Tuesday, September 28, 2010

The Guth-Vanchurin paradox




The kindergarten paradox goes as follows.

(The story was improved by L.M.)



All the pills are randomly mixed up in the factory.

Fine. So you - imagine that you're a random U.S. woman called Cynthia - emigrate to North Korea away from the evil capitalism. At some point, you feel sleepy and take one random pill from a big 50-50 reservoir, just like everyone else.

Then you wake up. The socialist sedatives are designed so that you can't determine how much you have slept. Now, the question is
What is the probability that you have slept just for 20 minutes?
Well, every nap is associated with a pill. And exactly 1/2 of them are 20-minute pills. So if you assume that the end-of-the-nap is chosen randomly, with a uniform distribution, from the set of all ends-of-the-naps, the probability that you have slept for 20 minutes is fifty percent.

As you may expect, they want to argue that there is a way to see that the probability of a short nap differs from 50%. Why should it differ?

Well, Guth, Vanchurin, Bousso, Freivogel, and others obviously believe that it must be more likely that the awaken people have just slept for a short time because the people who sleep too much are likely to be killed by the "doomsday cutoff", whatever it is, before they wake up. Too bad for them. They should have taken the 20-minute pill. The 12-hour pill is an evolutionary disadvantage.

In North Korea that approaches the doomsday, the survival of the fittest dictates that you should better find a 20-minute pill. But no one knows how to do so.

Now, if you randomly choose yourself from the set of the people in the whole spacetime of North Korea who woke up before the doomsday, and it is the only way to wake up, the 20-minute pill comrades will be given a boost. Their percentage of 20-pill comrades among the awaken citizens will increase above 50%. Can you calculate the exact number?

Well, you may be able to do so but the result clearly depends on the duration of the North Korean regime.

If their regime lasts for a very long time - a much longer time than 12 hours, and it unfortunately does - then the existence of a doomsday (that will arrive to the country, but it will fortunately be more localized than a collapse of the Universe) is pretty much irrelevant.

The number of 12-hour sleepers who will be killed while sleeping will be about 36 times higher than the number of 20-minute sleepers who will be killed while sleeping (assuming that everyone sleeps once a day; this point will be discussed later). But this number of naps that will be interrupted by the doomsday is much smaller than the overall number of 20-minute or 12-hour sleeps, so the effect of the interruption is negligible. The probability of a 20-minute sleep after you wake up will be close to 50% because throughout the Korean history, most sleeping episodes were survived by the people.

However, if the duration of the North Korean regime were shorter, for example 10 hours, and if the citizens knew about this speedy fate of their paradise on Earth, then the probability that you have slept for 20 minutes could go up to 100%. It's easy to see why: if you take a 12-hour pill, it's too bad because the judgment day (or judgment second) is guaranteed to arrive while you sleep and you will therefore never wake up again. So if you do wake up, it does prove that you have only slept for 20 minutes.

At any rate, if we know the history of the country and the approximate numbers of people who go to bed at various points of the history, we can easily calculate the probability that a randomly chosen person who wakes up sometime in the North Korean history is a 20-minute sleeper. It's just about the dividing of two numbers.

In particular, if the life of North Korea is essentially infinite, the probability will obviously approach 50%.

Frequency of sleep

It's useful to note that the probability that you were a 20-minute sleeper doesn't depend on the number of naps you have to take per day as a function of the type of the pill. For example, you may need to sleep 36 times a day if you're unlucky to get the 20-minute pills all the time - while the 12-hour pill recipients only sleep once. You could think that this asymmetry also increases the percentage of the 20-minute freshly awaken sleepers because "they" wake up more often.

Well, it doesn't because the number of 20-minute sleepers is dictated by the pills and only 1/2 of the pills are 20-minute pills. So the people who sleep for 20 minutes 36 times in a row may exist but they're rare. And the correct result that includes the existence of such people with the proper weighting still gives you "p=50%".

The Earth Day in North Korea. The country celebrates it every day.

Incomplete information available to the sleepers

Of course, if the citizens aren't told that the North Korean regime won't last forever, they may be unaware of the judgment day and it will influence their calculation of the probabilities. Most typically, they will be told that communism is a paradise on Earth that will last forever. After all, Kim is called the eternal president which means that his empire should be eternal as well. ;-)

Because of this propaganda, you will think that the regime exists forever which means that the judgment day is not real - or is "infinitely" far away. After you wake up, you will calculate the probability to be a 20-minute sleeper to be exactly 50% even though the more accurate result taking the unsustainability of communism into account exceeds 50%.

In the real world, North Korea has a beginning in time - much like this Universe that began with a Big Bang. Needless to say, if you imagined a hypothetical world where these things have existed for an infinite amount of time, it would also become "almost guaranteed" that you woke up "almost an infinite time" before the doomsday. The probability of a 20-minute nap would therefore go to 50%, too.

All this discussion actually assumed that all the citizens at all times are "equally likely" to be you. In this sense, I was accepting the "anthropic rules of the game". Even with the anthropic rules of the game, there can't possibly exist any paradox. For any kind of a history, one can easily calculate the probabilities.

Acausality in wrong calculations of the probability

It's important to note that all these calculations must start with some knowledge about the history and the timing of the doomsday - either accurate knowledge, or a probabilistic knowledge - and then they calculate the probabilities that the awaken people obey certain conditions.

If you are a dictator who makes a timing of the doomsday a function of the people's expectations, and people's expectations, calculated probabilities, and therefore their behavior may depend on the doomsday, then you are clearly going to see paradoxes. That shouldn't be surprising because you have designed a classic world with closed time-like curves and they do lead to logical paradoxes.

That's the ultimate reason why the anthropic folks see "paradoxes": their formulae are genuinely paradoxical because they imply that the probabilities of an outcome, XY, to occur now is affected by the future doomsdays and other events, and the probability of the future is clearly affected by the present, too. A classic grandfather-paradox setup.

However, such paradoxes never appear in the real world - neither in North Korea nor in the multiverse if it is real - simply because people's current experiences (and their rational expectations) can never be affected by the future that is unknown. They only depend on the past (plus the quantum "random generator"). That's the very point of future that it is unknown.

In the very same day, no valid calculation done by any beings throughout the history of the Universe may depend on non-tautological assumptions about the future simply because the future is unknown. All valid reasoning always has to boil down to the knowledge of the past and present. There's no other "primary" knowledge. All of our knowledge about the future - whether it is more or less certain - is deduced from our actual knowledge of the past.
Anthropic people misunderstand and deny causality: All the formulae by Bousso and dozens of others for the current probabilities of any measurement depend on properties of the Universe in the future (because they're based on the counting of "similar situations and objects" that will emerge in the future), which is an acausal rule, and all of their contradictions arise for the same simple reason as the contradictions in the grandfather paradox.

But the real world never admits any formulae where the current probabilities depend on the properties of anything in the future. And any valid argument and any physical law that produces some probabilities always starts with some known past/present data and it predicts the numbers - and the world behaves and evolves according to these rules. It's never the other way around: the future may never influence the "current" laws of physics as an independent variable.

The past is the input, the probabilities for the present and future is the output of all calculations based on legitimate laws. It's never the other way around. For example, the probability that your DNA suffers from autism is not affected by the abundance of the autistic people in the future. The latter may be huge but it is completely irrelevant for your predictions about the present (or test you will undergo today).

The probabilities are first calculated "by Nature", and then they're (statistically) realized. And scientists have to respect this ordering, a basic manifestation of the logical arrow of time. They can't envision a future Universe, count the number of objects satisfying some properties, and use the counting to predict what will happen now - simply because they don't know and can't know the future at the beginning of their reasoning because the future depends on the things they're just trying to calculate.
Instead of the grandfather paradox, it may be more accurate to talk about the paradox of a malicious Iranian prophet. If he knew who will be elected the U.S. president in 2012, he could deliberately kill her now (poor Christine O'Donnell: a murder is worse than masturbation), thus throwing America into a logical inconsistency (or the need to elect a dead person). Is that a real paradox or just a funny joke?

I think it is the latter because no one can know the future of a physical system that includes his own decisions. In this fictitious story, the prophet decides about his shooting by knowing the future president. In the real world, it will go the other way: the election of the president will be influenced by the U.S. politicians who will be shot by 2012. Be sure that any terror will help the right-wing candidates.

If a doomsday terminates the Universe soon and if no one can predict such an event using local physics, then the right assumption is that such a doomsday simply can't happen. It's obvious that such an assumption can never hurt you. It makes no sense to buy insurance because of a hypothetically elevated probability of a doomsday because you won't be able to spend the money from the insurance company, anyway.

So the potential doomsday is inconsequential for your financial planning. It's inconsequential for any other scientific or rational reasoning, too. The real reason is that not only rational people and scientists ignore such a possible "end of the world"; Nature ignores it, too. Whether there will be an end of the world has yet to be seen - but its possibility impacts neither events that take place anytime before the doom, nor their actual probabilities. The scientists only "copy" this independence that follows from causality.

Additional information affecting the probability estimate

So far, we have assumed that the information about the "number of naps of one kind or another kind" is the only piece of data that the awaken citizens could have used to determine what kind of a pill they have used. In other words, we have accepted the "uniform Copernican distribution of conscience" which is popular among the advocates of the anthropic principle. Among N people who woke up in one of the T mornings, your combination is a random one among the N.T choices.

Even in this setup, there can't exist any paradoxes as long as your formulae respect basic principles such as causality.

But I don't think that the assumption that you have to be a "generic citizen of North Korea in a generic morning" is rational in any sense. It's just a possible (but in no way inevitable) choice for the prior probabilities - one that is instantly superseded by non-uniform distributions as soon as you learn anything about the world and about yourself.

Moreover, this "uniform prior" is impossible if the number of possibilities (or "spacetime volume") is infinite. This impossibility is no paradox of any kind.

It just proves that if there are infinitely many choices, all possible prior probability distributions have to be clumped at some special places, determined by some special points - a self-evident mathematical conclusion. If I ask 10,000 people to invent a random positive integer, 99+ percent of them will come up with a number smaller than a googolplex - even though the fraction of positive integers smaller than a googolplex among all integers is vanishingly small.

There's nothing wrong whatsoever with such a non-uniformity of the prior probabilities and whoever thinks that non-uniform probability distributions are forbidden by mathematics or physics has misunderstood pretty much everything about probabilities he could have misunderstood.

But I want to return to the additional information that actually affects our estimates of the probability that we have slept for 20 minutes.

As we have already mentioned, if we know that the empire only lasts for a short time, relatively to 12 hours, and then it is nuked away, it becomes significantly more likely that once we wake up, we have slept for 20 minutes. Those who sleep too much are pretty likely to be killed while they sleep.

However, even if you avoid this extreme scenario - and if you avoid any doomsday whatsoever - there can exist information or principles that make "us" more likely to wake up as 20-minute sleepers (or, on the contrary, 12-hour sleepers).

Most people don't ask

One of the aspects of socialism is that it transforms the citizens into mindless machines who never dare to ask any questions. So if you ask the question whether you have slept for 20 minutes or 12 hours, it already makes you special. So you should evaluate the probabilities by counting the people-and-naps within the restricted set in which the people are actually able to ask this question after they wake up.

It's conceivable that there are no people in North Korea of 2010 who dare to ask whether their government gave them a 20-minute pill or a 12-hour pill. But North Korea hasn't been around since the Big Bang. It was created in 1948. While the sedative rules may have worked since the beginning, other things were changing.

For example, between 1948 and 1949, people may have still been able to ask questions. However, let's imagine that the march of the communism was faster and only the people in the first 10 hours of the history of the country were able to ask inconvenient questions.

If that it so, it clearly follows that the probability that you were a 20-minute sleeper goes to 100% once again. It's because if you had ever taken a 12-hour pill, you would already wake up into the era of North Korea in which independent thinking was impossible, so you couldn't have asked the question. Because you asked them, you may prove that you were a 20-minute sleeper.

This metaphor is meant to convey the point that there is nothing wrong or contrived whatsoever if we're among the first 0.0001% people who were born. Imagine that "p.100%" of the people were born before us and "(1-p).100%" people will be born after us. Is there any reason why "p" should be comparable to one?

My answer is a resounding No. In fact, "p" can be zero if infinitely many people will be born in the future. Also, "p" can be nonzero and very tiny.

The claim that "p" should be of order one is a hallmark of the anthropic reasoning. It is a dogma that is supported by no rational evidence. Could you try to find some rational evidence?

Saying that "p" is of order one looks like naturalness - the statement that e.g. gauge couplings "g" should be of order one unless you can prove otherwise. Well, the naturalness holds because "g" may be calculated from some fundamental equations with no unnaturally small or large parameters. And the solutions of such equations are likely to be numbers of order one.

However, the "p" defined above - measuring how "early" our generations are - is clearly not a solution of any fundamental equations without parameters, because "you" (or "I" or "we") are not a canonical well-defined object that can be isolated from the first principles, so the argument that is applicable for the gauge couplings manifestly fails in the case of "p".

But even in the absence of such an argument, you could just say that you "feel" that "p" should be of order one. It's a "natural" assumption for you, a kind of "truthiness". However, there exist hundreds of other arguments that we do know - and thousands of additional possible facts that we don't know yet - that show, and much more reliably, that "p" is very likely to be much smaller than one. Such arguments may even estimate "p".

Quadrillions of people

For example, it's plausible that one can make a detailed model of the future of our civilization, showing that we ultimately have to control the resources of the Milky Way which will increase the number of humans born in the future to quintillions (10^{18}) or more.

I can't show you a rigorous version of this argument now. But I can show you approximate arguments that indicate that the probability that people will be able to do such things in the future - and their population exceeds a quintillion - exceeds 0.1%. Only a few "surmountable obstacles" may prevent the mankind from such a future - so such a future can't be quite excluded. Even with this modest number, and it is not too unrealistic, the expectation value for the number of people who will live in the future is guaranteed to exceed a quadrillion (10^{15}), making "p" smaller than 10^{-5}.

You may say that by your unjustified naturalness criterion applied to "p", it's unlikely for "p" to be smaller than 10^{-5}. The probability of such a thing is around 10^{-5}, you may (irrationally) claim. But even if we considered your argument to be legitimate, it is an argument that only carries something like 11 units of evidence.

There may exist - and there probably do exist - arguments that "p" is smaller than 10^{-5} and these arguments may carry dozens or hundreds of units of evidence or more. These arguments are based on the actual detailed laws of physics, economics, and the society - the usual things about particles, fields, collisions, GDP growth, wars - science as we have known it. And because these arguments are stronger, they simply beat your irrational expectation that "p" should be of order one which was only supported by a nominally very weak argument - that was moreover completely unjustified.

The whole anthropic reasoning that promotes various uniform measures is based on the complete denial of the rest of science and the rest of evidence - because a key result of any evidence and any science is that the measures are (highly) non-uniform.

And that's the memo.

P.S.: I am really curious what Alan Guth and his Vitaly Vanchurin collaborator (whose kids are modestly called Cosmos, Spartak, and Antonina) may write about this trivial thing.

P.S. II: The following example explaining the situation by Anonymous Cosmologist is witty and wise:
Suppose the U.S. will exist for billions of years, and all future presidents will be white. The anthropic principle implies that Barack Obama is white (with probability essentially approaching unity).

The old-fashioned, causal reasoning goes the other way: Since America's first black president has been such a disaster, no black person will be elected president ever again and all the remaining presidents will be white. :-)
Note that there are several key mistakes in the anthropic argument. It starts by "supposing" something about the future that one can't "suppose" because it is unknown - at least at the beginning of the reasoning. Then it "derives" something about the present which is an acausal reasoning. Moreover, this reasoning is based on a fallacy called "typicality" that leads, in this case, to a wrong conclusion about Obama's skin color.

The actual story, described in the second paragraph, follows the proper logical reasoning that respects the logical arrow of time. As a by-product, the actual story shows a very good reason why some features of some presidents - such as their skin color - could not only fail to be "guaranteed to be typical"; in the story, Obama's skin color is "guaranteed to be atypical" by a simple mechanism or a simple explanation. In this case, the mechanism is that "intelligent" objects or countries may avoid repeating mistakes - which is why the objects classified as "mistakes" (and their characteristics) will inevitably be atypical.

These and similar mechanisms and their discovery, verification, and application in the reality is what science and rational reasoning is all about; the anthropic reasoning is based on a complete refusal of such a type of reasoning. It is a denial of logic and a denial of any detailed knowledge.

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