, one-hundred-twenty-seventhers') incorrect answer grows exponentially large if you prolong the experiment. If the babes are allowed to live and split for 30,000 days (or infinitely many days), thirders would say that the probability it's the first day is \(P=1/2^{30,000}\) or so: they could basically (or strictly) eliminate the chance that it's the first day – or any other early day. Because they only deal with the inverse proportionality, halfers will not eliminate it.

There can't be a probability calculus consistent with the laws of physics that would allow you to eliminate the probability that "we're close to the first generations" because this reasoning is acausal. (This reasoning would also be catastrophic for science because it would allow you to prove that cosmology, geology, evolution of species, or any "historical" natural science is wrong because we can't possibly be finitely close to the important early events.) The statement "we're at most the \(X\)-th generation" has a probability that simply cannot be influenced by the assumptions about the glorious future of the mankind: any such influence would be acausal (the future affects the past). The laws of physics are chronologically causal (the past affects the future) and if you assumed both methods to determine the probabilities, you would have the equivalent of closed time-like curves – the paradoxes that ruin naive science-fiction movies with time travel.

A modification of the Sleeping Beauty problem

Let me return to the Sleeping Beauty problem and modify it in one more way. Even if the coin lands tails, the Sleeping Beauty will only be interviewed once – on Monday or on Tuesday. The day of the interview is decided by another coin toss immediately after the first one when the first one gives tails (on Sunday).

So when the Sleeping Beauty wakes up, she can see that there's no interview, and immediately determine that the first coin landed tails. However, when the interview begins, she is uncertain whether the first coin toss was "heads" or "tails" again. In that case, what is her correctly calculated subjective probability of heads? Her one-day memory is still erased on Monday night.

With this twist, the results of halfers and thirders get sort of interchanged.

Now, when she sees that the interview starts, the Sleeping Beauty does receive a new information, and that matters. Up to the moment when she learned whether the interview started, she had learned no new (nonzero) information since Sunday when she was told about the rules of the game.

But as soon as an interview \(I\) starts, she is seeing an event that was predicted to take place with the probability \(P(I|H)=1\) in the case of heads, and \(P(I|T)=1/2\) in the case of tails. Because the "heads theory" made a more self-confident prediction of the observation that was just made (by a factor of two), its relative odds must have increased by a factor of two relatively to the "tails theory" that was uncertain whether an interview should begin.

So once the interview begins, a halfer decides – by Bayes' theorem – that the heads-to-tails odds are 2-to-1 and \(P(H)=2/3\).

What about the thirders' answer? Well, thirders draw a spacetime diagram and imagine that the same week-long experiment takes place on many weeks. The number of interviews on the heads weeks is exactly the same as the number of interviews on the tails weeks, and that's why the naively spacetime-frequentist logic of the thirders leads them to say \(P(H)=1/2\). Nominally, thirders become "halfers" but that shouldn't confuse you: they are still equally stupid.

Why is it stupid? Because in this modified experiment, the logic of the original Sleeping Beauty problem got reverted. In the original problem, the Sleeping Beauty learned no nontrivial information when the interview started but thirders behaved as if some information emerged. Now, it's the other way around: the Sleeping Beauty clearly did learn some new information when the interview started – because it didn't have to start – but the thirders seem to completely ignore the information.

The number of heads interviews and tails interviews may be the same if the same experiment is repeated many times (for many weeks). But the symmetry between "heads" and "tails" is clearly broken. It is not hard to see that the same wrong thinking of the thirders may lead them to ignore any evidence or all evidence in the world. If it doesn't affect their heads-tails odds if they learn about some event that was predicted to be much more likely by one hypothesis than another hypothesis, they may ignore any other evidence, too. I believe that for any kind of evidence \(E\) in any situation, you could construct analogous modifications of the Sleeping Beauty problem that shows that the thirders ignore the evidence \(E\).

For thirders, whenever the probability of a hypothesis \(H\) is suppressed by a factor of \(K\) by the evidence but they want \(H\) to be right, anyway, they may compensate this fact by thinking about the possibility that \(H\) is right, after all, \(K\) times. In that way, they increase the "number of events in the spacetime that follow from \(H\)" by a factor of \(K\), and \(H\) looks as good as it did before.

In 2012, ATLAS as well as CMS announced the 5-sigma discovery of the Higgs boson, i.e. nominally a 99.9999% probability that a new Higgs-like particle around \(125\GeV\) exists. A thirder may think about the evolution of the data in a Higgs-less Universe 1 million times (and only once about the Higgs-ful Universe). The ensemble (the denominator of his "version of a probability") is filled with lots of Higgs-less histories, so the Higgs vs no-Higgs ratio stays the same.

I guess that everyone at least with some traces of a brain understands why this approach to the Higgs evidence is illogical or biased. Whether the Higgs exists or not was discovered billions of years ago and because of basic causality (cause precedes its effects) you just can't affect the odds of this past decision in the Universe – while the details of your "measurements of the probability" would obviously influence the result.

But thirders seem to misunderstand why their approach to the Sleeping Beauty problem is isomorphic to the silly mistake (or scam) that "allowed" the guy to ignore the Higgs evidence. The frequentist measurement of the probabilities is only acceptable if the experimenter's quality standards are imposed. The experimenter has to have the control over the preparation of the initial state and the procedure must be actively repeated \(N_{\rm total}\) times. When the ensemble of \(N_{\rm total}\) events is obtained differently, this \(N_{\rm total}\) can't be considered to be a denominator of any notion of a probability.

Because every awakening of the Sleeping Beauty follows after a coin toss that is the only controllable beginning of the experiment, it's the number of coin tosses (and not the number of awakenings) that must act as \(N_{\rm total}\).

Supreme Court odds

Mutalik offers a funnily stupid "analogy" explaining why he says that both halfers and thirders are right. Up to recently, 6 and 3 U.S. Supreme Court judges were Harvard and Yale Law School alumni, respectively, so the thirders' reasoning gave (accidentally the same) numerical value \(P=1/3\) for the probability that a randomly chosen current Supreme Court judge is a Yale alumnus. On the other hand, Mutalik claims that the halfers would count the odds that some random Americans get to Yale – they would simulate the past, Mutalik thinks – which is why the halfers would conclude that \(P\ll 1/3\) because there are many schools etc.

But this is ludicrous. When we talk about a "judge randomly chosen from the current Supreme Court" and we're given the composition, we are clearly given totally different input information than in the different problem in which the members of the Supreme Court are unknown and have to be "simulated" by following some random students' movement through the U.S. schools institutions. The results are different because the problems are self-evidently different. Halfers would surely agree that if a judge is uniformly randomly chosen from the specified Supreme Court, we would get \(P=1/3\) for Yale.

On the other hand, in the Sleeping Beauty problem, there is no "contrast" between these two very different problems. The last information that the Sleeping Beauty actually knows is the information about the plan of the experiment she was told about on Sunday. Whether it's "analogous" to the information about demographics of the U.S. or the composition of the current Supreme Court can't be said clearly. Analogies can be constructed in many different ways. However, the point is that in the Sleeping Beauty problem, it's completely unambiguous what she's actually allowed to assume about the coin, herself, and the people around her. There is no ambiguity in the problem that would be analogous to the ambiguity in the Supreme Court example!

What may look analogous between the thirders' flawed Sleeping Beauty thinking and the \(P=1/3\) Yale thinking is that the thirders define their ensembles in the "future" relatively to where the halfers "cut the pie". But it's just wrong to make assumptions about her probabilities to be in one awakening or another – simply because she isn't getting any information about those and there is no law of physics that would make them equally likely etc. In the Supreme Court case, the random member of the court is obtained by running a (pseudo)random generator, some event that produces number 1-9 with equal probabilities, and those determine which judge we pick. But in the Sleeping Beauty case, there is simply no event in which a random generator yields a number between 1-3 that would determine which (coin_state, day) combination she is just experiencing. So there's simply no rational basis for her to conclude that the three combinations are equally likely. The actual rational justification exists for \(P(H)=P(T)\) because of the toss of the fair coin on Sunday. And in the case of tails, there may also be a rational basis for \(P(T,Mon)=P(T,Tues)\) because the two days "feel the same" in the given history of the Universe.

But the egalitarian assumptions of the style\[

P(H,Mon) = P(T,Tues)

\] comparing the probabilities of loci in two different possible histories are always wrong – this kind of egalitarianism is always an example of mixing apples and oranges. The pie is sliced to alternative histories in the past, during the coin toss in this case, and no extra "quotas" on the equality of different probabilities across the spacetime and/or across its alternative possible histories can ever be correct.

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