Friday, May 25, 2007

The Doomsday Argument

Yesterday, I was stuck on a six lane highway. I mean six lanes in each direction, somewhere in LA, cars bumper to bumper, people sweating and cursing and tapping fingers on their rooftops in the Californian afternoon heat. Picture this, and then tell me how you can not conclude that the extinction of the human race must be near. But I have always had my problems with the doomsday argument.

One ingredient is the Copernican principle, which states that you should not expect to be special in any regard: our galaxy is not the center of the universe, the earth is not the center of the Milky Way, and Waterloo is not in the middle of nowhere. A typical example is the number of planets in our solar system. The Copernican principle tells you that we are an average solar system in an average galaxy, and the number of planets is not exceptional but rather common. You then expect other solar systems to have a similar number of planets.

It is quite natural for human beings to adopt the Copernican principle. Typically, we assume everybody else experiences the world similar as we do. Needless to say, this is the cause of countless problems, and the reason why the men in my life state frequently they don't understand me... ah, sorry, getting distracted...

To briefly summarize the doomsday argument: consider the lives of humans as distributed over time, until doomsday, after which there are no humans any more. Then the Copernican principle comes into play. Neither you nor me is special in any regard, and the conclusion is that with equal probability I could be any of these human beings. The most common versions of the doomsday argument that I know are

A) If one assumes that population is growing, the largest number of humans will live directly before doomsday. Thus, if I am a random choice among all the humans that will ever been born, the probability is the highest that doomsday is within my lifetime.

B) I consider I am a random choice and want to know how much time is left before the end of the world. I assume a confidence of typically 95% with which I am among the last human beings that will ever been born. Since I roughly know how many people have lived before me this gives an estimate about the total number of people that will ever been born, and the time left to doomsday - with a confidence of 95%.

Okay, nice mathematical trick. Now here is what I don't understand about it.

One of the lessons from stochastic I can recall is that coincidence has no memory. Consider I flip a coin, the probability for each outcome is 1/2, and repeated flips are completely uncorrelated. I flip and it's heads. I flip again and it's heads again. I do that, say, 120 times, and the result is always the same. Then you have to make a bet on the next flip. What would you bet?

Well, okay, by now you have either fallen asleep or decided I am cheating somehow and wouldn't want to bet with me (thereby employing the Copernican principle which states that I am rather average... okay, okay, being somewhat cynical here). But let's assume this is a fair game, and I am totally unable to cheat. You'd be tempted to bet next flip does not continue this awfully unlikely series of heads, no? After all, the saying goes 'lightning never strikes twice'.

However, the probability for the next flip to show heads is (drums please) 1/2. As it was all the time. Yes, the probability for the 120 heads is a tiny (1/2)120, but the probability for the next flip is still 1/2. Coincidence has no memory. Think about Mike who enters the room after flip # 119. He sees # 120 showing heads, and the whole room goes 'ooooooh'. Mike would conclude these people are totally nuts.

Things are different if you take a bag and place in it 120 red and 120 blue marbles and pick them blindly without putting back. If you picked 120 red ones, you know the next one has to be blue. In this case, the probability depends on how many marbles you have already picked.

Now let's come back to mankind. To be clear we will state the following assumption:

1) There is a probability p that doomsday is tomorrow.

And we specify: we don't know when it is, but it's some universal property. That is to say, I am having one of my better days and I am willing to assume the presence of mankind on this planet does not increase the probability of the world ending tomorrow.

Okay. Then quantify the number of all humans that will ever been born as N and distribute them over the time prior to doomsday with function N(t), consider you are number n out of N. Take the derivative of N(t), multiply it with the speed of light, integrate it, subtract two, invert it and turn the paper upside down. Think hard about it and then tell me what is the probability that the world will end tomorrow?

Well. By assumption 1), the probability is p.

And it is in no relation with the number of people on that planet whatsoever. How come the doomsday argument suggests it is related with the number of people alive? To make the argument, the number of people living is treated as a random variable distributed over time with a probability (density), out of which you 'pick' your live span. You might picture it as a bag that contains all the CVs of all the people that will ever live, and you have to draw one. However, the way that the Copernican principle is used, there is no mentioning of how many CVs already have been distributed, so your pick has to be understood as independent of this.

This one can do for stuff like number of planets but not for events with a causal connection - like lives, or everything that has a time evolution. Even though the time evolution itself does not have to be deterministic, the total number of people at a given time is not a random variable over time. For example, if one just assumes some probability and distributes all the N people according to it, there is the small but non-vanishing probability that all the N people are born in the years 0 +/- 10 - which is impossible because it is in conflict with the evolution law for the population growth.

What one has for the population is is not a probability distribution over time, but a stochastic differential equation, that yields the probability for so-and-so many people living at a timestep t from that at timestep t-1. If one knows stochastics better than I, one can calculate a time-dependent average value for the population, around which there are random deviations, or the probability of N having a specific value at a given time. Yet, the question what is the probability of being born at a certain time is not a meaningful question to ask, because it is not independent of the times prior to that.

To come back to the two doomsday arguments.

A) Relies on the number of humans being a random variable instead of having a stochastic evolution. For an evolution law however, the Copernican principle does not make sense. Obviously, the number of people living at time t does depend on the number of people living before that. That is, you can't pick your CV out of the bag without making sure that your parents were born before you. There are causal correlations between the elements in that distribution.

B) Is a psychologically very interesting reformulation. One replaces one unknown parameter, the time when doomsday is with another unknown parameter, that is the confidence of being among the last humans that will ever live. Both parameters are related to each other. But both are still unknown. The conclusion doesn't yield any insights, it just sounds surprising.

Hmm. Running late. Have to get on that highway again. Sigh. A nice weekend to all of you :-)

71 comments:

CapitalistImperialistPig said...

The doomsday argument, and all similar arguments, rest on an untestable hypothesis. There are probably reasons to consider doomsday to be near, but I don't think they can be plausibly based in that kind of argument.

I am pretty sure that anthropic arguments have a similar flaw.

CarlBrannen said...

One problem with the doomsday argument is that it only takes into account the human population. An analysis by the last few Neanderthal would convince them that things were about to get much better. Which reminds me, latest word is that Europeans are, in fact, partly descended from Neanderthals.

QUASAR9 said...

Hi Bee, another paradox
Most of the Time when we are looking at space we are looking at the distant past, a supernovae that happened hundreds or thousands of years ago millions or billions of light years away.

Some of the stars we see in the sky, may no longer be there 'today' for we are seeing the light they emitted hundreds of thousands or millions of years ago.

So even assuming there may be other life in space, it would be more accurate to say that there may well have been other life in space, but they no longer exist in our time frame - or we could be their 'descendants' - and myths of alien space ships having travelled to our solar system long ago (or even only thousands of years ago) may well be possibly be true, but they were on a one way ticket.

Of course that no evidence of alien craft have been found is not proof that no alien craft landed on earth, and that such craft were hidden or dismantled, or simply took off again to 'discover' or colonise other worlds.

Now, I'm not suggesting that these myths are either true, probable or credible, but they are in the realm of the possible.

But the real issue is that life in a distant corner of a galaxy if it did exist, would have existed in some distant past, and need no longer exist in our 'present' day or timeframe.

Moreover I ask when we look at space why do we always talk of the past - are we (the solar system) the 'epitomy' or the last moment of Time in the fabric of spacetime.
Why is it all points in space tell me what happened before, why is there no point of space, where I can see what may have occurred after our galaxy or solar system or planet earth(?) -

well there is, the dynamics of the universe means that in some distant places new stars are being born, but because of the distances involved we (terrans) could actually be looking at 'parallel' events to the formation of our sun, simply looking at it thru the distorted glass of 'distance' in spacetime.

Here's wishing you a mighty fine day!

Arun said...

The loopholes in the doomsday argument become clear, if you imagine some group of species of dinosaurs that existed for millions of years attempting the same computation, i.e., doomsday could come after a very long steady state. Carl Brannen already pointed out the analysis by the last few Neanderthals as well - i.e, doomsday could simply be a long and slow decline into extinction.

Uncle Al said...

Fusion of I-5/I-405 in Orange County is 26 lanes wide at one point. Ten times/week some $280 million in cars are parked there.

http://www.octa.net/pdf/mmannual06.pdf
26 lanes - no exaggeration!

The world ended on 01 January 1901. The celestial paperwork is FUBAR.

Bee said...

Hi CIP,

what is the untestable argument you are referring to?

Hi Carl, Hi Arun,

yes, in principle I agree with you. There are several weak points in the details. For example it doesn't take into account evolution, as Carl has pointed out, where exactly does 'mankind' start and where does it come from. Or, as Arun mentions, population does not have to grow all the time (this however, only spoils the version A).

But anyhow, what I have tried to point out is that these details - though interesting to discuss - don't matter because the question itself doesn't make sense. One can't apply the Copernican principle because human life is not a probability distribution over time, out of which one picks randomly the own life.

Best,

B.

Bee said...

Dear Quasar,

yes, this is true. And isn't it kind of sad to think that we might eventually receive signals of another civilization just to find that they must have died millions of years ago?

But things would be dramatically different if one could travel faster than the speed of light. You know, when I was a kid I had the fantasy that once mankind could find out how to send and receive signals that travel with (potentially) infinite speed, we could connect to the communication among all the civilizations in the universe. much like you can log into a wireless when you have your laptop with you. sounds nice, but the devil is in the details since its rather unlikely communication would function by broadcast rather than directed signals. (besides the fact that there's still the issue with special relativity). Best,

B.

Bee said...

Hi Uncle,

gee, I keep thinking there must be a smarter way to move people around than in these metal boxes. could someone please work out the details of transmitters? I wonder why people keep predicting the end of the world. It's a loose-loose situation. Either your prediction is wrong, or it's right but then who cares? Best,

B.

paul valletta said...

Hi Bee, regardless of the probabilities, when the Universe initially came to be, there were no lifeforms anywhere. I am pretty certain that when the Universe comes to reboot itself, there will be no lifeforms anywhere?

The Universe has an inbuilt observer non dependancy paramiter at the start and end points!

Best paul.

wolfgang said...

> what is the untestable argument you are referring to?

I don't know what CIP had in mind, but I can tell you what I think is the implicit, untestable hypothesis of the doomsday argument.

>consider the lives of humans as >distributed over time, until >doomsday, after which there are no >humans any more.

Depending on what you assume here, your answer will vary:

i) If you assume that the population simply grows exponentially (and we will conquer the moon and then colonize space 8-), then there is no prediction.

ii) If you assume exponential growth for t less than T and then sudden collapse, then it is indeed likely that you find yourself near T.

iii) if there is exponential growth, followed by flattening and an eternal steady state, the doomsday argument again predicts nothing.

In other words, the prediction of the doomsday argument depends on the (hidden) assumption about the population you are dealing with.

You (or whoever came up with the D.A. first) have chosen distribution ii), but it is untestable for you (or us) as a member of the population.

wolfgang said...

I also agree with CIp on this one

> I am pretty sure that anthropic arguments have a similar flaw.

Indeed, in order to use anthropic reasoning one would need to know the probability distribution(s) in the 'landscape' of M-theory. In other words, one would need to know what M-theory actually is.

CapitalistImperialistPig said...

First, I agree with Wolfgang's points, but what I specifically had in mind was the so-called Copernican Principle, and especially it's application to any individual member of the human race.

Feynman had a joke that went something like this: "On the way to work today I saw a car with license plate CBK 495. Imagine the odds against that!"

The point, or so I imagine, is that each individual circumstance is in some respects absolutely atypical even if it seems unremarkable.

All of us alive today are descended in purely maternal line from a single woman who lived about 70,000 years ago. She had no reason to consider herself atypical from the standpoint of the CP, and she probably wasn't, but one of her female line descendents was one of the very few homo sapiens to survive (say) the catastrophe of the eruption of the Toba supervolcano.

The CP has no predictivity that can be tested - so far as I can see.

CapitalistImperialistPig said...

Also, my study of SF and religious literature has convinced me that nearly all intelligent species in our universe, not to mention all the gods, look a lot like humans.

How special is that?!

Bee said...

Hi CIP,

well, the whole post I wrote explains why the use of the Copernican Principle is inappropriate. Maybe you should read it.

Best,

B.

PS: I think the reason why 'gods' in illustration look often like humans is a) the principle of finite imagination and b) that we want them to look appealing.

Bee said...

Hi Wolfgang,

regarding your i - iii: what I have tried to point out is that these details - though interesting to discuss - don't matter because the question itself doesn't make sense. One can't apply the Copernican principle because human life is not a probability distribution over time, out of which one picks randomly the own life.

Best,

B.

Bee said...

Hi Paul,

true. But the problem with the observer is that we are observers. There is no science without scientists. No matter what was going on in the early universe, if we want to learn something about it we have to take into account our own place in this universe. Best,

B.

CapitalistImperialistPig said...

Bee,

I wasn't intending to disagree with you. You showed why the CP didn't make much sense for the doomsday argument. I don't think it makes much sense for most arguments.

The exception comes when something you expect to be typical isn't, or doesn't seem to be. Early estimates of the Hubble parameter seemed to indicate that the Milky Way was very large compared to similar appearing galaxies and that some of its stars were older than the universe. Here the CP should suggest to us that there was a flaw in the measurements.

Gordon said...

"Somtimes I think the surest sign that intelligent life exists elsewhere in the Universe is that none of it has tried to contact us."
-----------Calvin to Hobbes in "Weirdos From another Planet"

Bee said...

Hi CIP,
the question I was addressing is not whether the principle makes sense, but whether it's appropriate to use it even if it made sense in one or the other circumstances. You, Carl, Arun, Wolfgang are trying to explain why the car doesn't drive. I am saying, there is no car to begin with.

But to come back to the CP: it certainly does not suggest any flaw in a measurements. It's a probabilistic argument that can be useful for an estimate, but it's not a prediction. It can neither falsify a theory, nor can it tell you a measurement is wrong. If I look out of the window I see three Starbucks on this block (still in LA). The CP would lead me to expect it's the same everywhere. (Indeed, if one looks at American tourists, that's exactly what they expect - the same as home everywhere else.) Of course this expectation can be completely wrong.

The argument I was making is that probabilistic distributions over time-like variables don't make sense if there are causal connections. Sorry, I did not assume you disagree with me, but your mentioning of the CP lead me to think you didn't get the point. Best,

B.

Cynthia said...

Gordon, hate to say it, but as long as communications in the Universe are held to a speed of light limit, then your argument about the rarity of intelligent life is a bit flawed...

Plato said...

There are events within the larger universe that would look much like our universe in expression. yet these local events can they not be coisdered in context of the genus figure of the sphere to a donut?

Entropically there are also results, why not in terms of probabilty statistics?

Cynthia said...

Oh sure, if the Universe was in a *steady-state*, then the Copernican principle would be true - across all of spacetime. But since the Universe has been proven to be expanding (now accelerating), then this modified picture of the Universe in an *evolutionary state* requires that the Copernican principle be modified as well. Therefore, the Copernican principle no longer holds true across all of *spacetime*, but it still pretty much holds true within *slices* of spacetime!

CapitalistImperialistPig said...

Bee - ...probabilistic distributions over time-like variables don't make sense if there are causal connections.

OK, but I thought it was the variables with time-like separations that were causally connected ;) Or are we talking inflation here?

CapitalistImperialistPig said...

Bee,

Since you keep reminding me that I didn't understand your argument, i reread it and will try to summarize.

Case A: You assume population increases until it collapses to zero.

Case B: You assume population is a stochastic variable.

In case A, the doomsday argument plus typicallity (or the CP) implies that you are living close to doomsday.

In case B, the CP implies nothing special unless you assume some kind of stochastic dynamics.

The history of our planet suggests that if humans are a typical species, neither A nor B is likely to be the case. It also suggests that humans are an atypical species.

It doesn't seem surprising to me that such different sets of premises lead to different conclusions. But I'm not stranded on 405.

CapitalistImperialistPig said...

At some risk of overstaying my welcome, let me just add that I don't think that statement like "there is a probability p that the world will end tomorrow" make any sense except in the context of a model or theory. Statements like "there is a probability p that it will rain tomorrow" only make sense because we have some theoretical models of how the world behaves with respect to rain. Those models may be simple (it always rains in May here) or complex (5 million lines of FORTRAN on a 2048 processor computer), but it's the models that make the prediction interpretable.

Anonymous said...

The real world is more like the bag of 120 red + 120 blue marbles, where a mischievous kid removes and/or inserts in red and blue balls at random without anybody watching or knowing. So we never know what the true number of red and blue balls are.

Occasionally the kid also inserts in a green or yellow colored ball into the bag, in a completely unknown random manner.

In more real world terms, the red and blue balls represent things which we know, but for which we do not always know the probability of them happening. The green and/or yellow colored balls are events which we never could have predicted by any statistical a priori means.

Gordon said...

Cynthia: Read again! You were assuming I was quoting the Fermi argument. Calvin's argument is that they are not rare, just intelligent :)

gordon said...

Bee: Waterloo MAY not be in the middle of nowhere, but you can't get there from here :)

Bee said...

Hi CIP,

nah, I never assumed you didn't understand. I just had the impression you didn't read the post long enough to find the CP in the second paragraph (well, this seems to happen frequently in the blogosphere... you must have noticed). Therefore very special thanks for your detailed comment above- you are probably the only one who indeed read more than the title :-)

Regarding the cases A and B: these merely serve to clarify what I am talking about. It's not 'my assumption' but just the two versions of the argument that I came across.

Unfortunately your conclusion did not include the point I was trying to make. The doomsday argument in either version does not make sense because there is no appropriate probability distribution of humans over time for which one could use the CP. With this I do not mean that the assumption for the evolution growth may be inappropriate but that this function - whichever you chose -is not a probability distribution for which you can use the CP.

Since the whole argument in both cases A and B rests on the CP, conclusions in both cases are simply not valid. In addition, case B does not make any sensible prediction anyhow, it just replaces one unknown parameter with another.

Let me rephrase it in terms of the example I used above. The CP can be used if picks out of a sample of are uncorrelated. If you pick one planet, then you pick the next, these picks are independent of each other. My referral to causal connections serves to explain that this is just not the case with a distribution (human lives over time) in which there are causal connections. Picking one CV out of the back is definitely NOT independent of the others. There is no probability distribution, but a stochastic evolution equation.

At some risk of overstaying my welcome, let me just add that I don't think that statement like "there is a probability p that the world will end tomorrow" make any sense except in the context of a model or theory

You are always welcome :-) The statement makes sense in the context of every theory we know, even if it is non-deterministic as QM (including the measurement process). The probability could be zero though. It is the absolutely minimal assumption one needs for making an argument like the doomsday argument. If one assumes that this probability is not affected by the presence of us humans, then this already includes that the doomsday argument is nothing but a big cloud of fog.

Best,

B.

Bee said...

Hi Gordon,

Bee: Waterloo MAY not be in the middle of nowhere, but you can't get there from here :)

Ah, as long as my blog posts get from here to there I feel sufficiently causally connected.

The emphasis in my sentence was on middle of nowhere. Makes me wonder whether nowhere has a middle or whether it's completely homogeneous and isotropic?

-B.

Bee said...

Hi Cynthia,

I am not completely sure I understand your comment, but if I get it right I mostly agree. Surely one can't apply the CP for comparing systems in different stages of evolution. E.g. you wouldn't expect a young galaxy to have a similar number of planets as an older one. One has to take this into account, and be careful when using the CP. This dynamics is of course related to the expansion of the universe, but the issue also exists without specifying dynamics of the whole universe.

Oh sure, if the Universe was in a *steady-state*, then the Copernican principle would be true - across all of spacetime.

The Copernican principle can't be true, because it's not a conjecture waiting to be proven. It's neither right nor wrong. It's an expectation that might be useful if applied wisely. I guess you mean to say under certain conditions the CP can be expected to work, in others it will be completely off? (Like in my example with the Starbucks? See my comment At 11:15 AM, May 27, 2007) Best,

B.

Cynthia said...

Bee, thanks for the rely! And please know you really helped to clear things up for me. :-)

Gordon, I guess you're right! A comic strip writer's argument is just as valid as most (if not all) arguments presented by world-renowned physicists! ;-)

Arun said...

Carl, Arun, Wolfgang are trying to explain why the car doesn't drive. I am saying, there is no car to begin with.

If there is no car, then what are you doing stuck on the freeway?

:)

gordon said...

Cynthia: Lots of physicists are
cartoonists---look at George Gamow,
Piet Hein, Alex Vilenkin etc.
BTW you still dont seem to have got the joke. It must be one of those things that male brains can comprehend.

Gordon said...

The Doomsday argument is Bayesian--one can easily make unwarranted assumptions--I remember reading Gott ( not God) on his take on this, and feeling uneasy.

Gordon said...

Looks like a very interesting talk at Princeton by Jeff Weeks on the Shape of Space---I remember when he and Neil Turok were quite excited by the WMAP and were trying to see if the
the universe is smaller than the Last
Scattering Surface and then the LSS wraps around the universe and intersects itself ( cicrcles of intersection) I dont think they found that, but the data may have been inconclusive.
http://www.math.princeton.edu/Thurston60th/lectures.html

CarlBrannen said...

A problem with the doomsday argument is that it is like the clock whose hands are perpetually stuck at the same time. It never gives an "other than doom" prediction. This in itself should be worrisome to a scientist.

The world's population dynamics are driven by urbanization and wealth. These are two trends that are universal across this planet, and as people get more urban and more wealthy they have fewer babies.

The more advanced nations have negative population growth, some mildly, some (i.e. Japan) wildly. The natural prediction is that the world population will top out at some level within the next 50 years, and then slowly decrease.

Eventually we could dwindle to nothing, or there could be social pressures to have children. Or there could be advances in technology so that the species could reproduce without having to change dirty diapers &c.

In the usual S curve population growth and later collapse, the growth rate decreases because of disease or food pressures. Neither of these exist significantly on this planet today.

50 years after I read that we would soon starve and be going to war for food, most of trade talks are just the opposite. Nations want others to accept their food exports. Starvation only happens in the places where war prevents population movements and economics from curing it.

In any case, the current population is no longer showing signs of high exponential growth into the indefinite future, nor is it showing population growth constrained by disease or food. There is no S curve driven by inexorable limits to population that presage a collapse. The world population is happier and healthhier and wealthier and more free now than it has ever been before.

The commentary (not excluding my own) reminds me of something that I believe I read that Feynman said at dinner before getting a Nobel prize. A film star said that since Feynman was here, the party could talk about physics. Feynman said that, no, since he was there, physics was one of the things they couldn't talk about.

Arun said...

Regarding the probability that the world will end tomorrow, Bee says The statement makes sense in the context of every theory we know, even if it is non-deterministic as QM (including the measurement process). The probability could be zero though.

Then, in the same vein, we can also assign a probability to whether a particular observer is exceptional in some regard (e.g., in a solar system with an unusual number of planets, in an different-from-usual galaxy, in an unusual era of her solar system's star, etc., etc.) and thus quantify the C.P.

Cynthia said...

Okay, Gordon! I'll admit, I failed to get the Calvin joke the first time.... But I certainly got it the second time around! I simply chose to reply to your reference to Fermi's argument rather than to Calvin's argument. And please know my female brain can grasp Calvin's humor with minimal effort! ;-)

Funny you mentioned a talk at Princeton. Just yesterday as I was watching one from the Princeton archive of public lectures entitled, "Escher and the Droste Effect", you happened to cross my mind... But then no surprise here, after all, I know you're a huge fan of Escher. And if you (including that male brain of yours;-)) don't already know this, the term "Droste Effect" was named after a Dutch brand of cocoa. On the box of cocoa, there's a recursive picture of a handmaiden carrying a serving tray with a cup of hot chocolate. However, I'd be extremely hesitant to take a drink from any lady with her creepy looks!

Bee said...

Hi Arun,

well, I was trying to think the cars away, the outcome was this blog post. Don't you sometimes wonder what people might think in some thousand years if they look at what is left of our highways? Will they wonder how we were driving, what traffic regulations we had, will they wonder what the street signs mean and how we could live with something so ugly - before they push a button on their portable transmitter and go for a cup of coffee on the other side of the planet?

Best,

B.

island said...

Interesting, I was recently discussing this subject, and my typical contribution included that the missing "evolutionay" feature to the doomsday argument is the "Goldilocks Enigma", which defines the difference between a Copernican extension and what is actually observed.

The physics notes that we exist at a specific time and region of the observed universe that occurs balanced "between" extremes. In this case, we exist in the goldilocks zone that occurs between the beginning and the end of the universe, not before, nor after, it's a fine "layer" of time.

You can nail it down by including the rest of the relevant "enigmi".

The Goldilocks Enigma

This also resolves the alleged, Fermi "Paradox".

stefan said...

Dear Bee,


very interesting - and perplexing... I'm not sure if I've understood your point...

Suppose all CVs of all humans that have ever lived in the past, are living now, and will ever bee living in the future are put into one big bag. Exactly one of these CVs is picked at random. Then, the probability to locate this CV on the time axis at some time t is proportional to the number N(t) of humans alive at time t? There are some subtleties if the time scale on which N(t) evolves is smaller than the life expectancy of a human, but I guess this can be ignored to begin with. Also, the shape of N(t) (slowly growing, exponential growth with sudden decline, or whatever) doesn't matter for the argument. Do you agree so far?

Now, I could be tempted to suppose, for simplicity, that my CV is an example of one CV picked at random out of this bag. That's what they call the CP in this context. At this point, you say that this reasoning assumes some "sampling with replacement" from the big bag of CVs, and that this assumption is flawed? Is that your point? That there should be used a much more complex model for the choice of CV?

All this statistical reasoning is always so confusing ;-)...

Best, stefan

Bee said...

Dear Stefan,

No, I am afraid you got me wrong. I understand what you are saying about sampling with replacement, but this was not the point I was trying to make. Maybe my example with the marbles was pretty confusing. I just used it to show how important it is to clarify the conditions of the sampling.

The point is if you pick one CV out of the back, you need to pull out the whole evolution that comes with it. If you look at the Brownian motion of a particle, the particle doesn't just appear in a place with a certain probability, but it has to get there on a path for which there is a stochastic process. As a very simple example, take a one-line motion starting from zero, each step has probability 1/2 to either the right or the left in each timestep. Now you pick out of the distribution the position n steps to the right at the n-th step, then you know what happened in each single step. There is no randomness left. Each step must have been a step to the right. Now you could have assigned a probability for the particle to be there in that place at step n, but once you've picked it there's no more random sample and the second pick is completely determined. The use of the CP is completely inappropriate for the distribution over time. You can say that among all the points that can be reached in the n-th step you are average. But saying you are average among all the points that have been passed in the n steps doesn't make sense because the steps are correlated. You can't go from one arbitrary point to another. If you had a random distribution over time and position for your particle, you could pick a configuration that simply does not form closed paths, and is thus not compatible with your evolution law.

At this point, you say that this reasoning assumes some "sampling with replacement" from the big bag of CVs, and that this assumption is flawed? Is that your point? That there should be used a much more complex model for the choice of CV?

Yes, even though you came there on a different line of thought I am saying this assumption is flawed. I am not saying though there should be a more complex model. As I've tried to argue above, if one assumes the probability is p and independent of mankind, then the probability is p and independent of mankind. It doesn't matter how complex and sophisticated you pick your CVs.

Best,

B.

wolfgang said...

I just typed and lost this comment (perhaps now it will appear twice).

Stefan,

> the probability to locate this CV on the time axis at some time t is proportional to the number N(t) of humans alive at time t

Yes, the probability is *proportional * to N(t), but it actually *is* N(t)/Ntot.
With the total number Ntot the "shape" reappears, because it makes a big difference if N_tot is some finite number or arbirarily large.

As I wrote alread, the confusion about the D.A. results from the (hidden) assumption that N(T) = 0 at some point T and at the same time using exponential growth.

It seems also clear to me that a member of the population cannot determine N(t) for all t and thus cannot know Ntot. Therefore, it makes no sense to discuss probabilities.

Everything else is just rhetoric.

stefan said...

Dear Bee,

thank you for the explanations... I'll have to think about it...


Dear Wolfgang,

I guess I see your point.. So even if one ignores Bee's objection, in order to apply to CP one has to assume some shape for N(t) yielding a finite N_total at the first place. Which means that all one could say is "if there is a doomsday, it will happen then and then with such probability"...

Best, stefan

Bee said...

Dear Wolfgang,

That argument never made sense to me. For one, you can perfectly well look at the limit T -> infinity, T = infinity meaning there is no doomsday. But besides this, even if you have a problem with the normalization you could still talk about ratios of probabilities i.e. N(t_1)/N(t_2).

But besides this, what I am trying to say is that the assumption of there being such a probability distribution out of which one random picks lives and can apply the CP does already make an assumption. It assumes that the random picks are uncorrelated and this is just not true. I could as well ask the question: what is the probability to pick a red marble? The answer is that it depends on the previous picks. The question is just not appropriately asked.

Similarly, you can't ask what is the probability for being born at time t without specifying the conditions for that pick. The appropriate specification would be the condition at timestep t-\delta t because it's a stochastic evolution. That is to say, the probability of n people being born at time t depends on the probability of people living at time t-\delta t and your assumption about the evolution. Without that input the question just doesn't make any sense.

See, if N(t)/N_tot was a density for the random variable 'birth of human being' and you pick N_tot birth-times according to it, then there is the small but non-vanishing probability that all are born at the same time. This of course can not be the case, it's in conflict with the evolution law. The reason is that the N(t) you are talking about all the time is NOT a probability density for human birth, but the average value of the evolution that your stochastic process should yield.

I think one could use the doomsday approach for Boltzmann brains though. They are truly randomly created, and not causally connected. Best,

B.

Bee said...

Hi Gordon,

BTW you still dont seem to have got the joke. It must be one of those things that male brains can comprehend.

Ah, I finally found one of those things that male brains CAN comprehend... what much of a difference the word 'only' can make...

Best,

B.

wolfgang said...

> That argument never made sense to me...you could still talk about ratios of probabilities i.e. N(t_1)/N(t_2)

But then the (original) D.A. does not work, which claims to give a probability that the end T is n years away.

island said...

Ah, I finally found one of those things that male brains CAN comprehend...

ZING!!!

oUch!... that was a good one!

Gordon said...

Bee:
Hmm... I didn't want to confuse the women by using adjectives.

Gordon said...

From "the Devil's Dictionary":
FEMALE, n.
One of the opposing, or unfair, sex.
http://www.alcyone.com/max/lit/devils/

Bee said...

Hi Wolfgang,

> That argument never made sense to me...you could still talk about ratios of probabilities i.e. N(t_1)/N(t_2)

But then the (original) D.A. does not work, which claims to give a probability that the end T is n years away.


As I've argued in great length above the DA doesn't make sense one way or the other. But to explain why your argument that the normalization spoils the argument doesn't apply: in case

A) talking about ratios in a monotonically increasing distribution means you can still state that the probability is higher the later you are born. this doesn't mean there is a final time T, T could as well be at infinity. Still the probability to be born closer to doomsday is larger the closer you are (you'll have a renormalization problem here since the distance to infinity is infinite, consider differences between finite times instead)

B) as explained above does not contain any new insights whatsoever, since it only replaces one unknown parameter (the confidence with which doomsday is not within your lifetime) with another (how much time till doomsday). The confidence belonging to 'doomsday is never' is 100%. If you are 100% certain doomsday is not within your lifetime you either don't live (measure zero) or doomsday doesn't take place. You can't spoil this 'argument' because it's empty to begin with.

Best,

B.

wolfgang said...

> the probability to be born closer to doomsday is larger the closer you are

Very true 8-)

Bee said...

? Not sure if I used a sensible formulation for that statement. If there was a probability density and it was not monotonically increasing, then the probability to be born closer to doomsday was not necessarily larger the closer the time of doomsday.

rillian said...

Bee, I hate to continue the discussion at this point, but I still don't get your point.

In your random walk example, the case you've chosen to illustrate causal connection is highly atypical. There are only 2 paths (out of 2^n for n timesteps) whose history is completely contrained. If you instead ask for the probability that the particle is at position 0 at step n, you will find a large number of histories consistent with that choice, and so that choice has a (much) higher probability of occurring.

Likewise, if you model population growth as a differential equation, you get a set of solutions which give a count of human lives vs time. If you model it as a stochastic differential equation, don't you just get a probability distribution of sets of solutions? Either way you end up with something you can look at as a probability distribution vs. time.

You seem to be arguing that because the individual histories are contingent that one can't make a random selection from that resulting distribution. Isn't the point of such models to average over the microdynamics and overall track the effective behaviour of the majority of the histories?

Aren't the histories within any model, stochastic or not, contingent in this way? Are you then arguing that the CP can never be applied, not just that it's inapplicable to the DA?

Andrew Thomas said...

I think the Doomsday argument has been successfully refuted: http://www.lrb.co.uk/v21/n13/gree04_.html

Bee said...

Hi Rillian:

you are right with your argumentation. The point is that what you pick in the DA using the CP is not a history which one could assign a certain probability for, but the argument rests on a probability for picking only a section of such a history - alas, one life. I am not arguing that one can't pick histories, I am saying this is not what the DA rests on.

Aren't the histories within any model, stochastic or not, contingent in this way? Are you then arguing that the CP can never be applied, not just that it's inapplicable to the DA?

The CP can be applied if choices out of a sample are uncorrelated. In the usual cases one applies it to a configuration at a given time for which there ought to be a distribution. Like e.g. the size and shape of our galaxy. There is (to a very high degree) no correlation between these. You could imagine a universe with all galaxies being shaped like our own - just that it's extremely improbable. There is no evolution that would explicitly forbid this. In contrast to this, a human population in which everybody would be born in the same year is not only unlikely, it is impossible because it's in conflict with the evolution law.

Best,

B.

Aaron said...

God forbid I add to this even more but....

One has to be carefull with their words when working in statistics.

For example with galaxy size and shape distributions, the more accurate phase would be "if all the galaxies are the same then it is highly unlike that our model of random uncorrelated sizes and shapes is correct" Subtle difference but very important. I see these same errors being made all the time in health research.

The second point regards your brownian motion example, your reasoning using moments (average position) can't be used because brownian motion is described by a non-ergodic distribution, so its moments are not well defined. However its logarythmic moments are defined

Bee said...

Hi Aaron:

the more accurate phase would be "if all the galaxies are the same then it is highly unlike that our model of random uncorrelated sizes and shapes is correct" Subtle difference but very important.

I totally agree. If we would observe a configuration that was extremely unlikely if our assumption about the distribution was correct, we should instead take into account that our model is unlikely to be correct. Which brings us to the question: if our present understanding of the universe lets the conditions we observe seem extremely unlikely, shouldn't we take into account that our understanding about the universe's evolution is very insufficient instead of adapting anthropic reasoning?

Reg Brownian motion: it was an example to explain that the path has to be connected and the position is not a random distribution. It is an insufficient analogy in many regards but I found it useful. Thanks for the clarification.

Best,

B.

Bee said...

Dear Arun:

Then, in the same vein, we can also assign a probability to whether a particular observer is exceptional in some regard (e.g., in a solar system with an unusual number of planets, in an different-from-usual galaxy, in an unusual era of her solar system's star, etc., etc.) and thus quantify the C.P.

The question is not whether you can assign a probability but exactly what probability you assign to what. E.g. there might very well be a probability distribution for a human to be exceptional in some regard that allows the use of the CP - all of your examples sound good to me, none uses a distribution over time. I was trying to point out that the use of the CP as done for the DA is inappropriate, not to claim that a) the CP can never be used or b) there is never a probability for being an exceptional observer.

Basically, what I was trying to say is: one has to ask the right question to get a meaningful answer.

Best,

B.

fred said...

Reading the comments here, I couldn't help resurrecting this thread, since there was no strong defense of the DA.

Imagine a species of rat whose population grew (approximately) exponentially after if first appeared, until some disaster when the rat went abruptly extinct. Suppose each rat's skeleton was fossilized and eventually recovered by archeologists, and then put into storage, each skeleton with a unique catalog number.

Now you wish to study a skeleton. You go to the storage facility and you are assigned a random catalog number, and you take the corresponding skeleton.

Clearly, your skeleton will most likely have lived near the end of the species, in a way that can of course be quantified statistically. Importantly, any correlations that may have existed between rats (eg, your rat was the child of two other rats, etc) are irrelevant to this conclusion.

The essential point is that for the purposes of the selection of your skeleton, the relevant probability distribution is simply (proportional to) the number of rats that lived as a function of time. Being issued a particular skeleton certainly implies that its parents had lived and so on, but this is irrelevant to the conclusion since all of the skeletons are already in storage.

Now we can apply the same argument to humans. Suppose a similar exponential-then-extinction pattern. Pick a human at random from the set of all who have or will live. This set is completely analogous to the rat skeleton repository. This set of humans is thought of as given, so the fact "you picked Joe Bloe implies his parents existed" etc is irrelevant. Again, the relevant information is just the number of humans alive at each time. Therefore the same conclusion applies, irrespective of any correlations between humans.

Of course, there is a crucial assumption here, as others have commented above, that the population doesn't drop dramatically but not completely disappear. So we may not be headed for extinction just yet. But correlations play no role here.

Bee said...

Fred: Sorry, but you missed the point. If you put your skeletons in in storage etc and pick one, you're not making predictions about a time evolution to the future, which is what the DA purports to do. Please note the difference. In the one case you're asking for the next step in a time-evolution modeled by a simplistic random process. In the other case you are asking for the probability distribution of points in the paths of that process. It's simply a different quantity. Best,

B.

fred said...

OK, I see the difference, but why does it matter? Is it just that the DA claims to be making a prediction? Of course the "prediction" is only probabilistic. And it is completely analogous to the following "prediction" of the Copernican Principle (CP): Suppose the Universe is much as we see it in a sphere of, say, 1000 of our (current) Hubble radii, and then there's a roughly spherical shell of mild (not life threatening) bubble wall collisions. Applying the CP, we'd expect to not see a bubble wall collision within our Hubble volume, if our position is not special. This expectation could of course be quantified. Whether you wish to call it a "prediction" or not is a trivial matter of semantics. The DA is essentially a temporal analog of such a spatial argument.

Perhaps the misunderstanding arises because it's hard to think of the set of all humans who have or will live as analogous to the rat repository, since some of those humans haven't lived yet? Think of it as a "virtual" set if that helps. Imagine numbering each human uniquely by order of birth. Then consider the set of those numbers, together with the time of birth, instead. Apply the DA to that set of numbers. That should make it clear why any correlations are irrelevant. That the DA may claim to make a "prediction" is just an accident of the fact that the DA is being made at a time inside the span of human existance.

Finally, and conversely, suppose that humans continue to live as a species much as today, with a population of billions, for billions of years. Then do you not find it exceedingly odd that you find yourself amongst the first millionth (or whatever) of all humans, assuming you're not special?

Bee said...

Fred: The point is that the DA makes a wrong prediction by abusing a sloppy wording what the prediction actually is about. Look, we are considering a partly random process with a probability p that doomsday is tomorrow. By assumption p doesn't depend on the number of people on the planet or anything, ie it's totally uncorrelated with what previously happened. Then the probability that doomsday is tomorrow is p. The catchy phrase my prof used for that is "randomness has no memory." The rest of the DA is entirely irrelevant. You already know the probability that doomsday is tomorrow and it's p. There's nothing more to explain!

As to the question of drawing people from a temporal distribution, I already told you that there's causal correlations between the draws and it's simply a wrong statement. You can't be "drawn" unless your ancestors are also all "drawn" and your children can't be "drawn" without you being "drawn". The only thing you can meaningfully "draw" is a full history (ie one path of the random process). Best,

B.

fred said...

OK, now I think I've (finally!) gotten your criticism. I'll lay out an explicit counter-argument to the DA; please tell me if this is your point:

Consider a human-like species, called the Zog, that first appears in year 0, and quickly reaches and maintains thereafter a nearly constant population, N(t) \simeq const (at least until doomsday). [It's straightforward to generalize the argument to more realistic N(t).] Suppose there are asteroids, which, if one hits the planet, will cause global catastrophe and extinction of the Zog. Suppose that with the Zog's best instruments, they can't determine when any asteroid will hit, only a probability per unit time, p(t), that one will hit. Suppose that they determine that p(t) is constant, with value p \sim 10^{-9}/yr. Then they'd expect the Zog to survive on order a billion years.

Now a particularly clever Zog, called Zog1, in year 1000, applies the DA. Assuming N(t) constant, he'd conclude that the odds are very slim that the Zog will last a billion years, in particular 1000/1 billion \sim 10^{-6}, and that the Zog are expected to last of order 1000 years, with p \sim 10^{-3}/yr. So how do we explain this discrepancy in probability estimates?

To see the answer, recall that both estimates of p are the best the Zog can say about the future given incomplete knowledge of the present (and past). They are epistemological, in that they depend on the Zog's current knowledge. The two estimates of p differ simply because they're based on different knowledge. In one case, on measurements of asteroids, and in the other, on the time since the Zog species began. It's not surprizing that different knowledge leads to different conclusions, and that some are more accurate than others.

Similarly, in year 1 million, a Zog would estimate p \sim 10^{-6}/yr using the DA. He gets a very different answer than Zog1 did using the DA, simply because he has different information. Namely, he knows that the Zog have survived a million years rather than just a thousand.

Again, causal correlations between the Zog (parents/children/etc) are irrelevant here. To estimate p using the DA, only the current time since the Zog appeared needs to be used.

Bee said...

Fred: No, I think you still don't get it. You have assumed the probabilities are uncorrelated. Prior knowledge does not matter by assumption. You are still confusing two different things, let me use the example with the coins I used in the blogpost. I'm going to flip a (fair) coin, what's the probability of heads? Well, it's 1/2. Now I tell you I've been flipping this coin for the last 3 hours and it's always been heads. What's the probability of the next flip being heads? Well, it's 1/2. What has a small probability is the three-hour series of having heads. But that's a different question. If your Zog gets up one morning, then the probability that an asteroid will hit that day is [something]. Knowing that no asteroid has hit the day before doesn't matter by assumption. The question how long it will take on average till an asteroid hits etc, is a different question. Now go back to the DA. If the probability that the world ends any particular day is p, what's the probability that the world ends tomorrow. Well, it's p. You're just confusing yourself with all sorts of irrelevant information. A similar argument would lead you to conclude that the longer you play in the lottery the higher are your chances you win in the next draw, which is also wrong. The probability that you win is always the same. It is true that there are cases in which you can update information to obtain a better probability estimate, but this only has an effect if the information is indeed relevant to the probability. (Let's say you know my coin isn't a fair coin, then you could use prior flips to obtain a better estimate on the next flip.) Best,

B.

fred said...

Bee, we can certainly construct scenarios where the DA provides no useful information. Suppose that p is unity. Eg, say the Zog know their planet will be destroyed in year 1 million. Then if Zog1 applies the DA in year 1000, he will have to conclude that the DA isn't telling him anything, and that he is simply very unusually early in the span of his species.

But our real concern here isn't whether the DA can be applied to certain scenarios involving the Zog! We want to know if we can apply the DA to ourselves. And for ourselves, we certainly don't know when our species will end! Nor can we even come anywhere close to providing a meaningful estimate for p, the probability/yr that it will end! So if we don't know p, there can be no conflict with the DA.

That's why the DA is useful. In the absence of any other reliable estimates of our fate, the DA is all we have.

I suspect that your objection will be that even if we can't estimate p in practice, that some value for p exists in some sense nevertheless. But it doesn't, since classical probabilities simply characterize our lack of knowledge. They are not ontological. In this case, we have essentially no relevant knowledge to make a meaningful estimate of p (apart from the DA itself). All we can really say (without applying the DA) is that our species will end sometime between now and eternity (or some heat-death age of the Universe or whatever). Therefore the "coincidence has no memory" argument does not apply.

Bee said...

Fred: I have just told you why the Doomsday argument, in the form I stated it, is a logically faulty argument. I don't think, and I doubt anybody in their right mind thinks, this argument does actually provide some insight about the future of the human race. It is just an example to create a story around the maths. (Like the twin paradox, Schroedinger's cat, etc etc). To begin with the probability that the human race goes extinct arguably does depend on the population. No, that isn't my objection. My objection is simply that this isn't the argument I've been talking about. Best,

B.

fred said...

Alright, Bee, it's certainly perfectly fine to criticize one particular form of the argument. But I think it's even more interesting to consider the validity of the DA more generally, which is what I've had in mind here.

Certainly no one should claim that the DA says that we're doomed within a lifetime, say. There are assumptions involved, that we're typical and about the shape of the future population vs t curve.

But what's very cool about the DA is that it tries to quantify (in a surprizing way) the sense of extreme oddness we get when we contemplate the following possibility: That our species continues for many billions of years with a population of many billions, yet we find ourselves exquisitely close to the beginning. And that sense of oddness I think we can all appreciate.

Bee said...

Fred: That's the form of the argument I know. If you want to claim some other version makes some statement about something, I'm not able to follow because you haven't stated your assumptions. Best,

B.

fred said...

Bee, I just did state the assumptions in my last post ("that we're typical and about the shape of the future population vs t curve") as well as earlier. These are standard and shouldn't need any elaboration.

As I understand it, you've been considering the additional assumption that we know (independently of the DA) a probability/yr p(t) for doomsday. Realistically, we of course can't say anything about p(t) in practice (apart from applying the DA!): predicting the fate of a civilization is a horribly complex piece of physics! So then it appears that your "coincidence has no memory" argument doesn't apply.