Is there a strong, formal version of the Doomsday argument not based on cubicles and such? The obvious flaw in that version is that it is based on finite possibilities: that is, finding oneself in this or that kind of universe out of so many possible ones.
For starters, we have no idea of how likely our universe is; we know what numerator is 1, but what goes on the denominator? Is there even such a number? If so it is something so huge that the probability is vanishingly close to zero.
Then what about the parallel universes hypotheses: what if the universe is constantly fragmenting into multiple futures. In one future, I will submit this comment and close the tab, then check my e-mail. In another, I keep the tab open and go make a coffee.
This proliferation of multiple futures from any moment would tend to provide a way for doomsdays to be circumvented.
If you are cloned into N futures in this moment, and you're killed in N-1 of them, then you don't know it; by the anthropic principle itself, only the surviving future matters; there is no consciousness in the others. The suriviving future has no idea about the size of N, either.
The big gaping hole I see is that the blog post at least doesn't treat the earlier epochs with the same question. If a person in 1000 CE took up the same question, would they predict Doom Early or Doom Late? If everyone for the last ten thousand years would predict Doom Early, that seems like a piss poor predictor.
That's just a special case of a general feature of probabilistic arguments: a small but positive fraction of the times they are used, they will give wildly misleading predictions. If we assume that everyone uses the doomsday argument with a uniform improper prior on the total number of people who will ever live, then all of them will conclude with 95% probability that they are not one of the first 5% of people to ever live (that is, that no more than 20 times the past population will be created in the future), and 5% of them will be wrong, which is exactly how probabilities are supposed to work.
> The obvious flaw in that version is that it is based on finite possibilities: that is, finding oneself in this or that kind of universe out of so many possible ones.
As far as I know, that's a requirement for any thought experiment involving SSA vs. SAA.
> Then what about the parallel universes hypotheses: what if the universe is constantly fragmenting into multiple futures.
Actually, that's the many-worlds hypothesis of quantum theory. Parallel universes refers to something else.
> This proliferation of multiple futures from any moment would tend to provide a way for doomsdays to be circumvented.
Not exactly circumvented. In the many-worlds picture, all outcomes occur, including the one predicted by the Doomsday argument.
> If you are cloned into N futures in this moment, and you're killed in N-1 of them, then you don't know it ...
True but not compelling, because quantum theories don't require each of the many worlds to be occupied by a conscious observer. Were this not so, the universe couldn't have evolved to the point where consciousness was first possible.
That philosophers take their arguments too seriously?
I mean, this is fine as an interesting thought exercise. To the degree that you take it seriously as applying to the real world, to that degree you need... something. Perspective? To not take fine-sounding argument so seriously? To take a long walk? To get a life?
Not everything that cannot be logically disproven is true. Not everything that cannot be logically disproven should be acted upon as if it were true. It is useful to know when to smile at the philosophical earnestness and then just go on with your life.
there isn't a trivial mapping from one to the other, but they both do have the quality of "knowing what we know of the universe, what happens next?" and they both seem to conclude "if we aren't doomed that's because something novel is happening in the universe"
> The last step is to transpose these results to our actual situation here on Earth
Yeah, that's the trick, isn't it? Is the metaphor truly valid? This how philosophers differ from scientists: they see no need to test their hypothesis with real-world measurement. This is also why philosophy needs to go the way of alchemy and astrology - as an amusing but archaic belief.
My favourite answer to the doomsday argument: The many worlds hypothesis is correct and we haven't created our "world ending" p=np computers yet. (Literally halt if randomly generated solution is incorrect and catch fire to every sentient being in the world).
Another possibility is that we're very close to building realistic simulations, and by induction, it makes sense that we're at this stage of development.
>Now we modify the thought experiment a bit. We still have the hundred cubicles but this time they are not painted blue or red. Instead they are numbered from 1 to 100. The numbers are painted on the outside. Then a fair coin is tossed (by God perhaps). If the coin falls heads, one person is created in each cubicle. If the coin falls tails, then persons are only created in cubicles 1 through 10.
>You find yourself in one of the cubicles and are asked to guess whether there are ten or one hundred people? Since the number was determined by the flip of a fair coin, and since you haven’t seen how the coin fell and you don’t have any other relevant information, it seems you should believe with 50% probability that it fell heads (and thus that there are a hundred people).
The conclusion in the last sentence is incorrect is an incredibly subtle way. Since 10 times more people are in cubicles in the heads case, the probability that you find yourself in a cubicle at all is ten times higher in that case, which affects your prior. By Bayes' theorem:
The reason why this is so subtle is that it's not a given that you're in a cubicle. It's more likely for you to be put in a cubicle the more people who are put into cubicles.
> The reason why this is so subtle is that it's not a given that you're in a cubicle.
The way the problem is presented, it is a given that you're in a cubicle. If you wanted to be more explicit about it, you could say that God, before creating humanity, flipped a coin, and then created either 10 or 100 humans and placed them into cubicles. You wake up and have to estimate the probability of that coin flip.
It is given that you are but it's not given that that was a sure thing to happen. Imagine this:
God flips a coin, creates 10 cubicles. If the coin is heads he puts you in one of them if it tails the God flips another coin and if it tails again he puts you in one of the cubicles but if it heads he doesn't.
You wake up in a cubicle. It's given that you are in one! what are the chances the first coin landed heads ?
The probability isn't 50% because there is underlying probability that you would be in cubicle at all (1 vs 0.5 after first flip). There is the same problem with doomsday argument just better masked.
Ah, I see. It makes more sense if you forget about cubicles and just call the final outcomes A or B. Call the outcomes of the two coin flips H1/T1 and H2/T2. We have: H1->A, T1,H1->A, T1,T2->B. P(H1|A) is therefore 67% if the coin is fair, but in any event P(H1|A)>P(H1).
So the doomsday argument is actually correct. :-) But it's really nothing more than the (IMO vacuous) observation that (say) half of the people who ever live will have a birth order number greater than the median, and so the odds are 50-50 that fewer people will be born after you than before you. More generally, the odds that N times as many people will be born after you as before you is equal to 1/2N.
>>No. That you are in a cubicle is a given fact, so the probability that you find yourself in a cubicle is unconditionally 1.
But it's not 1 that you would be in a cubicle at all. The paradox disappears if the problem was formulated like this:
"God shuffles DNA and there is 0.0000000000001 chances tha it shuffles up you and he does that for every cuebicle". Or:
"God create you first, then flips a coin and put you at random in one of the created cubicles".
Formulation of the problem should mention which one is it. If it doesn't we are back to guessing what God does (similarly to 2 envelope problem which comes down to guessing what the sponsor's preferences for amounts are). As there are infinitely many ways God could decide to create humans in cubicles you can't answer that without giving some guess (priori) for probability distribution over those choices.
There are only finitely many ways to arrange matter in a finite volume under a finite temperature. If it's safe to assume that anything we could classify as "human" will fit inside a sphere of radius 1 light-year and has a temperature under 10^20 K, there are finitely many humans.
Yeah but it's not given that God chooses at random who is going to cubicle. He might have for example created you first and then put you in a cubicle regardless if it's 100 of them or 10.
I agree with your resolution of that paradox though. It's reasonable to assume - having no information about the process - that God creates humans randomly and it's 10x more likely that you are created in a world with 100 cubicles than in a world in 10 cubicles.
In my mind there is no paradox at all there.
And I don't understand why we ignore that human beings created the cubicles and there is no random god involved in any of it. Whether there is a human in the cubicle or not is entirely a result of physical processes. Any statement about probability is nothing but an explanation of our ignorance. Yeah, we might not be capable of knowing the totality of physical history of all the people who build the cubicles and where they are now and such... so what? That says nothing about the history of their species.
Other than the fact that you can't coherently reason based on your own existence (existence is not a predicate), one should rightfully be wary of an argument that would have been valid and false at every point in the past.
The Doomsday Argument is not an argument in favor of its conclusion -- it's a reductio ad absurdum that demonstrates why this type of reasoning is invalid.
In a general sense, I don't believe so. While "there exists an experimenter" is an inherent assumption in many deductive chains, there's no attempt to use the existence of the experimenter as a prior in a Bayesian calculation. The logical flaw in the DA comes when you say "I, the reasoner, am likely to exist at a random point in the history of humanity", because the reasoning could not take place without your existence and therefore the hypothesis is not-falsifiable (no matter when in the history of humanity the reasoning occurred).
Imagine you have 100 ordered rooms. All initially are filled with gold and ponies. God tosses a coin and depending on the result of the flip God strips 10 or 100 rooms of ponies and gold.
You are in one of the rooms, open your eyes and see that there is no gold and ponies around. You take a look at your room number and see 7. Then you know that best decision is to operate on assumption that there is 91% chance that only 10 rooms were stripped of gold and ponies. So lots of gold and ponies should await you very soon as you move to further rooms of increasing numbers.
Reassured with that bulletproof argument we may look into the future with hope that rather sooner than later there will be times where we won't be confined to this dull rock we evolved on, to this lame rapidly decaying bodies with machines that push those clunky electrons around that can barely add 0 to 1 while overheating immensely.
Because near future is (likely) full of gold and ponies.
I hope this illustrates why purely philosophical arguments are empty of any utility. If you reason without experimental data to anchor your reasoning to something real you end up with reasoning that works just as well when applied to the thing you had in mind when it wandered as to thing opposite to your intention giving opposite conclusions and teaching us absolutely nothing about world.
Ah. First two dowvotes and no comment from fans of realworld importance of philosophy in this day and age. How classy!
The Doomsday Argument works because the result (Doom Soon or Doom Late) affects the number of observers, and we can work backwards to update our beliefs on which one will be.
That is why replacing the words with ponies and gold won't work unless you are a pony or sentient gold.
For my argument I just replaced "existing" with "not having gold and ponies" and drew parallel not between "specific rooms being occupied" and our civ existing at specific time but between "specific rooms not being emptied of gold and ponies" and "our civ having decent tech at specific time".
It works. Use your own substitutions.
Phisolophical reasonings often latch onto inexplicit meaning of words. If you replace them with different words that fullfill same role in language but have different meaning argument becomes much less apparently profound or even false.
Imagine that the coin that we're flipping isn't fair, and comes up heads with prior probability p. Further, suppose that the ratio between the small population and the large population is R (in the article this is R = 10 / 100 = .1). In this situation, upon discovering that you're inside the small population, the posterior probability of heads ends up as
p / (R - p * (1 - R))
So what if the coin is very unlikely to be heads instead of 50/50? If p = .001 (and we leave R = .1), our estimate of the probability heads after we observe that we are in the small population only comes up to about 1%.
Thus, the real fallacy in the argument is that the choice of prior is unimportant. (If anyone tells you the prior is unimportant in any situation, they are wrong.) With a suitable estimate of the prior likelihood of Doom Soon, the posterior likelihood of Doom Soon is still low enough.
I played around with the model a bit and based on a couple assumptions it can be scaled to
P(Doom Soon) = x/(1+x) where
x = B/b * d
where B is the number of humans born in the Doom Late scenario, b is the number of humans born so far, and d is the prior probability of Doom Soon.
Assumptions: a) the number of humans born between now and Doom Soon is negligible and b) the Doom Late scenario has many more humans than Doom Soon (B + b ~= B).
Notice I made no assumptions about the prior, d. Of course d does matter, but the point is that as long as the number of humans in Doom Long is assumed to be large enough, the probability will go close to one even for very small d.
For example, if we use a Doom Late population of 200 trillion like in the post, then we have 95% probability of Doom Soon even if d ~= 0.0001.
That being said, I am still fairly unconvinced by this argument. It would have every intelligent species continually concluding that they are about to go extinct, right up until the moment that they either do go extinct or they achieve immortality and stop reproducing.
Haha nice. Although, in this case we are looking ahead, basing our short-term survival on what a long-term survival would theoretically look like. So the more people there are in the hypothetical Doom Late, the more likely Doom Soon becomes. The more I think about this the more absurd it seems.
Yeah, all this analysis (both what we've done and in the original article) are facile--a proper Bayesian treatment would have continuous priors/posteriors that would be a little more informative than an either-or.
The problem you're seeing here is that if you make your possibilities "Humans live forever (even past the heat death of the universe)" or "All humans die in the next 10 minutes", you'll find that the chance that humans die in the next 10 minutes is really absurdly high.
Perhaps I'm ignorant of some deeper concept... but it seems that this argument is deeply flawed on the basis that they are presuming that finding yourself in cubicles 1-10 somehow indicates that cubicles 11+ are probably vacant.
In both doom soon and doom late, cubicles 1-10 are occupied. There's no aspect of doom late that would be made less likely as a result of cubicles 1-10 being occupied.
The reverse works well, though. Being in one of cubicles 11+ /does/ eliminate the possibility of doom soon. So only passing the cubicle threshold gives any meaningful information about which of the two scenarios are true.
That's an Aristotelian viewpoint, operating logic on the basis of True and False, in which case the argument presented is indeed a fallacy. However, when you operate on probabilities, Bayes' Rule does indeed allow you to drive the logic in a way that seems "backwards" from the Aristotelian point of view.
This, by the way, has nothing to do with "Bayesian" vs. "Frequentism"... this is just the straight application of a simple and non-controversial thing call Bayes' Rule. Assuming you know the probabilities with the requisite precision you are perfectly justified in reasoning that way.
It's also worth reading the end of the article carefully too... for instance, "doom" may be a bit leading, since "doom" could include a full Singularity after which we are no longer "humans" but nobody is upset about that, or other such scenarios that are still in some sense "doom" for The Human Race (imagine those glorious calligraphy letters for that) but aren't necessarily doom in a bad way.
There's also just the casual observation that if society goes on for billions of years there are still simply some people who are born early. Sometimes people win the lottery, and we would not generally be too impressed if they reasoned backwards from there to the conclusion that winning the lottery was more likely than they would have initially thought. (That is not exactly a parallel, since a lottery is part of a much bigger net of information than the deliberately simplified version presented in the argument.) One of my objections would be that it is not clear that we can take a non-temporal line of logic as given in the beginning of the argument and directly apply it to a situation with time. And when I mean "not clear", I mean precisely that it is not clear to me, not that I'm saying it's impossible.
In general I've observed that I'm not impressed with our attempts to apply our local probability rules at cosmological scales. Is Occam's Razor really applicable at cosmological scales? It's hard to say what's "simplest" when we don't even have any reasonable priors (or translate that to whatever math framework you may like, like "have no information about"). The orders of magnitude involved even with merely the visible universe, which we have no reason to believe is even the whole of reality, IMHO can easily swamp the simple probability of error in the argument too easily. (That is to say... there's an old observation in computer science that running a "probabilistic" algorithm with a probability of failure lower than the probability of hardware failure is indistinguishable from running a deterministic algorithm. It is not clear to me that we can overcome our probability of having something wrong in our argument.)
The reasoning in the article is correct. Cubicles 1-10 are occupied in both scenarios, but the observation of (self <= 10) is much less likely in the Doom Late scenario. This means that if you observe (self <= 10), the conditional probability of Doom Soon is much higher.
They key point is that, if Doom Late occurred, then the probability of ending up in cubicle 1-10 is only 10%. So if the only observation that you have is that your cubicle is numbered 1-10, which is more likely: Doom Soon (50% absolute probability), or Doom Late (50% absolute) and you happened to end up in 1-10 (10% of the 50%)?
Therefore, the conditional probability of Doom Soon is higher than 50% based on the observation of self <= 10.
The premises in the argument can abstracted into the following:
if doom_soon, then the probability of myself being in cubicles 1-10 is 100%;
if doom_soon, then the probability of myself being in cubicles 11-100 is 0%;
if doom_late, then the probability of myself being in cubicles 1-10 is 10%;
if doom_late, then the probability of myself being in cubicles 11-100 is 90%;
???
Therefore if I am the cubicles 1-10, the probability of doom soon is 81%;
In any logic class, this argument would quickly be pointed out as a fallacy. The statement of "if p, then q" says nothing about the truth of p, given q. Which is exactly what this argument seems to do, reason backwards from the conclusion when none of the premises are able to be logically inferred from the conclusion.
I thought of a better way of explaining the invalidity of the argument.
What would happen if, instead of inferring from our own presence in the cubicle, we take an external role. Furthermore, instead of taking the probability that a randomly selected person exists in of the two sets, we look at the probability that /someone/ exists in one of the two sets.
If doom soon occurred, then the probability of cubicles 1-10 being occupied is 100%;
If doom soon occurred, then the probability of cubicles 11-100 being occupied is 0%;
If doom late occurred, then the probability of cubicles 1-10 being occupied is 100%;
If doom late occurred, then the probability of cubicles 11-100 being occupied is 100%;
So given those two probabilities, we have a 100% chance of cubicles 1-10 being occupied and a 50% chance of cubicles 11-100 being occupied.
In either scenario, the occupancy status of cubicles 1-10 is assured. So checking in cubicles 1-10 gives us no meaningful information about which of the two scenarios occurred. Checking in cubicles 11-100 on the other hand /would/ affect our results. The occupancy of cubicles 11-100 on the other hand would tell us exactly which scenario occured.
It's probably not worth worrying about the fate of humanity till you've figured out the rooms example. Since you're on this site I'm sure you can trivially knock up a script to simulate what happens. Or just draw the probability tree. In any case the relevant question is,, out of those who find their room number is less than ten, what is the probability that the coin toss meant only ten rooms are occupied. The answer is 90.9% - we could go round in circles trying to choose words that everyone understands, but the numbers speak for themselves.
10 people have a room number less than or equal to 10.
Scenario 2: Doom late
10 people have a room number less than or equal to 10.
---
Given that each scenario has an equal probability in isolation, there are 20 possible positions for the the <=10 individual to find themselves in (10 in doom soon, 10 in doom late).
Of those 20 possibilities, 10 of those are in the doom soon scenario. The remaining 10 are in the doom late scenario.
10 doom soon to 10 doom late is a 50/50 probability.
Edit: writing up a script as well, right now. Will add to link to it in my original comment when complete.
You're wrong. Not being mean, you've just arrived at the wrong numerical answer to the maths question.
(Let's leave aside mentions of doom and stick to the rooms: according to the article no-one has quite nailed down yet whether there are subtleties about unbounded future populations). Rather than try to draw a tree I'll write it out flat:
10 rooms (0.5) & you're in 1-10 (1.0): Probability 0.5
10 rooms (0.5) & you're in 11-100 (0.0): Probability 0.0
100 rooms (0.5) & you're in 1-10 (0.1): Probability 0.05
100 rooms (0.5) & you're in 11-100 (0.9): Probability 0.45
Work out any relative probabilities using the right-hand column. For instance, when you're in room 1-10, the chances of the coin having been heads is 0.5/0.55. Even if this bugs you, this approach of using the tree will let you get the right answers to tricky teasers about medical tests with false positives and false negatives.
The situation you gave me was "out of those who find their room number is less than ten, what is the probability that the coin toss meant only ten rooms are occupied".
In my example, I composed my tree of solely those who with a room number less than or equal to 10, since in your question, we're only concerned about those who are in rooms 1-10.
In your example, it seems you've included those who are in rooms 11-100... If we are only concerned with rooms 1-10, what relevance does the probability of 11-100 play in this role?
More specifically, your seems to be what the odds are of /being in/ room 1-10, rather than the odds of each scenario occurring, assuming you are in room 10.
494644 results in rooms less than 10: 449838 had 10 rooms (90.9%), 44806 had 100 rooms 9.1%
Note that anyone who finds themselves in a high-numbered room can draw the obvious conclusion. The question is, what can you infer about the situation if you find yourself in a low-numbered room?
To hammer the point, let say the coin toss decides between 1 room and ten million. If you find yourself in room One, do you still think the chance the coin has come up heads is 50%? If you do, don't tell me, write a letter to Monty Hall.
My vague ponderings about the actual doomsday problem is that it holds up, i.e. it's a reasonable argument that it makes eventual populations of say 10^400 simulated souls using the whole universe as a computer highly unlikely. But the sheer bigness of the possible-but-unlikely population leaves plenty of room for astronomical-sized future populations and doom isn't necessarily imminent in human terms.
In response to rm445's challenge in a reply, I wrote up a script simulating the two scenarios, and in situations where a room in 1-10 occupied, added up the totals of which scenario the individual is in.
>>So you're in the room. Conditioned on that fact, how worried should you be? How likely is it that you're going to die?
A: 1/36.
Scott: OK. That would be one guess.
>>One answer is that the dice have a 1/36 chance of landing snake-eyes, so you should be only a "little bit" worried (considering). A second reflection you could make is to consider, of people who enter the room, what the fraction is of people who ever get out. Let's say that it ends at 1,000. Then, 110 people get out and 1,000 die. If it ends at 10,000, then 1,110 people get out and 10,000 die. In either case, about 8/9 of the people who ever go into the room will die.
But it's not really a problem.
If you flip a coin, then 10 coins, then 100 coins and continue until you get all heads then vast majority of flipped coins will be heads while still any particular coin have 1/2 probability of landing heads. There just isn't any paradox or even anything surprising.
Similarly here it's in my opinion obvious that chances of dying are 1/36 and they are the same for all people in the situation. The fact that most people die isn't a paradox at all.
I think it's still a "problem." Which one is the "correct probability" that you're going to die? If, instead of dying, a red light in the room might turn on after a minute, how would you place a bet about whether the red light would turn on?
"Corresponding to the prior probability (50%) of the coin falling heads or tails, we now have some prior probability of Doom Soon or Doom Late."
Nice to just magically know this probability. I get that the actual number doesn't matter, but I worry about a predictor that doesn't care about any real probabilities based on observation.
In the cubicle example discovering that you are in cube 1-10 makes the likelihood of the 1-10 scenario much more likely than it was before you discovered you were in one of the first 10 cubes using a simple application of Bayes theorem.
With the 100 billion or 100 trillion people example, you are person 60 billion. Theoretically making 100 billion much more likely than if you didn't know where you fell. That probability approaches one if work off the assumption that doom(late) means hundreds of millennia of humans spreading across the galaxy at our current growth rate. the higher the Total possible humans in doom(late) the higher the probability that being in the first 100b indicated that there will only be 100b. (or similar numbers)
One difference I see between the two scenarios is time. The 100 cubes are not filled in sequence. In the 100billion/100trillion example, every single person that ever lived is in the 100billion, until they're not, then everyone is not in the 100 billion. I don't know that it affects the math, but it affects my thinking about the problem.
The argument assumes that the only possible end-state for humanity is doomsday; that is, there are a finite number of humans, and when there are no more doomsday has occurred. The article touches on this with the infinite humans / transhuman stuff at the end.
The argument assumes exponential growth in number of humans until the terminal state, but this is not a foregone conclusion. While this doesn't change the reasoning in a strict sense, it may put the "soon" cutoff millions of years in the future.
Finally, the prior probability is chosen arbitrarily. It does matter quite a bit. Take the exponential argument for evidence as to why: For any given probabilty p that doomsday will occur on a given day, it is more likely today than on any single other day! This is simply because if doomsday occurs today, it cannot occur on any future day, making it less likely on each successive day.
Except that if p=1E-10, doomsday is fantastically unlikely to occur this year. The Bayesian argument has the same problem, you just start seeing it at a different value of p.
Here's a simpler way to see what is going on. Number all the humans that will ever live in order of their birth from 1 to N. The odds of your birth order number (call it B) being in the range of 1..M<N is M/N. The larger N is, the less likely it is that your B is as small as it is. So the odds of N being large (relative to B) are small.
Another way to look at it: half of the humans who ever live will have fewer people born after them than before them. Your a priori odds of being such a person are exactly 1/2.
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[ 3.3 ms ] story [ 123 ms ] threadFor starters, we have no idea of how likely our universe is; we know what numerator is 1, but what goes on the denominator? Is there even such a number? If so it is something so huge that the probability is vanishingly close to zero.
Then what about the parallel universes hypotheses: what if the universe is constantly fragmenting into multiple futures. In one future, I will submit this comment and close the tab, then check my e-mail. In another, I keep the tab open and go make a coffee.
This proliferation of multiple futures from any moment would tend to provide a way for doomsdays to be circumvented.
If you are cloned into N futures in this moment, and you're killed in N-1 of them, then you don't know it; by the anthropic principle itself, only the surviving future matters; there is no consciousness in the others. The suriviving future has no idea about the size of N, either.
As far as I know, that's a requirement for any thought experiment involving SSA vs. SAA.
https://en.wikipedia.org/wiki/Self-Sampling_Assumption
https://en.wikipedia.org/wiki/Self-Indication_Assumption
Actually, that's the many-worlds hypothesis of quantum theory. Parallel universes refers to something else.
> This proliferation of multiple futures from any moment would tend to provide a way for doomsdays to be circumvented.
Not exactly circumvented. In the many-worlds picture, all outcomes occur, including the one predicted by the Doomsday argument.
> If you are cloned into N futures in this moment, and you're killed in N-1 of them, then you don't know it ...
True but not compelling, because quantum theories don't require each of the many worlds to be occupied by a conscious observer. Were this not so, the universe couldn't have evolved to the point where consciousness was first possible.
That philosophers are not statisticians?
I mean, this is fine as an interesting thought exercise. To the degree that you take it seriously as applying to the real world, to that degree you need... something. Perspective? To not take fine-sounding argument so seriously? To take a long walk? To get a life?
Not everything that cannot be logically disproven is true. Not everything that cannot be logically disproven should be acted upon as if it were true. It is useful to know when to smile at the philosophical earnestness and then just go on with your life.
http://en.wikipedia.org/wiki/Fermi_paradox
http://en.wikipedia.org/wiki/Doomsday_argument
Yeah, that's the trick, isn't it? Is the metaphor truly valid? This how philosophers differ from scientists: they see no need to test their hypothesis with real-world measurement. This is also why philosophy needs to go the way of alchemy and astrology - as an amusing but archaic belief.
Another possibility is that we're very close to building realistic simulations, and by induction, it makes sense that we're at this stage of development.
>You find yourself in one of the cubicles and are asked to guess whether there are ten or one hundred people? Since the number was determined by the flip of a fair coin, and since you haven’t seen how the coin fell and you don’t have any other relevant information, it seems you should believe with 50% probability that it fell heads (and thus that there are a hundred people).
The conclusion in the last sentence is incorrect is an incredibly subtle way. Since 10 times more people are in cubicles in the heads case, the probability that you find yourself in a cubicle at all is ten times higher in that case, which affects your prior. By Bayes' theorem:
P(HEADS| WAKE UP) = (P(WAKE UP|HEADS) * P(HEADS)) / ( P(WAKE UP |HEADS) * P(HEADS) + P(WAKE UP|TAILS) * P(TAILS) )
= (10 * P(WAKE UP|TAILS) * P(HEADS)) / ( 10 * P(WAKE UP|TAILS)* P(HEADS) + P(WAKE UP|TAILS) * P(TAILS) )
= 10 / (10 + 1)
= 10 / 11, or about 91% chance of heads given that you find yourself in a cubicle.
No. That you are in a cubicle is a given fact, so the probability that you find yourself in a cubicle is unconditionally 1.
The way the problem is presented, it is a given that you're in a cubicle. If you wanted to be more explicit about it, you could say that God, before creating humanity, flipped a coin, and then created either 10 or 100 humans and placed them into cubicles. You wake up and have to estimate the probability of that coin flip.
So the doomsday argument is actually correct. :-) But it's really nothing more than the (IMO vacuous) observation that (say) half of the people who ever live will have a birth order number greater than the median, and so the odds are 50-50 that fewer people will be born after you than before you. More generally, the odds that N times as many people will be born after you as before you is equal to 1/2N.
But it's not 1 that you would be in a cubicle at all. The paradox disappears if the problem was formulated like this: "God shuffles DNA and there is 0.0000000000001 chances tha it shuffles up you and he does that for every cuebicle". Or: "God create you first, then flips a coin and put you at random in one of the created cubicles".
Formulation of the problem should mention which one is it. If it doesn't we are back to guessing what God does (similarly to 2 envelope problem which comes down to guessing what the sponsor's preferences for amounts are). As there are infinitely many ways God could decide to create humans in cubicles you can't answer that without giving some guess (priori) for probability distribution over those choices.
The Doomsday Argument is not an argument in favor of its conclusion -- it's a reductio ad absurdum that demonstrates why this type of reasoning is invalid.
You are in one of the rooms, open your eyes and see that there is no gold and ponies around. You take a look at your room number and see 7. Then you know that best decision is to operate on assumption that there is 91% chance that only 10 rooms were stripped of gold and ponies. So lots of gold and ponies should await you very soon as you move to further rooms of increasing numbers.
Reassured with that bulletproof argument we may look into the future with hope that rather sooner than later there will be times where we won't be confined to this dull rock we evolved on, to this lame rapidly decaying bodies with machines that push those clunky electrons around that can barely add 0 to 1 while overheating immensely.
Because near future is (likely) full of gold and ponies.
I hope this illustrates why purely philosophical arguments are empty of any utility. If you reason without experimental data to anchor your reasoning to something real you end up with reasoning that works just as well when applied to the thing you had in mind when it wandered as to thing opposite to your intention giving opposite conclusions and teaching us absolutely nothing about world.
Ah. First two dowvotes and no comment from fans of realworld importance of philosophy in this day and age. How classy!
That is why replacing the words with ponies and gold won't work unless you are a pony or sentient gold.
It works. Use your own substitutions.
Phisolophical reasonings often latch onto inexplicit meaning of words. If you replace them with different words that fullfill same role in language but have different meaning argument becomes much less apparently profound or even false.
Imagine that the coin that we're flipping isn't fair, and comes up heads with prior probability p. Further, suppose that the ratio between the small population and the large population is R (in the article this is R = 10 / 100 = .1). In this situation, upon discovering that you're inside the small population, the posterior probability of heads ends up as
So what if the coin is very unlikely to be heads instead of 50/50? If p = .001 (and we leave R = .1), our estimate of the probability heads after we observe that we are in the small population only comes up to about 1%.Thus, the real fallacy in the argument is that the choice of prior is unimportant. (If anyone tells you the prior is unimportant in any situation, they are wrong.) With a suitable estimate of the prior likelihood of Doom Soon, the posterior likelihood of Doom Soon is still low enough.
Assumptions: a) the number of humans born between now and Doom Soon is negligible and b) the Doom Late scenario has many more humans than Doom Soon (B + b ~= B).
Notice I made no assumptions about the prior, d. Of course d does matter, but the point is that as long as the number of humans in Doom Long is assumed to be large enough, the probability will go close to one even for very small d.
For example, if we use a Doom Late population of 200 trillion like in the post, then we have 95% probability of Doom Soon even if d ~= 0.0001.
That being said, I am still fairly unconvinced by this argument. It would have every intelligent species continually concluding that they are about to go extinct, right up until the moment that they either do go extinct or they achieve immortality and stop reproducing.
I'm reminded of the Fight Club quote: On a long enough timeline, the survival rate for everyone drops to zero. :)
The problem you're seeing here is that if you make your possibilities "Humans live forever (even past the heat death of the universe)" or "All humans die in the next 10 minutes", you'll find that the chance that humans die in the next 10 minutes is really absurdly high.
In both doom soon and doom late, cubicles 1-10 are occupied. There's no aspect of doom late that would be made less likely as a result of cubicles 1-10 being occupied.
The reverse works well, though. Being in one of cubicles 11+ /does/ eliminate the possibility of doom soon. So only passing the cubicle threshold gives any meaningful information about which of the two scenarios are true.
This, by the way, has nothing to do with "Bayesian" vs. "Frequentism"... this is just the straight application of a simple and non-controversial thing call Bayes' Rule. Assuming you know the probabilities with the requisite precision you are perfectly justified in reasoning that way.
It's also worth reading the end of the article carefully too... for instance, "doom" may be a bit leading, since "doom" could include a full Singularity after which we are no longer "humans" but nobody is upset about that, or other such scenarios that are still in some sense "doom" for The Human Race (imagine those glorious calligraphy letters for that) but aren't necessarily doom in a bad way.
There's also just the casual observation that if society goes on for billions of years there are still simply some people who are born early. Sometimes people win the lottery, and we would not generally be too impressed if they reasoned backwards from there to the conclusion that winning the lottery was more likely than they would have initially thought. (That is not exactly a parallel, since a lottery is part of a much bigger net of information than the deliberately simplified version presented in the argument.) One of my objections would be that it is not clear that we can take a non-temporal line of logic as given in the beginning of the argument and directly apply it to a situation with time. And when I mean "not clear", I mean precisely that it is not clear to me, not that I'm saying it's impossible.
In general I've observed that I'm not impressed with our attempts to apply our local probability rules at cosmological scales. Is Occam's Razor really applicable at cosmological scales? It's hard to say what's "simplest" when we don't even have any reasonable priors (or translate that to whatever math framework you may like, like "have no information about"). The orders of magnitude involved even with merely the visible universe, which we have no reason to believe is even the whole of reality, IMHO can easily swamp the simple probability of error in the argument too easily. (That is to say... there's an old observation in computer science that running a "probabilistic" algorithm with a probability of failure lower than the probability of hardware failure is indistinguishable from running a deterministic algorithm. It is not clear to me that we can overcome our probability of having something wrong in our argument.)
They key point is that, if Doom Late occurred, then the probability of ending up in cubicle 1-10 is only 10%. So if the only observation that you have is that your cubicle is numbered 1-10, which is more likely: Doom Soon (50% absolute probability), or Doom Late (50% absolute) and you happened to end up in 1-10 (10% of the 50%)?
Therefore, the conditional probability of Doom Soon is higher than 50% based on the observation of self <= 10.
if doom_soon, then the probability of myself being in cubicles 1-10 is 100%; if doom_soon, then the probability of myself being in cubicles 11-100 is 0%;
if doom_late, then the probability of myself being in cubicles 1-10 is 10%; if doom_late, then the probability of myself being in cubicles 11-100 is 90%;
???
Therefore if I am the cubicles 1-10, the probability of doom soon is 81%;
In any logic class, this argument would quickly be pointed out as a fallacy. The statement of "if p, then q" says nothing about the truth of p, given q. Which is exactly what this argument seems to do, reason backwards from the conclusion when none of the premises are able to be logically inferred from the conclusion.
What would happen if, instead of inferring from our own presence in the cubicle, we take an external role. Furthermore, instead of taking the probability that a randomly selected person exists in of the two sets, we look at the probability that /someone/ exists in one of the two sets.
If doom soon occurred, then the probability of cubicles 1-10 being occupied is 100%; If doom soon occurred, then the probability of cubicles 11-100 being occupied is 0%;
If doom late occurred, then the probability of cubicles 1-10 being occupied is 100%; If doom late occurred, then the probability of cubicles 11-100 being occupied is 100%;
So given those two probabilities, we have a 100% chance of cubicles 1-10 being occupied and a 50% chance of cubicles 11-100 being occupied.
In either scenario, the occupancy status of cubicles 1-10 is assured. So checking in cubicles 1-10 gives us no meaningful information about which of the two scenarios occurred. Checking in cubicles 11-100 on the other hand /would/ affect our results. The occupancy of cubicles 11-100 on the other hand would tell us exactly which scenario occured.
Scenario 1: Doom soon
10 people have a room number less than or equal to 10.
Scenario 2: Doom late
10 people have a room number less than or equal to 10.
---
Given that each scenario has an equal probability in isolation, there are 20 possible positions for the the <=10 individual to find themselves in (10 in doom soon, 10 in doom late).
Of those 20 possibilities, 10 of those are in the doom soon scenario. The remaining 10 are in the doom late scenario.
10 doom soon to 10 doom late is a 50/50 probability.
Edit: writing up a script as well, right now. Will add to link to it in my original comment when complete.
(Let's leave aside mentions of doom and stick to the rooms: according to the article no-one has quite nailed down yet whether there are subtleties about unbounded future populations). Rather than try to draw a tree I'll write it out flat:
10 rooms (0.5) & you're in 1-10 (1.0): Probability 0.5
10 rooms (0.5) & you're in 11-100 (0.0): Probability 0.0
100 rooms (0.5) & you're in 1-10 (0.1): Probability 0.05
100 rooms (0.5) & you're in 11-100 (0.9): Probability 0.45
Work out any relative probabilities using the right-hand column. For instance, when you're in room 1-10, the chances of the coin having been heads is 0.5/0.55. Even if this bugs you, this approach of using the tree will let you get the right answers to tricky teasers about medical tests with false positives and false negatives.
The situation you gave me was "out of those who find their room number is less than ten, what is the probability that the coin toss meant only ten rooms are occupied".
In my example, I composed my tree of solely those who with a room number less than or equal to 10, since in your question, we're only concerned about those who are in rooms 1-10.
In your example, it seems you've included those who are in rooms 11-100... If we are only concerned with rooms 1-10, what relevance does the probability of 11-100 play in this role?
More specifically, your seems to be what the odds are of /being in/ room 1-10, rather than the odds of each scenario occurring, assuming you are in room 10.
http://jsfiddle.net/3f49ry5u/
So... unless I did the simulation wrong, it supports my conclusion.
If I did do it wrong, please show me a corrected simulation which supports your position.
http://pastebin.com/79rmjmKr
Typical result:
494644 results in rooms less than 10: 449838 had 10 rooms (90.9%), 44806 had 100 rooms 9.1%
Note that anyone who finds themselves in a high-numbered room can draw the obvious conclusion. The question is, what can you infer about the situation if you find yourself in a low-numbered room?
To hammer the point, let say the coin toss decides between 1 room and ten million. If you find yourself in room One, do you still think the chance the coin has come up heads is 50%? If you do, don't tell me, write a letter to Monty Hall.
My vague ponderings about the actual doomsday problem is that it holds up, i.e. it's a reasonable argument that it makes eventual populations of say 10^400 simulated souls using the whole universe as a computer highly unlikely. But the sheer bigness of the possible-but-unlikely population leaves plenty of room for astronomical-sized future populations and doom isn't necessarily imminent in human terms.
http://jsfiddle.net/3f49ry5u/
The script seems to support the conclusion that given the premises of the scenario, the odds for each scenario are about 50/50
http://www.scottaaronson.com/democritus/lec17.html
The whole course is also very good, and will probably be interesting to many HN users.
>>So you're in the room. Conditioned on that fact, how worried should you be? How likely is it that you're going to die?
>>One answer is that the dice have a 1/36 chance of landing snake-eyes, so you should be only a "little bit" worried (considering). A second reflection you could make is to consider, of people who enter the room, what the fraction is of people who ever get out. Let's say that it ends at 1,000. Then, 110 people get out and 1,000 die. If it ends at 10,000, then 1,110 people get out and 10,000 die. In either case, about 8/9 of the people who ever go into the room will die.But it's not really a problem. If you flip a coin, then 10 coins, then 100 coins and continue until you get all heads then vast majority of flipped coins will be heads while still any particular coin have 1/2 probability of landing heads. There just isn't any paradox or even anything surprising. Similarly here it's in my opinion obvious that chances of dying are 1/36 and they are the same for all people in the situation. The fact that most people die isn't a paradox at all.
Nice to just magically know this probability. I get that the actual number doesn't matter, but I worry about a predictor that doesn't care about any real probabilities based on observation.
In the cubicle example discovering that you are in cube 1-10 makes the likelihood of the 1-10 scenario much more likely than it was before you discovered you were in one of the first 10 cubes using a simple application of Bayes theorem.
With the 100 billion or 100 trillion people example, you are person 60 billion. Theoretically making 100 billion much more likely than if you didn't know where you fell. That probability approaches one if work off the assumption that doom(late) means hundreds of millennia of humans spreading across the galaxy at our current growth rate. the higher the Total possible humans in doom(late) the higher the probability that being in the first 100b indicated that there will only be 100b. (or similar numbers)
One difference I see between the two scenarios is time. The 100 cubes are not filled in sequence. In the 100billion/100trillion example, every single person that ever lived is in the 100billion, until they're not, then everyone is not in the 100 billion. I don't know that it affects the math, but it affects my thinking about the problem.
The argument assumes that the only possible end-state for humanity is doomsday; that is, there are a finite number of humans, and when there are no more doomsday has occurred. The article touches on this with the infinite humans / transhuman stuff at the end.
The argument assumes exponential growth in number of humans until the terminal state, but this is not a foregone conclusion. While this doesn't change the reasoning in a strict sense, it may put the "soon" cutoff millions of years in the future.
Finally, the prior probability is chosen arbitrarily. It does matter quite a bit. Take the exponential argument for evidence as to why: For any given probabilty p that doomsday will occur on a given day, it is more likely today than on any single other day! This is simply because if doomsday occurs today, it cannot occur on any future day, making it less likely on each successive day.
Except that if p=1E-10, doomsday is fantastically unlikely to occur this year. The Bayesian argument has the same problem, you just start seeing it at a different value of p.
Another way to look at it: half of the humans who ever live will have fewer people born after them than before them. Your a priori odds of being such a person are exactly 1/2.