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:
P(Short)=P(Long)=1/2
Assuming that you are a random person among all the people destined to be born:
P(6|Short)=1/10
P(6|Long)=1/10000
According to the Law of Total Probability:
P(6)=P(6|Short)P(Short)+P(6|Long)P(Long)=0.05005
Therefore by Bayes’ Theorem:
P(Short|6)=P(6|Short)P(Short)/P(6)>0.999
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.
.
.
.
.
.
.
.
.
.
Did you do it?
.
.
.
.
.
.
.
.
.
Well, regardless, there will be more time for it. First, let’s discuss the fact that both major anthropic theories SSA and SIAaccept 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.
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 Thirdermodels 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...
.
.
.
.
.
.
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.
.
.
.
.
.
.
And this assumption is...
.
.
.
.
.
.
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.
P(6|Long)=P(6|Short)=1
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.
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:
As short and long history are a priori indistinguishable and mutually exclusive:
P(Short)=P(Long)=1/2
Assuming that you are a random person among all the people destined to be born:
P(6|Short)=1/10
P(6|Long)=1/10000
According to the Law of Total Probability:
P(6)=P(6|Short)P(Short)+P(6|Long)P(Long)=0.05005
Therefore by Bayes’ Theorem:
P(Short|6)=P(6|Short)P(Short)/P(6)>0.999
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.
.
.
.
.
.
.
.
.
.
Did you do it?
.
.
.
.
.
.
.
.
.
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:
The solution is totally analogous to the Doomsday Inference above:
P(First)=P(Second)=1/2
P(6|First)=1/10
P(6|Second)=1/10000
P(6)=P(6|First)P(First)+P(6|Second)P(Second)=0.05005
P(First|6)=P(6|First)P(First)/P(6)=0.05/0.05005>0.999
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:
P(First)=P(Second)=1/2
P(Find|First)=1/1000
P(Find|Second)=1
P(Find)=P(Find|First)P(First)+P(Find|Second)P(Second)=0.5005
P(First|Second)=P(Find|Second)P(Second)/P(Find)=0.5/0.5005>0.999
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...
.
.
.
.
.
.
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.
.
.
.
.
.
.
And this assumption is...
.
.
.
.
.
.
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.
P(6|Long)=P(6|Short)=1
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.