Doomsday Argument and the False Dilemma of Anthropic Reasoning
Doomsday Inference
Can we use probability theory to estimate how many people there will be throughout the whole human history? Sure. We can build a probability model, that takes into account birth rates, possible existential hazards, ways to mitigate them and multiple other factors. Such models tend not to be very precise, so we would have pretty large confidence intervals but, we would still have some estimate.
Hmm… this sounds like a lot of work for not much of a result. Can’t we just use the incredible psychic powers of anthropics to circumvent all that, and get a very confident estimate just from the fact that we exist? Consider this:
Suppose that there are two indistinguishable possibilities: a short human history, in which there are only 100 billion people and a long human history, in whichthere are 100 trillion people. You happen to be born among the 6th 10-billion group of people. What should be your credence that the history is short?
As short and long history are a priori indistinguishable and mutually exclusive:
Assuming that you are a random person among all the people destined to be born:
According to the Law of Total Probability:
Therefore by Bayes’ Theorem:
We should be extremely confident that humanity will have a short history, just by the fact that we exist right now.
This strong update in favor of short history solely due to the knowledge of your birth rank is known as the Doomsday Inference. I remember encountering it for the first time. I immediately felt that it can’t be right. Back in the day I didn’t have the right lexicon to explain why cognition engines can’t produce knowledge this way. I wasn’t familiar with the concept of noticing my own confusion. But I’ve already accustomed myself with several sophisms, and even practiced constructing some myself. So I noticed the familiar feeling of “trickery” that signaled that one of the assumptions is wrong.
I think it took me a couple of minutes to find it. I recommend for everyone to try to do it themselves right now. It’s not a difficult problem to begin with, and should be especially easy if you’ve read and understood my sequence on Sleeping Beauty problem.
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Did you do it?
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Well, regardless, there will be more time for it. First, let’s discuss the fact that both major anthropic theories SSA and SIA accept the doomsday inference, because they are crazy and wrong and we live in an extremely embarrassing timeline.
Biting the Doomsday Bullet
Consider this simple and totally non-anthropic probability theory problem:
Suppose there are two undistinguishable bags with numbered pieces of paper. The first bag has 10 pieces of paper and the second has 10000. You were given a random piece of paper from one of the bags and it happens to have number 6. What should be your credence that you’ve just picked a piece of paper from the first bag?
The solution is totally analogous to the Doomsday Inference above:
But here there is no controversy. Nothing appears to be out of order. This is the experiment you can conduct and see for yourself that indeed, the absolute majority of cases where you get the piece of paper with number 6 happen when the paper was picked from the first bag.
And so if we accept this logic here, we should also accept the Doomsday Inference, shouldn’t we? Unless you want to defy Bayes’ theorem itself! Maybe the ability to predict the future might appear counterintuitive for our sensibilities, but that’s what the math says. And if your intuition doesn’t fit the math—that sounds like your problem. Change your intuition, duh.
This is the position of Self-Sampling Assumption. According to which you should reason about your own existence the exact same way you reason about this paper picking example—as if you are randomly sampled from all existent people.
Now, I’m nothing if not in favor of changing one’s intuition to fit what the math says. But it’s important to make sure that the mathematical model in question is appropriate to the problem at hand. And here, it doesn’t really look like this.
For once, consider that picking a piece of paper with number 6 from the second bag is an extremely rare event, which on average happens only once per 10,000 tries. This naturally justifies the severity of the update in favor of of the first bag, when seeing number 6 on the paper.
But applying the same logic towards one’s existence would mean that every person who has ever lived observed an extremely rare event, that predictably updated them in favor of short history! There has to be something very different about these two cases.
Biting the Infinity Bullet
Now you might have heard that an alternative anthropic theory, Self Indexing Assumption, provides an opportunity to evade the doomsday conclusion. This is true.
SIA doesn’t directly challenges the Doomsday Inference, after all—math is math. But it claims that there is one extra compensatory factor.
Consider this modification of our non-anthropic paper picking-problem:
You pick the piece of paper from the bag yourself. The bags are of such sizes that when you put you hand in the second bag you always find a piece of paper immediately at the bottom of the bag, while when you put your hand in the first bag there is a lot of empty space so you may not find the piece of paper immediately. You immediately find a piece of paper in the bag. What should be your credence that you are picking from the second bag?
This is the exact same update as previously but in the opposite direction. Upon immediately finding a piece of paper, you have to very strongly update in favor of it being the second bag. So then, after learning that the number on the paper is 6 and making a similarly strong update in favor of the first bag, both updates cancel each other out, and the situation adds up to normality.
This is the position of Self Indexing Assumption. According to which you should treat your existence as this paper picking example—as a random sample from all possible people, and with a very high probability your existence wouldn’t even happen.
In a sense there is some elegance to it. But I can’t help but see it as a completely unsatisfying band aid over a broken bone. We haven’t really addressed the general problem, we just found a clever hack to make the result fit our intuition in this particular case. Actually, now the situation is even worse. Instead of one extremely low probability event that every person is guaranteed to experience upon realization that they exist, there are two of them, pointing in opposite directions.
And the thing about dramatic updates is that once you’ve updated in one direction you really do not expect to update back. In our idealized scenario where a person simply started to accept that they are among 6th ten-billion group of people, the situation appears normal.
But under SIA, you are supposed to believe that the existence of N people is more likely than n people for any N > n. SIA’s median prediction of the number of people who exist is infinity. So no amount of evidence can realistically persuade you that you are merely n-th person. SIA followers are extremely confident that there are aliens/other universes/simulations—anything that would explain why there are more people than it seems, regardless of actual, non-anthropic evidence.
False Anthropic Dilemma
And so this is our dilemma. Either you accept SSA or SIA with all the weird consequences they entail. Either you believe that humanity is doomed to a short timeline or that there are infinite people. Either we can predict the future, or have the knowledge about the infinite present even without looking around. All anthropic theories are inevitably presumptuous. We are simply left to choose which kind of presumptuousness is more to our liking. Because, surely, there is no conceivable, non-ridiculous alternative, right?
You may notice the obvious parallels between Doomsday argument and Sleeping Beauty. How it at first may seem that there is no reasonable option except Lewisian Halfism and Thirdism. How Lewis’s model expects the Beauty to be able to predict the outcome of a future coin toss, if she knows that she is woken on Monday, while Thirder models are more confident in Tails just from the completely expected fact of awakening at all.
And just like in Sleeping Beauty there is this core wrong assumption, that we need to get rid of...
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Okay, this is as many hints as I’ll give, so it’s your last opportunity to solve the problem yourself, if you haven’t done so already.
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And this assumption is...
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That you are randomly sampled throughout time, of course!
Third Alternative
Just like Sleeping Beauty shouldn’t reason about her current awakening as being somehow randomly sampled between three awakening states, you shouldn’t reason about your existence as being a random sample from all the people throughout time. Just like the previous and next awakenings of the Beauty are not, in fact, independent from each other, neither are past and future people. Because some are actually ancestors of others.
If we didn’t know anything about human reproduction or causality, it would be understandable why we could assume that being born is a random sample. But we know.
If we had some serious evidence that souls not only exist, but also precede the existence of a person, that a soul is somehow chosen to be instantiated in newborn, then it would be understandable why we could assume that being born is a random sample. But we don’t have such evidence.
If we trace your causal history, in accordance with our modern knowledge about the way the universe works, It looks like you were born because your parents had a particular sexual act at a particular time. Your body, brain and, therefore mind are downstream of it. Saying that you’re randomly sampled between all people throughout time is utter nonsense. Such a probability model simply ignores a significant part of our knowledge about the world, smuggling in idealist philosophy and a naive idea of souls.
You couldn’t possibly have come into existence in the distant past, before your parents, or in the far future, after they already died. Neither could your parents exist in a different time for the same reasons. Therefore, our knowledge about the universe leaves us a very limited time frame for your possible existence, precisely among the 6th ten-billion group of people, regardless of how many people will exist in the future.
And, therefore, the Doomsday Inference is wrong. Not because we should suddenly abolish Bayesian reasoning, when talking about anthropics, but because when we do it with the right assumptions everything adds up to normality. No precognitive powers, no certainty in infinities, no observation of extremely rare events, no huge updates compensating each other. If humanity indeed is doomed to have a short history it will be for completely mundane reasons, not because you are your parents’ child.
Meanwhile, lets not doom ourselves to the perpetuation of the Anthropic False Dilemma throughout the whole human history, be it short or long. There is a non-crazy way to reason about the matter, instead of biting one or the other ridiculous bullet. Just use basic probability theory, while making sure that your mathematical model makes sense in the context of the real world problem you are talking about.
The main argument against the idea that “I can’t think about myself as of random sample’ is that it can be experimentally tested.
For example, I can use the month of my birth to get reasonable estimation of the number of months in year: I was born in September, so it is reasonable to give 50 per сent credence to 18 months in a year, which is close to 12.
Similarly I can use the distance to equator of the location where I was born to estimate the total diameter of the Earth.
You may respond that in this example there is no adding up of subsequent observers. But I can estimate median human life expectancy just using my age.
I just imagined and performed a new experiment—I want to learn the duration in hours of a day, using just one sample—I look at my clock and it is now 15.14. It gives 50 per cent that total number of hours in a day 30 that is reasonably close to 24.
The idea is not “I can’t use the dates of births of people as a way to generate pseudo-random numbers”. It’s not even “I can’t think about myself as of a random sample in any circumstances”. The idea is that “I’m not a random sample from all the people throughout history in the context of Doomsday Argument”. And it indeed can be experimentally tested. The discovery of the fact that some people are parents of the others proves it, for every case not involving souls.
Consider some disease, which 2% of humans have. How likely you should expect yourself to have it? Well, as long as you do not have any other information, you can start from the assumption that it’s randomly spreaded throughout all the human population and therefore you have 2% chance to have it. But as soon as you learn that, say, 90% of cases are in the Third World, the situation changes dramatically.
For Doomsday argument we should use only competent observers who at least can think about complex math of Doomsday argument. In other words, I randomly selected from the set of competent observers.
However, my parents was not the members of this class. They were clever, but never look at that direction. So your argument about parents doesn’t work for DA. Also, first competent observers appeared around 1970, so the Doomsday is in next few decades.
Also, I still not getting the main idea of your argument. For example, if there are three people in a room who are my grandmother, mother and I. I know only that I am one of them—how it changes that I am grandmother?
Irrelevant. The fact that your parents didn’t think about DA doesn’t mean that they are not your parents. You could only exist as a result of a particular sexual act between your parents. And the same logic applies to them and their parents and so on. Therefore, in order for you to exist at some other time all your chain of ancestry has to exist at some other time as well, which is impossible. Therefore, you could exist only in a very limited timeframe. The fact that you decided to additionally restrict your “reference class” to only people who think about DA, doesn’t remove the time restriction.
If you know that there are three people in the room, one of them you and the other are your mother and your grandmother, then naturally you know that you are neither your own mother nor your own grandmother.
What you are saying reminds me of FNIC—full non-indexical conditioning. It was discussed on LW. It still may have some anthropic effects—like panspermia is more likely, but may be used against DA.
However, most facts about my personality are irrelevant to my thought process about anthropic. There was a good term “people in my epistemic situation” used in the SIA-thread here in LW.
another example: I can use my age as random sample of human ages and predict that median human life expectancy based on it. This works despite my ages being subsequent from one to another.
For three person in the room—I assume that some amnesia drug removed the knowledge of who exactly I am.
As the originator of Full Non-indexical Conditioning (FNC), I’m curious why you think it favours panspermia over independent origin of life on Earth.
FNC favours theories that better explain what you know. We know that there is life on Earth, but we know very little about whether life originated on Earth, or came from elsewhere. We also know very little about whether life exists elsewhere, except that if it does, it hasn’t made its existence obvious to us.
Off hand, I don’t see how FNC says anything about the panspermia question. FNC should disfavour the “panspermia difficult, independent origin also difficult” possibility, since it makes our existence on Earth less likely (though it does make the non-obviousness of life elsewhere more likely, so the overall effect is unclear). But between “independent origin easy” and “independent origin hard, but panspermia easy”, FNC would seem to have no preference. Both make life on Earth reasonably likely, and both make lots of life elsewhere also seem likely (one then needs to separately explain why we haven’t already observed life elsewhere).
My reasoning is the following:
1.My experience will be the same in the planets with and without panspermia, as it is basically invisible for now.
2. If Universe is very large and slightly diverse, there are regions where panspermia is possible and regions where they are not—without any visible for us consequences.
3. (Assumptions) Abiogenesis is difficult, but potentially habitable planets are very numerous.
4. In the regions with pasnpermia, life will be disseminated from initial Edem to millions habitable planets in the Galaxy.
5. For every habitable planet in the non-panspermia region there will be million habitable planets in panseprmia-region.
6. As there is no observable differences between regions, for any my exact copy in non-panspermia region there will be million my copies in paspermia regions. (May be I am wrongly understand FNIC, but this is how I apply it.)
What do you think?
You’re forgetting the “non-indexical” part of FNC. With FNC, one finds conditional probabilities given that “someone has your exact memories”, not that “you have your exact memories”. The universe is assumed to be small enough that it is unlikely that there are two people with the same exact memories, so (by assumption) there are not millions of exact copies of you. (If that were true, there would likely be at least one (maybe many) copies of people with practically any set of memories, rendering FNC useless.)
If you assume that abiogenesis is difficult, then FNC does indeed favour panspermia, since it would make the existence of someone with your exact memories more likely. (Again, the non-observation of aliens may provide a counteracting inference.) But I don’t see any reason to think that abiogenesis is less (or more) difficult than panspermia, with our present state of knowledge.
Abiogenesis seems to depend on the random synthesis of a 100-pieces long stand of RNA capable to self-replicate. Chances of it on any given planet is like 10E-50.
Interstellar panspermia has much less variables, and we know that most of its ingredients are already in place: martian meteorites, interstellar comets. It may have like 0.01 initial probability.
Non-observation of aliens may be explained by the fact that a) either p(intelligence|life) is very small or b) we are the first of many nearby siblings and will meet them soon (local grabby aliens).
I think you’re overly-confident of the difficulty of abiogenesis, given our ignorance of the matter. For example, it could be that some simpler (easier to start) self-replicating system came first, with RNA then getting used as an enhancement to that system, and eventually replacing it—just as it’s currently thought that DNA (mostly) replaced RNA (as the inherited genetic material) after the RNA world developed.
Actually, it looks like from this that FNIC favors simpler ways of abiogenesis—as there will be more planets with life and more chances for me to appear.
I didn’t explore it in depth but as far as I know FNIC thirds in Sleeping Beauty which I do not find acceptable. Thinking that pamspermia is more likely for no reason also seems quite bad.
I find this “people in my epistemic situation” thing to be pretty unhelpful. Being in a particular epistemic situation is itself may or may not be evidence in favor of particular hypothesis, depending on the setting you are in. We should just be thinking in terms of probability experiments and its causal structure—do the thing we do for every non-anthropic probability theory problem. There is really no need for an extra complication.
It works as long as the causal process that leads to you selecting the age for the eastimate can be approximated as low probability occurance that can happen at any age and so the sequential nature of age doesn’t matter. On the other hand, if you simply tried to approximate median human age by doing that estimate at every year, then your results would be pretty bad. Most of the estimates would be very off.
Specify the conditions of the experiment in details then. What information does the participant have? Probability of which event should be estimated?
The case for panspermia seems simpler than Sleeping beauty, as it doesn’t include possible worlds. Imagine that there are two regions of the Universe, in one of which panspermia is possible and in another is not. The one where it is possible has, 100 times more habitable planets per volume. This suggests that we are more likely to be in the region in which panspermia is happening.
Most of the estimates will not be very off. 90 per cent of them will give the correct order of magnitude.
Since you conclude that both SIA and SSA are flawed because we know a lot about our parents, let’s see if that works. Imagine a world where people spend their first years not knowing much about their parents, or about human reproduction. Suppose they each live in a kind of egg, and receive newscasts from outside only carrying information about the particular anthropic problem we want them to solve (e.g. “the world currently contains N people”, or “there are two theories about astronomy” and so on). How should people solve anthropic problems under such conditions? Should they use SIA or SSA? Or should they still reject both, but based on some other argument, and the “we know a lot about our parents” argument was a red herring?
This is probably a good situation to abstain from judgement all together. But if we really want some number, just for the sake of it, then adopting equiprobable prior between SSA, SIA and No Randomness seems to be pretty decent. Or confidence interval wide enough so that all the three estimates fit into it. This is not going to be a particularly helpful estimate, but what else did you expect from reasoning based on your own ignorance?
SIA doesn’t compensate SSA for Doomsday argument. SIA predicts infinite number of civilizations in the universe, but not the median duration of the existence of typical civilization.
The problem is underspecified, specifically the “you were given a random piece of paper from one of the bags” step. That could be either “one bag was chosen at random, and then one piece of paper was chosen at random from within the bag” (in which case “the observed number was 6” gives you information about which bag the paper was pulled from) or “the papers from both bags were mixed together, then a piece was pulled from the mixed pile” in which case “the observed number was 6” gives you no information about the original bag the piece of paper came from.
I think the closest analogy to the question people are trying to answer with anthropic reasoning, though, is something like “N bags with the numbers 1 − 10 and M bags with the numbers 1 − 10000 were mixed together, and then a piece of paper was chosen at random from that combined pile. That paper was a 6. How does this update your estimation of the ratio of M to N?”
That said
Mostly agreed. If you fail to condition on the factors which caused you to ask the question, you will get garbage results.
Concretely, my scenario from before becomes something like “A strange man comes up to you and tells you that N bags with the numbers 1 − 10 and M bags with the numbers 1 − 10000 were mixed together, and then a piece of paper was chosen at random from that combined pile. He further says that he would only tell you his observation if it was between 5 and 9, and otherwise he would not have said anything. He tells you that the paper was a 6. How does this update your estimation of the ratio of M to N?”
I struggle to see how the latter is the natural reading of the condition of the experiment. Should I explicitly specify that no mixing of bags was involved, to clear the confusion?
You can, but then you’d need to justify why “choose a bag, then choose from within the bag” is analogous to the correct way to do anthropic reasoning, despite failing in an obvious way in non-anthropic scenarios (e.g. you don’t have an equal probabilityof choosing any human on earth by following the procedure “pick a random country, then pick a random person within that country”, so at least to me it would be surprising if that was the correct procedure across universes).
On the contrary. This is exactly the way we do reasoning about probability every time, the idea of “mixing bags” suddenly comes up only in anthropics. Which, I suppose, can be somewhat justified by the fact that this case is isomorphic to picking from a bag with possibility not to find anything, but the point is that its what requires extra justification, not the other way around.
When we evaluate alternative theories based on our available evidence we do not combine their possible outcomes together into one set and assume equiprobable distribution over the elementary events for no reason. We assign equiprobable prior to the theories and then evaluate their likelihood conditionally on available evidence via Bayes Theorem.
We are not supposed to have equal probability. I really don’t see where are you going with this example.
I’m not entirely sure what you mean by this. Can you give a concrete, non-anthropic example?
Alternatively, in your “one bag with numbers 1-10 and one bag with numbers 1-10000” example, would you change your answer if there were instead 100,000 identical small bags with numbers 1-10 and 100 identical large bags with numbers 1-10000? If that would change your answer, then we don’t have to do any mixing at all—the anthropic argument as I understand it says that observing a 6 gives you informaron about the ratio of small bags to large bags, rather than information about your bag.
Well, the paper picking is exactly this kind of uncontroversial non-anthropic example. I specifically started from it in order for the subject to be less confusing, so its a bit ironic.
You have two alternative hypothesizes about the bag. Either it contains 10000 paper pieces or just 10. You don’t have any additional information which of the hypothesizes is more likely. Therefore, you assign equal prior probabilities to this two hypothesizes. Then, after you received the paper with 6 written on it you update your credence according to Bayes Theorem. Now the theory that there are only 10 pieces of paper in the bag seems much more likely in light of this new evidence.
It would mean that prior probabilities between the two hypothesizes are not equal but are 1000:1 instead. Naturally this would affect the resulting credence but not the logic of the update.
Now let’s say we put all of the small bags in one sack, and all of the large bags in a different sack. Now we have a sack containing 100,000 small bags with numbers 1-10, and a sack containing 100 large bags with numbers 1-10,000. Does this now change your prior when choosing a random piece of paper?
What if, instead of physically putting the small and large bags inside of sacks, we just put them in the categories “small” and “large” in our head?
Basically what I’m saying here is that “we choose at random” or “we have a uniform prior” is underspecified. Which is why I’m asking for a concrete example where the “choose at random” procedure is more clearly defined, and a justification for why we should expect that specific “choose at random” procedure to map onto the question of whether we should expect to find ourselves in a world where we are probably not among the very first or very last observers asking that kind of anthropic question.
As a note, this should all add up to normality. If everyone who evaluates the DA guesses “I am not in the chronological first 1% of people in my world to ask this question, and I am also not in the chronological last 1% of people in my world to ask this question”, we should expect 98% of those people to be correct in their guess.
Edited to add: One more intuition pump for the original scenario with the small bag of 1-10 and the large bag of 1-10,000: Instead of saying “one piece of paper is drawn and handed to you, and it happens to be a 6” you say “every piece of paper is drawn and handed to a person—each paper is handed to exactly one person, and every person has exactly one paper—and you are one of those people. None of the people can communicate with each other in any way. What probability do you assign that the paper came from the small bag? Now you observe that the number on your piece of paper is a 6. What probability do you assign that the paper came from the small bag?”. I think that intuition pump maps better to the anthropic arguments I’ve seen.
Here we are back to the 1:1 prior between theories. It doesn’t matter how many equal bags you put together. 1⁄10 = 100000/1000000.
Unless we explicitly specify it, like I did in the previous comment where I said that uniform prior is over two alternative hypothezises which we do not have any evidence for or against.
I feel that there is some confusion about fundamentals of the probability theory is going on and that’s why we are talking past each other. Lets take a step back. Forget about anthropics. Where does the idea of uniform prior even comes from? What is its justification?
Suppose you have a bag about which you know that it either contains 9 red marbles and 1 blue marble or 5 blue marbles and 5 red marbles. You have no idea which is more likely. You blindly pick one marble from the bag and its blue. How should you reason about this scenario?
Should you assume that the two “possible bags” are mixed together and there is uniform prior over the marbles from the united bag? Or should you assume that there is uniform prior over two alternative theories? Or should you assume that there are some M and N such as M bags with mostly red marbles and N bags with equal number of marbles are mixed together? Or should you assume that there is uniform prior over M+N bags? What should be different about the setting of the experiment or your knowledge state about it so that the answer was different?
Oh, you believe there is a difference between these two scenarios, don’t you? I suppose I’ll have to make a post about it as well. For now consider this problem:
There are 100 students and 100 tickets. Every student blindly picks a ticket and then its removed from the pool of available tickets, therefore every student gets a unique ticket. Students can decide the order they pick tickets among themselves. You’ve learned only half of the tickets. What is your optimal strategy to maximize your chances to pass the exam? Should you go first? Should you go last? Should you wait till there are only 50 tickets left? What are you probabilities to pass the exam when you follow these strategies?
For clarity, what I think in the case of “scenario starts with one small bag containing 1-10 and one large bag containing 1-10,000, each piece of paper goes to exactly one person, each person gets exactly one piece of paper, you are one of the people” is that, prior to seeing what number is on your piece of paper, you should think there’s a 10,00010,000+10≈0.999 chance that the paper you received came from the large bag—i.e. not a uniform prior of “50/50 small bag or large bag”.
Once you observe that your piece of paper has a 6 on it, you now know you either are the person who got the single 6 from the small bag or the person who got the single 6 from the large bag, so your posterior is 11+1=0.5 that the paper you received came from the large bag. Had you instead gotten a piece of paper that had e.g. 7194 on it, you would now be sure that your paper did not come from the small bag.
This is why I am skeptical of your choice to use the randomization procedure of “first choose the bag, then choose within the bag”—that choice of randomization procedure only works the first time you run it, but if you try to distribute the papers one per person that randomization procedure breaks down.
As a note, you could rescue your 50⁄50 prior on small bag vs large bag by saying that the small bag has 1,000 pieces of paper with a 1 on them, 1,000 pieces of paper with a 2 on them, [...], 1,000 pieces of paper with a 10 on them. But in that case, if you observe that your paper has a 6 on it, you know that there are 1,000 people who got a 6 from the small bag and 1 person who got a 6 from the large bag, and so your posterior is that there’s a 11001≈0.001 chance that your paper came from the large bag.
In both cases, observing that your paper had 6 written on it causes you to increase your expectation that the paper you received came from the small bag.
You need to plug some number in for the prior if you want to get a posterior out, and usually you have enough evidence that the number you put in for the prior doesn’t really matter. “A uniform prior” seems like a pretty good one in the complete absence of any other information (note, though, that “there are bags containing different numbers of pieces of paper” is nonzero information). There’s some theoretical justification in the form of e.g. Laplace’s Rule of Succession but honestly the main justification I use is “probabilities are facts about my model of the world, not facts about the world itself, and so the mechanics of my model dictate the probability, and the prior is no exception”.
Absent any further information, there is no benefit of going earlier or later—you have a 50⁄100 chance of passing the class regardless of your strategy, because none of the choices given give you any information about the distribution of remaining tickets.
Counter-scenario for you:
There are 100 students and 100 tickets. The tickets are distributed between two identical bowls which are shuffled around prior to each student’s turn. The first bowl contains the numbers 1 − 10, and the second bowl contains the numbers 11-100. Any student who picks a ticket numbered 51 or higher will pass the class. Each student will pick a bowl. If that bowl contains any tickets, they will blindly pick a ticket from within that bowl. If the bowl they choose contains no tickets, they will blindly pick a ticket from the other bowl.
What is your optimal strategy to maximize your chances to pass the exam? Should you go first? Should you go last? Should you wait till there are only 50 tickets left? What are you probabilities to pass the exam when you follow these strategies?
Do you mean a situation where every piece of paper from each of the two bags is given, therefore 10010 people got a piece of paper? If so this is very much not what I’ve been talking about. We are dealing with an either/or case, where only one bag is used to give papers, but you have no idea which one. This should be obvious in the context of doomsday argument, because humanity doesn’t simultaneously has long and short history.
Anyway, as I said, lets forget about anthropics for now and deal with the marble picking example from the previous comment.
Oh sure, probabilities are about the model. But we want our models of the world to correspond to the way world actually is, so that our models were useful. You want a model that systematically produces correct answers in reality, not just being self-consistent.
It sure does. I encorage you to think about why. Anyway, lets, for now, simply accept that if we do not have any information about some alternatives we assume equiprobable prior. Now consider these scenarios. I think They highlight the crux of disagreement very well:
1. There is a bag on your table. You know that it’s either bag 1 or bag 2 and you know nothing else. What should be your credence that it’s bag 1?
Here we have uniform prior between bags. P(Bag 1) = P(Bag 2) = 1⁄2
2. A marble was picked. You know that there are 20 possible marbles that could be picked: 14 red and 6 blue. What is your credence that the marble blue?
Here we have uniform prior between marbles: P(Blue) = 3⁄10
3. There is a bag on your table. You know that it’s either bag 1 which contains 9 red and 1 blue marbles or bag 2 which contains 5 red and 5 blue marbles. You see a person blindly picked a marble from the bag. What’s the probability that the marble the person picked is blue.
Are we supposed to be using uniform prior over bags or are we supposed to be using uniform prior over marbles here? This is an important question because even though In this case both produce the same answer:
Over bags:
P(Bag 1) = P(Bag 2) = 1⁄2
P(Blue) = P(Blue|Bag 1)P(Bag 1) + P(Blue| Bag 2)P(Bag 2) = 1⁄20 + 1⁄4 = 6⁄20 = 3⁄10
Over marbles:
P(Blue) = 3⁄10
If the arrangement of the marbles was different, say, bag 1 one contained 10 red and 2 blue while bag 2 contained 4 red and 4 blue, the situation changes:
Over bags:
P(Blue) = P(Blue|Bag 1)P(Bag 1) + P(Blue| Bag 2)P(Bag 2) = 1⁄22 + 1⁄4 = 6⁄20 = 13⁄44
Over marbles:
P(Blue) = 3⁄10
So how do we decide what to do? What is the principled position here? Notice that this case has absolutely nothing to do with anthropics and is basic probability theory.
I think people are just choosing comparison sets wrong? Using the bag anology, there are two pieces of paper labeled “6” so upon being handed a random piece of paper labeled “6″ I should be 50⁄50 as to whether it came from the first bag or the second bag, No?
Eh I didn’t think you can just ignore facts like the components of the bag. You could actually do this experiment, and the probability won’t be 50%.
In the Laplace’s sunrise problem you are not assuming that you are randomly sampled in time, but the prediction is basically the same as in Doomsday argument—if applied to the same period of time.
Typically the sunrise problem is applied only to the next day, and DA is applied to the time several times longer than the whole history of existence of qualified observers, and also sunrise problem is discrete. But if we apply them to the same period of time, they give the same probability.
Could you explain the resemblance between Laplace’s sunrise and Doomsday argument and why the Laplace’s sunrise prediction is problematic?
In the Laplace’s sunrise problem the question is: what are the chances that Sun will rise again after it has raised 5000 previous day. Let’s reframe the problem: what are the chances that a catastrophe will not happen in the year 5001 given that it didn’t happen in the previous 5000 years. Laplace gives chances of no catastrophe as 1 − 1/(5000 +2). ’+2″ appears here because we are a) speaking about discrete events, and the next year is 5001.
So simplifying we get that Laplace gives 1/(5000) chances of catastrophe for the next year after 5000 year of no-catastrophe.
If we take Gott’s equation for Doomsday argument, it also gives probability of catastrophe 1/(5000) for the situation when I survived 5000 years without a catastrophe BUT was randomly selected from that period. Laplace and Gott achieved basically the same equation but using different methods.
I do not see Laplace’s problem as problematic, it is another version of Doomsday argument and both are correct. But is shows us that “random sampling’ is not a necessary condition of for having Doomsday argument.
I think there is another assumption (or another way of framing your assumption, I’m not sure if it is really distinct), which is that all people throughout time are alike. It’s not necessarily groundless, but it is an assumption. There could be some symmetry breaking factor that determines your position that doesn’t simply align with “total humans that ever exist”, and thus renders using the “birth rank” inappropriate to determine what number you have. This thing doesn’t have to be related to reproduction though.
It’s not a distinct assumption. It’s the same assumption, but formulated in a terribly confusing manner as a lot of things in anthropics are.
The “likeness” is relevant for SSA reference class which is somewhat of a free parameter in the theory. The correct reference class is the group of people from which you could’ve been anyone. And this question is very much related to reproduction in case of Doomsday argument. I’ll be talking about reference classes and how it’s technically possible to salvage SSA with them in a future post.
I think the factors that determine your reference class can be related to changes over time in the environment you inhabit, not just how you are built. This is what I mean by not necessarily related to reproduction. If I cloned a billion people with the same cloning vat over 1000 years, how then would you determine reference class? But maybe something about the environment would narrow that reference class despite all the people being made by the same process (like the presence or absence of other things in the environment as they relate to the people coming out of the vat).
In principle—yes. Different settings of probability experiments naturally lead to different reference classes. But in case of DA this seem to boil down to the question of reproduction.
A person created by this vat may start from assuming that you are a random person from that billion and then adjust their credence based on available evidence as usual. Like if you checked the year of your creation, now you treat yourself as a random person from the people created in this vat specifically in this year. You can frame is as change of reference class or as simple bayesian update on the available evidence.
While I agree with the notion that we cannot regard ourselves as random samples from all human beings past, present and future, I find the discussion wanting in vigorously defining the reference of “us”, or “me” or by extension “my parents”. Without doing that there’s always the logical wiggle room for arriving at an ad hoc conclusion that does not give paradoxical results, e.g. while discussion SBP, you suggested that “today” could mean any day, then attempting to derive the probability of “today is Monday” from there. That just doesn’t sit comfortably with me.
Similarly, in this post, while discussing the possibility of “my birth rank” you focused on the history of my family tree, arriving at the conclusion that I cannot be born much earlier/later than I really did (realistically speaking a few month earlier at most, otherwise I cannot physically be alive/exist). But that is not DA’s claim, the uniform distribution does not imply that your physical body could be born so prematurely such to predate the first ever human being. It simply says as an unknown prior, you being the son of your current parents and you being a different human being—e.g. someone born in the 2500s, ceteris paribus, are equally likely. And the fact that you are not someone born in the 2500s is the evidence of doom soon which drives the Bayesian update. The satisfactory rebuttal to DA should undermine that.
Let’s say I concede to your argument.: since I cannot be born much earlier or much later than I really am, therefore cannot regard myself as a random sample from all time. Then what? Is it reasonably to regard myself as a random sample of all human beings born within that short timeframe of my possible birth? Are you comfortable with that? Doesn’t it also carry the underlying assumption of preexistence of souls being incarnated into bodies?
Extra vigor is always nice but I don’t see how its necessary here. “I” am a downstream reasult of my parents having sex at a particular time. The same way “my father” is a downstream result of his parents having sex at a particular time. The same way “my mother” is a downstream result of her parents having sex at a particuler time and so on and so forth. This level of vigor is already enough to see that DA is nonsense.
I was showing that “Today” is a poorly specified term in the setting of Sleeping Beauty and if we try to define it as “Monday xor Tuesday”—the way we usually define it in such situations—we clearly observe that this event doesn’t happen in 50% of iterations of the experiment and so the popular claim that the Beauty always observes that she is “Awake Today” is wrong. As long as we stop thinking about “Todays” and instead make a model about the propbability experiment as a whole—as we are supposed to—everything adds up to normality.
Either DA does imply that my physical body could happen to exist in the distant past or distant future or it implies that there is something non-physical about my identity. Therefore, I mention that to rescue DA we need the concept of souls. Only this way the idea of me being a different human being is coherent.
All probabilistic models are approximations. This model is less unreasonable and, considering where the current sanity waterline for anthropics is, this is good enough. There is still room for improvement, of course.
As long as we are not talking about me being born to different parents but simply having a different birth rank—I’m born a bit earlier, while someone else is born a bit later, for example, - then no souls are required.
The point of defining “me” vigorously is not about how much upstream or physically specific we ought to be, but rather when conducting discussions in the anthropic field, we ought to recognize words such as “me” or “now” are used equivocally in two different senses: 1, the specific physical person, i.e. the particular human being born to the specific parents etc. and 2, just a reference to the first person of any given perspective. Without distinguishing which meaning in particular is used in an argument, there is room for confounding the discussion, I feel most of our discourse here unproductive due to this confusion.
This is interesting. I suppose being born to the same parents is just an example you used. e.g. are you a sample of all your siblings? And are other potential children your parents could have if a different sperm fertilize a different egg still you? In those cases there still would be the same problem of soul incarnations.
So my understanding to your position is that there is no problem as long as you consider yourself to be the same physical person, i.e. the same parents, the same sperm and eggs. The variation in birth rank is due to the events during pregnancy that makes the particular physical person born slightly earlier or later. And this is the correct way to thing about your birth rank.
If that’s your position, then wouldn’t your argument against regarding oneself as a random sample among all human beings (past, present, and future) ultimately be : “Because I am this particular physical human being”. And that there is no sense discussing alternatives like “I am a different physical human being”?
I don’t see how it’s happening here, but sure let’s try to be vigorous in this sense. Let me construct the argument, while tabooing the word “I” altogether:
Dadadarren is a result of a particular sexual encounter between dadadarren’s parents. Dadadarren’s mother is a result of a particular sexual encounter between dadadarren’s mother’s parents. Dadadarren’s father is a result of a particular sexual encounter between dadadarren’s father’s parents. And so on. Therefore, dadadarren is not a random person from all the people throughout human history. Therefore doomsday inference is incorrect for dadadarren.
Ape in the coat is a result of a particular sexual encounter between Ape in the coat’s parents. Ape in the coat’s mother is a result of a particular sexual encounter between Ape in the coat’s mother’s parents. Ape in the coat’s father is a result of a particular sexual encounter between Ape in the coat’s father’s parents. And so on. Therefore, Ape in the coat is not a random person from all the people throughout human history. Therefore doomsday inference is incorrect for Ape in the coat.
And so on. Its possible to construct this kind of argument for every person who has ever lived. Therefore, doomsday inference is incorrect in general.
Using “I” simply compresses billions of such individual statements into a shorter and more comprehensive form.
Yes, that’s exactly what it is. Unless there are souls—a non physical component to personal identity which can add additional uncertanity about the causal process, “I” is just a pointer that refers to a particular human being and therefore statement “I could have been a different human being” is as absurd as claiming that A != A.
The description of “I” you just had is what I earlier referred to as the physical person, which is one of the two possible meanings. For the Doomsday argument, it also used the second meaning: the nonphysical reference to the first-person perspective. I.E. the uniform prior distribution DA proposed, which is integral to the controversial Bayesian update, is not suggesting that a particular physical person can be born earlier than all human beings or later than all of them due to variations in its gestation period. In its convoluted way thanks to the equivocation, it is effectively saying among all the people in the human beings’ entire history, “I” could be anyone of them. i.e. “The fact that I am Ape in the Coat living in the earlier 21 century (with a birth rank around 100 billion), rather than someone living in the far future with a birth rank around 500 billion, is evidence to believe that maybe there aren’t that many human beings in total”. Notice the “I” here is not equivalent to a particular physical person anymore but a reference to the first-person perspective. This claim is what gives the DA a soul-incarnation flavour.
Therefore, if we take the first-person perspective and the physical person combination as a given and deny theorizing alternatives, there won’t be a Doomsday argument.”I am this particular physical person, period (be it Ape in the Coat in your case or Dadadarren for me). There’s no rational way of reasoning otherwise.” is what ends the DA. There really is no further need of inquiring into the particular physical person’s birth rank variations due to pregnancy complications. And as you said, this won’t fall into the trap of soul incarnations.
And that is also my long-held position, that there is no rational way of theorizing “which person I could be”, regard the first-person perspective as primitively given: “I am this physical human being” and be done with it, avoid the temptation of theorizing what the first-person could be as SSA or SIA did. Then there won’t be any paradoxes.
As always we completely agree in substance, while using different semantics.
Yes, that’s why I’m saying that it requires the existence of some non-physical entity, which I call souls.
You seem to imply that “first person perspective” itself is non-physical, but this sounds weird to me, Clearly physicalism is not debunked by the fact taht people have first person perspectives. There are seem to be very physical rules due to which mind in Dadadarren’s body has Dadarren’s first person perspective and not Ape in the coat’s.
The only way how “I” here can not be equivalent to particular physical person is if we assume that there is something non-physical about personhood. People do indeed implicitly assume it all the time. But this assumption is completely ungrounded, and that’s what I’m pointing out.
Yes, absolutely. “I” is just a variable, referencing different things depending on who says it. When Dadadarren says “I” it means “Dadadarren”. When Ape in the coat says “I” it means “Ape in the coat”.
Doomsday argument can be expressed in terms of birth ranks. So inquiring in the mechanism due to which physical people accure birth ranks seems to be only right thing to do.
Please do not take this as an insult. Though I do not intend to continue this discussion further, I feel obliged to say that I strongly disagree that we have the same position in substance and only disagree in semantics. Our position are different on a fundamental level.
Still don’t get how souls would get you psychic powers. Otherwise randomness and causality don’t matter—you may as well simultaneously create people in numbered rooms and people in low-numbered rooms would have the same problems.
If you are an immaterial soul then being born in a particular body can be conceptualized as receiving a jacket of a particular color—the idea that you could’ve been born in a different body doesn’t immediately appear as a complete nonsense. And therefore there is at least some chance that doomsday inference is correct.
Phrase “randomly sampled” means “everything” in probability theory, so it’s not a really an argument.
You don’t need the idea of “souls” to justify anthropics, you can just sample instantiated computations in universe, run on them is_observer, and get answer “yes’ with some probability, which is pretty much well-defined.
I don’t see how what you’ve just written justifies the assumption “I could have been born in distant past or in far future” which is the basis for the Doomsday Inference.
In a world where “I” am just a result of matterial causes and effects, I couldn’t have existed before the causes for my existence existed, and they couldn’t have existed before their causes existed and so on. Therefore the idea that I could’ve existed at some other time dosen’t makes sense.
On the other hand, if the body is a result of specific causes and effects, but “I” am not inseparable with this body “I” am a specific soul that could’ve been instantiated in any body, then the idea that I could’ve been born at some other moment makes sense.
In a world with material causes and effects, when you toss a coin, you are not tossing the platonic ideal of randomness. If you observe coin on table heads up and know how you are going to take it and how you are going to toss it you basically have all information necessary to predict which side it lands. If you are capable to avoid frequentist nonsense “well, probability of coin landing is either 1 or 0 and I can’t do anything about this” by saying “let’s pretend that I don’t have all this information and say that coin lands heads with probability 0.5″, you are also capable to say “let’s pretend that I have zero information about myself except my birth rank” and make proper update on this.
I agree that if you update on full embedded information you are going to get interesting anthropic results, but it’s another problem.
Sure. If you knew all the relevant information for the coin toss, you could predict the outcome. But as you do not know it, you can treat the situation as an iteration of probability experiment with two possible outcomes.
You do not pretend not to have all this information. You actually do not know it! When you reason based on the information available—you get correct application of probability theory, the type of reasoning that systematically produces correct map-territory relations. When you pretend that you do not know something that you actually know—you systematically get wrong results.
And this is exactly what you propose here. To reason based on less information than we actually have. We can do it, of course, but then we shouldn’t be surprised that the results are crazy and not correlated with reality which we wanted to reason about in the first place.
You have all relevant information tho. I’m pretty sure AIXI can predict coin toss if it has access to your vision field and proprioception data. You can’t compute outcome from this, but probability theory shouldn’t change from the fact that you can’t properly compute update.
Eh, no? Usually I can pretty much sensibly predict what I would think if I didn’t have some piece of information.
Then AIXI has the relevant information, while I do not.
A probabilistic model describes knowledge state of an observer and naturally changes when the knowledge state of the observer changes. My ability or inability to extract some information obviously affects which model is appropriate for the problem.
Suppose a coin is tossed and then the outcome is written in Japanese on a piece of paper and this piece of paper is shown to you. Whether or not your credence in the state of the coin changes from equiprobable prior depends on whether you know Japanese or not.
Of course you can. But this way of thinking would be sub-optimal in the situation where you actually has extra information.
I suppose you express skepticism that “being randomly sampled” is a meaningful statement. I’m planning an indepth dive into foundations of probability theory in order to clear these kind of confusions and arrive to a propper definition of what “randomness” even means.
For now, consider the two paper picking examples from the post. Even if you don’t exactly understand what randomness is it should be clear that it works different in these examples. In one case you can get any piece of paper from all the papers in the bag. In the other you may not get a piece of paper at all—empty spaces are added as possible outcomes of the experiment.
Likewise consider the third version of paper picking experiment, where the papers are not picked blindly and you always get the paper with number 6, regardless of the bag. Clearly, this situation is quite different from the previous two. In this experiment you have only one possible outcome, where previously there were multiple of them. So we say that in this case there is no randomness.
As saying goes, “constant is a random variable”.
And as another saying goes, “A rose by any other name would smell as sweet”.