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*Homo sapiens*in a single number – how many individuals of our species have existed so far.

*application*of this argument, which uses anthropic reasoning. Since anthropic reasoning is often very difficult to think about, it’s best to understand the general outline of the probabilistic argument first, if only in qualitative terms. That way we can be certain that any objections are coming from the important claims of the argument, rather than mere misunderstandings of probability.

**Critique 1: There is no fact of the matter about N (yet).**This objection asks whether it makes sense to say that there is some fact of the matter about what N is, given that N cannot be known without traveling to the future, and given that N depends on our actions in the here and now. Speaking loosely, it criticizes the seeming determinism of the DA.

*before*the bacteria divide (determinism) – probability allows us to reason about N anyway, whether or not it is determined.

**Critique 2: We somehow know a priori that we are among the earliest humans.**E.g., we expect most humans to be cyborgs, but we are not cyborgs, therefore, we are known to be early. “All else is not equal; we have good reasons for thinking we are not randomly selected humans from all who will ever live.” (Robin Hanson)

*assuming*that this objection fails.

**Critique 3: Self-indicating assumption.**The possibility of your existing at all depends on how many humans will ever exist (N). If this is a high number, then the possibility of your existing is higher than if only a few humans will ever exist. Since you do indeed exist, this is evidence that the number of humans that will ever exist is high.

**Critique 4: Richard’s objection.**An RS commenter named Richard got me thinking about the DA. His objection has to do with the proper assignment of priors in the DA, not directly with the validity of the Bayesian update. According to him, although there may be a 1000-to-1 likelihood ratio for DoomSoon over DoomLate, this effect is exactly counteracted by one’s priors on DoomSoon and DoomLate. DoomLate, on this view, should have a greater prior (200e12/200e9 = 1,000 times bigger, to be precise), essentially because you are likelier to find yourself in a larger group of people than a smaller one (thus, I think Richard’s objection ends up being equivalent to the self-indicating assumption).

*already*have been adjusted. I tentatively agree with Richard that it looks like there might be something circular going on here: we are using the DA to determine our priors, to which we then apply the DA?

**Critique 5: The problem of reference classes.**We are discussing the probability of “finding yourself” at rank such-and-such, but exactly what kinds of entities fall into the class that is being ranked? If we include prehistoric humans in our reference class, then our birth rank looks a little bit better and we get a different answer; if we include only people born on Mondays, the answer changes yet again; if we include all humans back to our last common ancestor, it changes yet again. And what if there are extraterrestrial observers? Couldn’t you just as easily have “found yourself” inside the body of Zaphod Beeblebrox? If so, how on earth can any conclusions be drawn about human population limits?

**Critique 6: Metaphysics of “random sampling” from the set of all observers.**One difficulty I have, which is rather hard to put into words, lies in the concept of “finding oneself” at a particular birth rank. This choice of language puts me in mind of nothing so much as a deck of immaterial souls being dealt by some cosmic Dealer to various previously lifeless bodies.

One objection I've always had relates to #5 and #6. It's not clear what it would mean to say that I "could have" been born as someone else. If you talk about multiple realities or multiple simulations thereof, there could be multiple "me"s, identical (at least at first) down to the electron or bit level.

ReplyDeleteBut there is (I think) no consistent counterfactual world in which I was born Abraham Lincoln. Anyone born when and where he was, with his genetics, would not in any meaningful sense be me. So it seems like I should consider the prior likelihood of that to be near zero.

Another thing I should point out is that the self-indication assumption gives what I believe to be the "obvious" answer to the Sleeping Beauty problem. If your goal is to maximize the expectation value of the number of times you form a true belief, then using the self-indicating assumption is provably your best option (in fact, you can explain the reasoning here and that in the Monty Hall problem) in the same way.

The only reason why you would reject this is if you don't actually place any *value* on the number of times you independently get the right answer (e.g. you don't consider it more wrong to believe a coin came up the wrong way 1000000 times rather than once). In this experiment, where there are no stakes, that might be fine. But if you lose $100 every time you guess wrong, I recommend the SIA!

>It's not clear what it would mean to say that I "could have" been born as someone else... there is (I think) no consistent counterfactual world in which I was born Abraham Lincoln. Anyone born when and where he was, with his genetics, would not in any meaningful sense be me.

DeleteYes, this tracks my intuition nicely.

Then again, Bostrom has another, easier thought experiment that goes like this:

>The world consists of a dungeon that has one hundred cells. In each cell there is one prisoner. Ninety of the cells are painted blue on the outside and the other ten are painted red. Each prisoner is asked to guess whether he is in a blue or a red cell. (And everybody knows all this.) You find yourself in one of the cells. What color should you think it is?

It seems pretty clear here that you should give odds 9:1 in favour of blue. I am pretty much okay with Bostrom's logic at this point.

The challenge is to come up with a principled distinction between this situation (where treating yourself as a randomly selected observer seems to make sense) and the DA.

I would argue that one distinction has to do with the fact that all cells are used, whereas in the DA, not all possible people are born. If the very fact that you are being used in the experiment changes the odds, a simple count of the various outcomes is no longer the best you can do.

DeleteSimilarly I could argue that the observation that I exist with birth rank X is not a matter of "finding" myself in arbitrary position X, but rather the observation that I exist. That is, I could not have had a very different birth rank and still existed, so the fact of my existence is an observation of the fact that this "me" exists, rather than the human race having died out years ago, and thus *increases* the expected lifespan of humanity. This is a way of sneaking off to appeal to the SIA.

It's worth adding that I am much more confident that the SIA gives the "right" answer in the Sleeping Beauty problem (for most people's purposes) than as a counterargument to the DA. If there's something fundamentally wrong with treating my own birth rank as an arbitrary number assigned to me, then neither the DA nor the SIA response to it would hold.

ReplyDeleteThe Sleeping Beauty problem is interesting. Ordinarily, I am very happy to identify probability roughly with the betting odds that maximize an agent's utility, but the SBP may be one case where they come apart. I agree that SIA maximizes Beauty's payout.

DeleteBUT one of the things that bugs me about it is that she has no reason to wait until after she is put to sleep, before she decides to declare 1/3 probability across the board. If her goal is to maximize her payout, she can precommit to that as a strategy before the experiment even begins. So it becomes unclear why we are talking about probability at all.

To me, the question the SBP should be asking is more purely epistemic: has Beauty gained any information she didn't have before the experiment, by virtue of waking up? Nope. Evens on the coin flip before, evens after. No update has occurred.

I agree with Ian (a rare thing!).

DeleteOn the question of the Sleeping Beauty Problem, the probability HAS to be 1/2!

Probability is based upon the information that you have. In this case, Beauty has NO information other than a coin is flipped and she may be waking up in one of three interviews (Heads/Monday, Tails/Monday, Tails/Tuesday). The problem tries to trick us into believing that the interviews are related to the coin flip. They are not – at least not from the point of view of Beauty.

Beauty is asked ONLY about the flip of the coin. She is NOT asked “which of three interviews do you believe this is”? (That WOULD be 1/3). If she knew more information about which interview it was, it would change her assessment of the heads/tails question.

But as the problem is stated, she has no information about which interview it is. Thus, the interview gives NO information regarding the outcome of the coin flip – which is the question being asked. She knows only that a coin was flipped, and she is waking up in SOME interview (which gives no further information).

Therefore, her assessment of the proposition that the coin landed heads can be no other than 1/2.

Another way to look at this is: what if they had continued MORE interviews after tails? Say, they did 999 interviews, giving the amnesia drug after each. Would the probability of heads become one in a thousand? Again, the question being asked is NOT “which interview is this?”, the question being asked is, “did the coin come up heads or tails”.

Any comments?

Agreed, Tom.

DeleteYet ANOTHER way of looking at it is to consider what happens when we remove the amnesia drug from the original thought experiment.

Let's say the coin toss comes up tails, and we wake Beauty on Monday. This had probability 1 of happening regardless of what the coin toss was, so her odds on tails are still 1:1 (50% prob).

Then she is awakened on Tuesday again, and remembering Monday, updates to a state of certainty that the coin came up tails.

Now consider: what was the role of the amnesia drug in the original gedanken? My answer is that it was an epistemic handicap imposed from the outside, pure and simple. And there is nothing particularly paradoxical about failing to come to the right conclusion if you have some handicap imposed on your brain.

Beauty was originally ignorant about the coin flip outcome (1:1 odds on tails), and she remains ignorant after waking (both days) because we don't allow her any information remotely relevant to the coin flip.

Ian, thanks for this article. I think it summarizes the arguments and counterarguments nicely. I agree that the objection I expressed in my comments under the Bayesian odds article is the same as the SIA. I hadn't heard of the Sleeping Beauty Problem before, but I agree with Sean -- the SIA gives the right answer.

DeleteRe your comment: "To me, the question the SBP should be asking is more purely epistemic: has Beauty gained any information she didn't have before the experiment, by virtue of waking up? Nope. Evens on the coin flip before, evens after. No update has occurred."

But if the experiment is repeated many times, she will be asked about the tails events twice as often as the heads events. Being awakened does *not* provide new information, *but* there was already sufficient information to deduce the 2:1 odds favoring tails before she was put to sleep. The odds on any given flip are 1:1, but the conditional odds for tails given that Beauty is being asked about a flip (without the knowledge of whether it is Monday or Tuesday) are 2:1.

Suppose instead of being awoken once for heads and twice for tails, we have a computer which holds a list of the outcomes of a million coin flips (of a fair coin). Suppose we both know that every time tails came up, that outcome was added to the list twice, as opposed to only once for each heads outcome. Now I offer you the following bet: The computer will randomly print an entry from the list. If it is tails, you pay me $100. If it is heads, I will pay you $150. Do you take the bet? Now I ask, how is this scenario different from the SBP?

Sean: I'm not sure I understand why you accept the SIA but not its applicability to the DA.

Tom: "Say, they did 999 interviews, giving the amnesia drug after each. Would the probability of heads become one in a thousand? Again, the question being asked is NOT “which interview is this?”, the question being asked is, “did the coin come up heads or tails”."

DeleteBut a rational (and Bayesian) Beauty would answer based on the probability of heads or tails *given the information available to her* -- she would give a posterior probability, not the 50% prior probability.

How about another version? Suppose if tails is the result, she is awoken once and interviewed, and the experiment ends. Suppose if heads is the result, we use a random number generator to decide whether to wake her and give the interview, setting the threshold to give 1/999 probability that we will do so. Otherwise she is not awakened, or perhaps (if the experimenters find that action to be ethically troubling) awakened and sent on her way without being interviewed. Now if she finds herself awake and being interviewed, what should her assessment of the outcome probability be? I imagine it would go like this: "I am awake and being interviewed. I know there is a 100% chance of that happening if the outcome was tails. I know there is one chance in 999 of that happening if the outcome was heads. The prior odds were 1 to 1. As a Bayesian I must adjust those odds to 999 to 1 in favor of tails."

Here she is giving posterior odds that incorporate the information gained from the knowledge that she is being interviewed. This is different from the other scenarios: In the other scenarios, she will *always* be interviewed, so she doesn't gain knowledge from the fact of the interview. But in those scenarios, she knows *in advance* that she will be interviewed more often in the tails case than in the heads case, so she has all she needs to arrive at the posterior *before* she goes to sleep. In the new scenario, she can deduce in advance that there is a slightly greater than 50% chance that she will be interviewed at all; she gains information when the interview starts.

"Yet ANOTHER way of looking at it is to consider what happens when we remove the amnesia drug from the original thought experiment."

Delete"Let's say the coin toss comes up tails, and we wake Beauty on Monday. This had probability 1 of happening regardless of what the coin toss was, so her odds on tails are still 1:1 (50% prob)."

"Then she is awakened on Tuesday again, and remembering Monday, updates to a state of certainty that the coin came up tails."

Yes, but this doesn't negate the use of the SIA. In this scenario, when she wakes up on Monday, she has information that she is lacking in the amnesia case: She knows that it is Monday, because if it were Tuesday, she would have a memory of Monday. Of the three possible outcomes (Mon/heads, Mon/tails, Tues/tails) she can eliminate the third. Instead of basing her reply on the posterior probability of heads given that she is being asked the question, she will base it on the posterior probability of heads given that she is being asked the question on a Monday.

"Beauty was originally ignorant about the coin flip outcome (1:1 odds on tails), and she remains ignorant after waking (both days) because we don't allow her any information remotely relevant to the coin flip."

I'm not so sure. The question she is to be asked is "What is your belief now for the proposition that the coin landed heads?" Are you saying it is not rational to modify her belief based on the knowledge that she will be asked this question more often if the toss comes up tails?

And what if the question is changed to "What is your belief now for the proposition that today is Monday"? Surely, if this experiment is performed 1000 times, that question can be expected to be asked 1000 times on a Monday and 500 times on a Tuesday.

Richard,

DeleteI believe you are right. I wrote that Beauty knows only that a coin is flipped and that she is in SOME interview. But I missed the fact that she ALSO knows that there are twice as many interviews after "tails" then after "heads". So, on any given interview, it makes sense to bet on "tails".

Have you also heard of "Bertrand's Paradox"?

> Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?<

Arguments have been made for 1/3, 1/2, and 1/4!

See:

http://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

http://www.academia.edu/1536034/resolving_bertrands_probability_paradox

Also, I once saw how to do a calculation for the expected lifespan of any event based solely upon the length of time the event has already existed. Do you (or anyone else reading this) recall that?

As Richard suggested, I believe that Sleeping Beauty *should* precommit to answering 1/3. This is not to say that she believes that the chance of heads is 1/3. Rather, it is to say that when she is woken up, the chance that it will be after heads is 1/3, so finding herself awake is evidence against heads (even though she will always experience it at least once).

DeleteBeing woken up doesn't give her new information, but she knows beforehand that the experiment is biased to make "tails" the right answer more often than heads.

Think of the Monty Hall problem. You know that switching is the best strategy before the game starts, because doing so allows you to systematically exploit information that the *host* has, but you don't.

As for my opinion of the SIA and DA, I've grown a bit more comfortable with it, but it has taken me a while to think through the very large number of variants and analogies.

The Shooting Room is interesting. I'm tempted to think that the answer depends on whether you magically know ahead of time that the experiment will last long enough to use you. It's very close to the DA.

I am a bit confused as to why 1/2 would be an answer to consider at all. Isn't the SBP equivalent to Monty Hall? In both cases, the experimenter knows sth extra and acts upon it, and the subjects knows that. In MH, it is the door that is opened that is guaranteed not to have a car behind it, in SBP it is the fact that only tails will induce another day of sleep.

DeleteTom: Yes, I've read about Bertrand's paradox. I think it arises because there is no single way to choose a chord at random (choose two points on the circumference and connect them; choose two points in the interior and connect them; choose a point in the interior and an angle; and so on). Probabilities for sets of reals are defined using measures on the sets we are interested in. For example, the probability of hitting a bullseye on a target with a random, unaimed arrow (given that you at least hit the target) is the ratio of the area of the bullseye to the area of the target. In the chords case, the measure varies with the selection method. A simpler analogy would be the probability of rolling a 7 with two six-sided dice, compared to the probability of rolling a 7 with 3 4-sided dice.

DeleteAs for the method of determining future lifespans based on lifespan to date, I think you are referring to Gott's delta-t argument, which is described in the "literature review" link that Ian posted at the end of the article. I actually heard about this argument before I heard about the Carter-Leslie argument, and initially thought it might have some very limited validity. But Carlton Caves published a refutation (http://arxiv.org/abs/astro-ph/0001414 -- warning, contains calculus) which, after the second or third reading, made sense to me. Essentially, Caves showed that a proper Bayesian derivation of Gott's argument implies that the "temporal Copernican principle" Gott relies on is only valid when you do not know how long the phenomenon has lasted (you are no longer J. Random Observer, you are Observer Present at Age X). But you need the temporal Copernican principle for the formula to be valid, and you need the age of the phenomenon to apply the formula. Caves showed that the actual posterior probability distribution of total duration upon learning the age of the phenomenon is equal to the prior distribution, but truncated to the left of the current age (we rule out total durations less than the current age, so the probability goes to zero for that interval), and renormalized so that the integral over all possible remaining durations is 1. And this result agrees with intuition (okay, it agrees with mine, at least).

The BSP is subjective and is based in a simple question;If in one experiment I guarantee you'll answer the question(s) correctly, which is better: one heads or two tails?

DeleteIf you think is better two tails because they are two questions and heads is only one, then the belief is actually 1/3.

But in my opinion they have the same value. If the coin came up heads and you'd get it right then the game is finished, but if came up tails then, suppose there is no amnesia, you get tails right TWICE and the game is finished.

So in the long run, if you stick to tails you'll always have the double of correct answers than playing always to heads, and there is no need of brainwashing to see that is not a fair game. For that reason I think 1 heads and 2 tails should have the same value and the belief in this case is 1/2.

I think if you're memory-less, and wake up to find that it's the 7301st day you've been alive, it is reasonable to assume that you won't be alive for much longer. But if you have memory, and know something about average human life expectancy etc., you should come to the conclusion that you'd be alive for much longer since you're only 20 years old.

ReplyDeleteSo I think Richard's objection is quite right. It is of course the case that you're more likely to wake up on 7301st day if your total lifetime was 7500 days, instead of 27,000 days. But that 7500 days hypothesis has very low prior probability.

In this case, the sun and earth has been here for billions of years. Great extinctions are rare, and homo sapiens have been here for at least 100,000 years. So it makes sense to think that we won't be driven to extinction by natural causes in the near future.

I'll repeat your disclaimer, I'm sure this has been raised before. But can you really use our species' past history of survival to provide evidence of future survival? I think you have to assume that all factors relevant to our survival in the past and future are the same. If you are a pessimist, you will note that our ability to destroy ourselves is much higher now than it was during almost our entire existence. If you are an optimist, you will note that we now have the knowledge and technology to overcome potential extinction events such as asteroids. Further, our extinction risk was fairly high at certain points in our past since our population levels dropped into the thousands. Perhaps our future extinction risk is lower than the historic average.

ReplyDeleteRegardless, I think the DA assumes constant hazards, which is dubious.

I think Dieks's refutation is on the money regarding the doomsday problem: http://philsci-archive.pitt.edu/2144/

ReplyDeleteIn a nutshell he argues that the hypothesis "Doomsday will be soon" is malposed in a Baysean analysis since "soon" already implies the current position of the observer in time. Instead a reasonable hypothesis would be "Doomsday will be before date X". If that is the case then the priors of this hypothesis and its negation are much different than our actual priors now, since before we know our position in time (or our birth rank) we could live anytime (even after date x if the hypothesis is false). The upshot is that once we take our position in time (or our brithrank) into account the priors are adjusted such that they fall back to the same probabilities we would devise for these hypothesis on the basis of our scientific insights up to now.

This also neatly dissolves the sleeping beauty problem, so I am a bit puzzled why his argument is not mentioned here. Pisaturo argues along similar lines (Pisaturo, R. 2009. Past longevity as evidence for the future. Philosophy of Science 76: 73–100.)

Thanks! I had not heard of this argument.

Delete>In a nutshell he argues that the hypothesis "Doomsday will be soon" is malposed in a Baysean analysis since "soon" already implies the current position of the observer in time.

I'm not sure I follow, just based on your comment (the paper is not in a format I can read here; I'll try to access later). "DoomSoon" is just a nickname for the hypothesis; its actual content is "humanity will last until the 60 billionth birth rank," which is not contaminated with anything indexical.

>If that is the case then the priors of this hypothesis and its negation are much different than our actual priors now, since before we know our position in time (or our birth rank) we could live anytime (even after date x if the hypothesis is false). The upshot is that once we take our position in time (or our brithrank) into account the priors are adjusted such that they fall back to the same probabilities we would devise for these hypothesis on the basis of our scientific insights up to now.

I wasn't able to follow this. If you're in the mood, spelling it out would be great; if not, I'll try to read that paper soon.

Sorry, for my inapt summary. Dieks's uses a time-based formulation, but I think that his argument runs equally well on birth-ranks. He proposes that the mistake "doom-sayers" make is based on their confidence that the probabilities we currently devise for a doomsday event should be taken as our bayesean priors for this event before knowing which birthrank we actually have.

DeleteBut this seems flawed on its face since if I knew my birthrank to be higher than the "60 billionth" I would know fur sure that "DoomSoon" is incorrect. In other words in our "common sense" probabilities about doomsday at rank X, we have already smuggled in our knowledge that our birthrank is actually lower than X. In order to get our priors before we know our birthrank we need to consider the possibility that knowledge of our birthrank alone invalidates DoomSoon.

Hmmm ... not sure whether this was any clearer. Anyway the article on the site I linked is a preprint tex file, so you should be able to read it with any text editor.

Reading Bostrom's primer, my initial reaction was that step I and II are fine, but III does not follow as the doors are temporally simultaneous whereas we are not. Also, why is Step I described as using the self-sampling assumption? SIA (as defined in Wikipedia) would arrive at the same result.

ReplyDeleteThird point (or rather question): not sure about the math, but even with the given probilitiy assumptions, aren't there infinite combinations of DoomSoon / DoomLate where DS1, DS2... all have the same odds over DL1, DL2...? I.e. even if the DA is correct, it only shows that Doom will happen sooner rather than later, but not that it will happen soon in absolute terms. (I.e. P(DS2)=P(DL1) etc.)

Cheers

Chris

Chris, I think Bostrom's primer fails at step 2. See my post from Dec. 6 (1:03 AM) under this article: http://www.rationallyspeaking.blogspot.com/2012/11/odds-again-bayes-made-usable.html#comment-form

DeleteThere I describe two thought experiments, let's call them TE1 and TE2. At first glance they seem equivalent, but on closer analysis we see that they are not.

Bostrum is assuming that his thought experiment is similar to my TE1. That is, God's flip of the coin is like my random selection of one of the two urns in TE1. I contend that Bostrum's experiment is analogous to TE2. To see why this is so, consider the possibility that God conducts the experiment 1000 times. Then, we expect that 500 times he will create 10 individuals, and 500 times he will create 100 individuals. I try to guess whether I am in one of the 500 instances of the first kind or one of the 500 instances of the second kind, on learning that I am in cell 7. God created a total of 55000 individuals in the 1000 trials. 50000 of those were in trials that created 100 individuals each, and 5000 were in trials that only created 10 individuals. So before I know my cell number, I have reason to believe it more likely that I am in my former group, at 10:1 odds -- this is my prior. When I adjust my prior based on knowing I'm in one of the first 10 cells (which is 10x more likely if I'm in the second group), I arrive at even odds. And that makes sense, because I know that 1000 individuals were/will be created in cell 7, and 500 of those were in the first group, 500 in the second group. That reasoning is in line with TE2; populating individuals into cells resembles handing everyone entering a room an envelope with a number on it and an "A" or "B" card inside. The number on the envelope corresponds to the number on the cell, and the card inside is indicating whether the recipient is a member of a small or a large population.

I can't see any reason to modify my reasoning if the number of trials is reduced to 100, or to 5, or to 1.

Makes sense

DeleteThe sampling is not instant, rather it is continuous. At any time, the only people available to consider this problem are those who are currently alive, and they are subject to the anthropic principle.

ReplyDeleteThe appropriate analogy in the Blam-O example goes something like this:

Your agents have in fact found an unknown number of fireworks. Since they are all practical jokers, they insist on first giving you the lowest serial number found among all the fireworks. The lowest serial number is 112. Estimate the number of fireworks.

For the benefit of those who have problems to access Dieks article, I thought I give a more precise description using the example Ian provides in the text. For ease of presentation I will only investigate the claim that DoomSoon is true given that I am informed that my birthrank is < than 200b (instead of 60b). The argument can be readily modified however.

ReplyDeleteIan assumes that P(DoomSoon)/P(DoomLate)= p_S / p_L = 1 / 50 and that we should use these estimates as our Bayesean priors since we did not incorporate knowledge about our birthrank.

Dieks argues that this is flawed and instead that the values p_L and p_S already reflect knowledge of our birthrank since we would only assume them as probabilities for the hypotheses GIVEN that our birthrank is < 200b. In other words Ian has so far only given us P(DoomLate|R<200b) and P(DoomSoon|R<200b).

If we are really to imagine that we could have randomly appeared at any time in history having any of the 200 trillion birthranks (if DoomLate is true), then we need to consider the possibility that we have a rank R>200b. But P(DoomLate|R>200b)=1 and P(DoomSoon|R>200b)=0.

The actual probability for DoomSoon is thus P(DoomSoon) = P(DoomSoon|R<200b) * P(R<200b) + P(DoomSoon|R>200b) * P(R>200b) = P_S * P(R<200b) + 0 * (P>200b).

The chance that I will actually be having a birthrank lower than 200b is the sum of two possibilities and crucially depends on whether DoomLate or DoomSoon is true. In case of DoomSoon I will certainly find myself among the first 200b citizens and in case of DoomLate my chance of finding me among the first 200b is quite low q = 200b/200tr.

It follows that P(R<200b)= P(DoomSoon) * 1 + P(DoomLate) * q. Inserting into the above equation we get:

P(DoomSoon)= p_S * P(DoomSoon) + p_S * P(DoomLate) * q <=>

(1-p_S) * P(DoomSoon) = p_S * P(DoomLate) * q <=>

P(DoomSoon) = p_S * P(DoomLate) * q / (1-p_S) <=>

P(DoomSoon)/P(DoomLate) = p_S * q / (1-p_S).

In other words the relationship between the "real" priors is:

P(DoomSoon)/P(DoomLate) = p_S/p_L * 200b/200tr and therefore P(DoomSoon) is much smaller without knowledge of my birthrank.

Once my birthrank is told to me (and it is indeed R<200b) I will update my priors and conclude that P(DoomSoon|R<200b)/P(DoomLate|R<200b) = p_S/p_L and sanity is restored.

I hope that helps!

Thanks very much! The objection is much clearer now, and it does seem to be a good one.

DeleteInteresting to see that the objection I was fuzzily waving at has come up a lot here. If I was unaware of my birth rank, then per the SIA, the fact that I am born at all weighs in favor of DoomLate, since more humans are born in that scenario.

ReplyDeleteDA pretends to weigh in favor of DoomSoon, but it assumes that birth rank is random, whereas in fact the only reason that we formulated DoomLate and DoomSoon in this way is *because* we live before DoomSoon. Anyone trying to use the DA to predict the future has a 0% chance of living after DoomSoon happens (however they define DoomSoon). Therefore there is an equal number of people to use the DA in each scenario, and they have the same birth ranks, so the fact that I am using DA and have a given birth rank gives me no information to distinguish the two.

So what I would say is that DA is invalid as a whole, and the SIA is valid, but gives no information that further predicts the future once I know my birth rank (like knowing that I am in cell 7 in the above example from Richard). If I did not know my birth rank, the SIA would argue in favor of realities with very many people. (Some variant may still argue in favor of the existence of aliens, since my birthday rank amongst all intelligent life is unknown!)

One more twist on the Sleeping Beauty problem. What if, instead of flipping a coin, they woke Sleeping Beauty three times, once with "Heads" written on a sign behind her, and twice with "Tails" written on the sign, and she has to say what probability it is that the sign says "Heads" this time. Is the probability 1/3? The only difference between this and the other example is that this happens in three trials in one world, whereas the original splits the everything into two possible worlds.

Sean, that's an interesting variation. I have a final (maybe) variation on the SBP. Maybe it will convince the halfers. Maybe not. Here goes anyway: Suppose you do the experiment with 100 beauties. Forget the amnesia and the Monday/Tuesday stuff. After the beauties are all sleeping, a coin is flipped. If heads, then one beauty is selected at random. She is awakened asked what her belief is about the coin toss. If tails, then two beauties are selected at random. They are awakened (but kept out of each other's presence, so neither knows there is another beauty awake) and asked what their beliefs are about the coin toss.

DeleteFrom the point of view of a Bayesian beauty who is awakened and interviewed, what is the probability that the coin landed heads?

In other words if it was 100 copies of Bostrums instead of beauties, and the experimenters killed a person every time a wrong answer was made, Bostrum would end up killing a lot of people.

DeleteI think what leads at least some halfers astray is the intuition that a fair coin's 1/2 probability is intrinsic to it. They say stuff like "1/1,000,000??? That's a lot of confidence when the coin isn't even tossed yet."

What bugs me is that when our question is how long into the future human species will exist, why don't we take into account other knowledge available to us, such as what we know about great extinctions, a star's life time etc., and instead rely solely on Doomsday Argument? I thought earlier that this was what you were talking about but I was skipping and had misunderstood.

brainoil: "What bugs me is that when our question is how long into the future human species will exist, why don't we take into account other knowledge available to us, such as what we know about great extinctions, a star's life time etc., and instead rely solely on Doomsday Argument? I thought earlier that this was what you were talking about but I was skipping and had misunderstood."

DeleteYes, I think that when you see that any kind of selection bias or whatever that you introduce gets cancelled out by an opposing bias when all of the conditionals are accounted for, the final expectation ends up the same as your prior expectations, which are due to the things you mention.

The SBP is a side track which is related to the study of the assumptions leading to the conditionals. In fact if I think about SBP the right way, I can see myself in agreement with the halfers, and then a little more thought swings me back into the thirder camp. What I think is going on is that the SBP is actually two distinct mathematical problems, with different formal statements, and with different correct answers, but when we try to express those problems in English, the language doesn't capture the subtle differences between the two. So they look like the same problem, but as we think about the problem, we get locked into one frame of mind or the other, corresponding to one of the two formal problem statements, so the opposing solution doesn't make sense to us.

That said, I think it is valid to apply the SIA in the case of the DA. That biases us toward longer-lived civilizations, because those civilizations produce more observers and so are more likely to produce each of us. Then the Carter-Leslie conditional can be applied and that takes us back to the original prior (before the SIA bias is applied), which is whatever we think it is based on pure empirical results such as frequency of extinctions, life spans of stars, etc.

The crux is that the Carter-Leslie conditional multiplier (the ratio of populations of long-lived and short-lived civilizations) is a *correction* for the SIA. If SIA is assumed, then you need the correction. If SSA is assumed, you do not need it.

If I got that statement right, then the SBP is actually a strong demonstration of the flaw of the DA. Under the SIA assumption, Beauty arrives at 2:1 odds for tails, because there will be (for her) twice as many "tails observation events" as "heads observation events". She doesn't know if it is Monday or Tuesday. "Monday" corresponds to "low birth number" in the Carter-Leslie DA. When this knowledge is applied to Beauty's 2:1 odds, it renormalizes back to 1:1 which is the original odds of a fair coin flip.

If instead you start with the SSA, then in effect all observations of the same event are treated as a single observation, so the relative probabilities of the alternatives are unchanged from the prior. In that case, it doesn't make sense to apply the Carter-Leslie conditional modification, because there is nothing to renormalize. In the case of the SBP: If Beauty takes the halfer position, which corresponds to the SSA, then she immediately gives equal odds to heads and tails. So, learning that it is Monday does not further inform Beauty's estimate of the odds in this case, because Beauty did not treat her potential Monday and Tuesday observations in the tails history as two separate observations.

In summary, we have two approaches that give the same answer:

Prior odds * SIA multiplier * Carter-Leslie multiplier = Posterior odds same as prior odds

or

Prior odds * SSA multiplier = Posterior odds same as prior

So the SSA multiplier is 1.

In other words you can start with either assumption (SIA or SSA), but you must choose the next step carefully to get the right answer. SIA and SSA are just different ways of thinking about the problem, one of which entails an additional step to reach the true odds, while the other does not.

(continued in next post)

(continuation)

DeleteI think the DA is calculating

Prior odds * SSA multiplier * Carter-Leslie multiplier = posterior odds *not* the same as prior odds

and I think this is wrong. You can see that this is wrong when you apply it to the SBP, where prior odds are 1:1, the SIA multiplier is 2 (2 observances of tails to 1 of heads), and the Carter-Leslie multiplier is 1/2 (ratio of "populations" -- i.e., events -- between the two alternate histories). The SSA multiplier is 1 as noted above. Both valid approaches I gave above give final odds of 1:1. The DA's approach gives an answer (after Beauty learns that it is Monday) that favors heads by 2:1!

Fascinating. This is the first time I'm learning most of these arguments so I might be wrong, but I think you're right.

DeleteIf the mere fact you're alive doesn't tell you anything, the same way the mere fact you're awake doesn't tell you anything in SBP, the fact that you're an early human shouldn't tell you anything either, the same way learning that it's Monday doesn't tell you anything in SBP. Doomsday thinks it does.

Great article. I think you are right to have your existential intuitions in #6. I hope you keep pursuing them. I think #6 is also the root of the problem with Boltzmann Brains. And it might have implications for Thermodynamics. Smuggled into the Anthropic Reasoning process is the notion that "finding oneself" at one point in time is a variable, that it has import to be here now rather than somewhere else at another time. But now has always stood outside of probability. Probability is about the future, not the present. Probability wave functions disappear in the now. No matter how unlikely the now can be, it immediately becomes the prior. "You happen to be winning the lottery now" is less likely than "You happen to be dreaming you have won the lottery right now" but people win the lottery every day.

ReplyDeleteI'm not sure if Boltzmann brains can be solved in the same way as the DA. Sure, the SIA favors the conclusion that we are Boltzmann brains (or rather, that I am a Boltzmann brain, since on that line of reasoning the rest of you are just figments of my Boltzmann imagination) -- assuming Boltzmann brains are not only possible but more common than actual biological brains.

DeleteI actually have a strong suspicion that Boltzmann brains are physically impossible. In other words, not every configuration of a collection of particles is physically possible -- some are prohibited. And among those that are prohibited are those that include self-aware entities that are not actually connected (in a sensory way) to a physical environment. Of course this is a conjectural solution: I don't think it can be proven or falsified. But then, the existence of Boltzmann brains is probably unprovable and unfalsifiable as well.

Richard, I disagree with your assumption about which configurations would be prohibited. Since the configuration of my brain is apparently not prohibited, it is difficult to see how an exact copy of it would be.

DeleteA problem I saw with a Boltzmann brain as I understood it was it assumed an infinite number of non-zero opportunities for a brain to come together. But calculus teaches me that an infinite series of non-zeroes does not necessarily go to infinity if the terms are decreasing. As entropy increases, wouldn't the likelihood of an appearance of a Boltzmann brain in a given time interval decrease? If that is the case, Boltzmann brains are not inevitable.

But it seems Boltzmann was talking about an infinite number of universes. Given worlds enough and time, I think they would have to happen.

"Richard, I disagree with your assumption about which configurations would be prohibited. Since the configuration of my brain is apparently not prohibited, it is difficult to see how an exact copy of it would be."

DeleteBut a Boltzmann brain (the "exact copy") interfaces with a wildly chaotic Universe, whereas (presumably) yours interfaces with a strongly patterned Universe. If not prohibited, I would at least say "far more improbable". I think the interface *must* affect the likelihood of a copy existing. This is an underlying assumption of any meaningful epistemology.

"Richard, I disagree with your assumption about which configurations would be prohibited. Since the configuration of my brain is apparently not prohibited, it is difficult to see how an exact copy of it would be."

ReplyDeleteI categorized it as a conjecture rather than assumption - more like food for thought than something you can build a testable theory on. The idea behind it, though, is that observations are a reflection of what's on the other side of the brain/Universe interface. A Boltzmann brain would have a boundary with something that is wildly chaotic, and these boundary conditions would have to be reflected inside the brain as well. But we observe a (mostly) orderly Universe. It's a view that meshes with my intuitions well, but as I said, not something that can be easily tested, if at all.

"But it seems Boltzmann was talking about an infinite number of universes. Given worlds enough and time, I think they would have to happen."

Either that, or an infinite Universe. I was thinking of one of the conjectural possibilities described by Sean Carroll in "From Eternity to Here" -- an infinite high-entropy sea of particles in which periodic fluctuations give rise to regions of low entropy such as Boltzmann brains or, in the more extreme cases, entire local observable Universes such as ours. I don't know if it's warranted to assume that those local fluctuations are completely unconstrained, and the thought of a highly organized entity interfacing directly to a chaotic environment, and internally producing an image of an environment that is not chaotic, seems like a potential violation of constraints.