# Doomsday Argument with Strong Self-Sampling Assumption

post by drnickbone · 2012-01-20T23:50:25.636Z · score: 9 (13 votes) · LW · GW · Legacy · 45 commentsHello everyone; I'm new to the forum, and have been advised to post this in the "discussion" section. Hope this is OK.

I've found some references to discussions here on Brandon Carter / John Leslie's "Doomsday Argument" and they seemed well-informed. One thing I've noticed about the argument though (but haven't seen discussed before) is that it can be made much sharper by assuming that we are making random *observations*, rather than just that we are a random *observer*.

For those who know the literature, this is a form of Nick Bostrom's Strong Self-Sampling Assumption as opposed to the (basic) Self-Sampling Assumption. Oddly enough, Bostrom discusses SSSA quite a lot in connection with the Doomsday Argument, but I can't see that he's done quite the analysis below.

So here goes:

In the "random observer" model (the Self-Sampling Assumption with the widest reference class of "all observers"), we discover that we are in a human civilization and there have been ~100 billion observers before us in that civilization. We should then predict (crudely) that there will be about ~100 billion observers coming after us in that civilization; also we should predict that a typical civilization of observers won't have much more than ~100-200 billion observers in total (otherwise we'd be in one of the much bigger ones, rather than in a smaller one). So typical civilizations don't expand beyond their planets of origin, and don't even last very long on their planets of origin.

Further, since there are currently ~150 million human births per year that would imply the end of the human race in ~700 years at current population size and birth-rates. Doom soon-ish but not very soon.

But what about the "random observation" model? One difference here is that a large portion of the ~100 billion humans living before us died very young (high infant mortality rate) so made very few observations. For instance, Carl Haub, who calculated the 100 billion number (see http://www.prb.org/Articles/2002/HowManyPeopleHaveEverLivedonEarth.aspx) reckons that for most of human history, life expectancy at birth has been little more than 10 years. By contrast, recent observers have had a life expectancy of 60+ years, so are making many more observations through their lives than average. This means that *observations* are much more concentrated in the present era than *observers*.

Working with Haub's population numbers, there have been about 1-2 trillion "person-years" of observations before our current observations (in January 2012). Also, that estimate is very stable even when we make quite different estimates about birth-rate. (The reason is that the overall population at different stages in history is easier to estimate than the overall birth-rate, so integrating population through time to give person-years is easier than integrating birth-rate through time to give births).

Under the "random observation" model, we would predict a similar number of person-years of observations to come in the future of our civilization. At a human population size of ~7 billion, there are only around 1-2000 / 7 or ~200 years until human extinction: doom rather sooner. And if population climbs to 10 or 14 billion before flattening out (as some demographers predict) then doom even sooner still.

What's also quite striking is that over 20% of all observations *so far* have happened since 1900, and under a "doom soon" model the *majority* of all observations would happen in the period of multi-billion population sizes. So our current observations look very typical in this model.

Now I'm aware that Bostrom thinks the SSSA is a way out of the Doomsday Argument, since by relativizing the "reference class" (to something other than all observations, or all human observatioons) then we get a less "doomish" prediction. All we can conclude is that the reference class we are part of (whatever that is) will terminate soon, whereas observers in general can carry on. I'm also aware of a number of criticisms of the whole SSA/SSSA approach.

On the other hand, it is quite striking that a very simple reference class (all observations), coupled to a very simple population model for observers (exponential growth -> short peak -> collapse) predicts more or less exactly what we are seeing now.

## 45 comments

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My problem with this is that as you add more and more information to the doomsday argument, it becomes more and more problematic that there's a ton of information you're leaving out, information that is in fact more strongly correlated with survival than just the total number of people born.

The doomsday argument with minimal information has a kind of symmetry, a nice aesthetic to it. But if you say "okay, now let's add lifespan data," the question is "how about war data, or murder rates, or progress on missile defense systems, or the stalling of manned space exploration, or research on bioweapons, or..."

The point about "adding more information" making the reference class more complex (and hence less plausible) is spot on. However, the interesting thing here is that counting person-years of observations actually uses strictly less information than counting births of observers.

To count person-years of observations, we just have to integrate population through time, and this simply requires information about population sizes at various stages of history. Whereas to count births, we also have to guess at the birth-rate per 1000 people as well as the population size, and integrate the product of population and birth-rate over time. See Carl Haub's article on this.

This is why, for instance, there is variation in the literature on the birth rank assumed for us right now; in Leslie's and Bostrom's papers, a birth rank of about 60 billion is assumed, whereas more recent estimates give a birth rank of about 100 billion. Even earlier estimates were for a birth rank of about 20-30 billion. We really don't know our own birth rank very well.

So that's totally true. If you get information about lives, you expect to be halfway through lives, if you get information about years lived, you expect to be halfway through years lived. I never really thought about how you can just get years lived from population, so you don't do anything weird like first learning about the number of lives and then throwing it away.

I guess it's just the implicit comparison of the merits of these two very minimal sets of information, on a subject about which we have lots of better information, that makes it a bit awkward for me.

A point I haven't seen made about the Doomsday Argument is that as time goes by, the predicted doom recedes into the future. It isn't like we just discovered that a specific catastrophe is awaiting us at a specific time, like the sun going nova, or even a constant probability per unit time of something, like an asteroid strike. Every year that goes by without doom makes the DA-predicted time until doom a year longer. And since DA also predicts that doom isn't going to happen immediately, one could conclude that there's nothing to worry about.

Touche - doom keeps receding, until suddenly it doesn't (because it just happened).

However, I think that one practical reason for "worrying about it" is the implied increase of our epistemic probability that Doom is going to happen. That could suggest all sorts of actions, such as: let's take particular Doom scenarios seriously, and try to mitigate them. Certainly, let's not scoff at doomsayers as "alarmist"(the traditional reaction), but think of why they might be right.

On the other hand, it is quite striking that a very simple reference class (all observations), coupled to a very simple population model for observers (exponential growth -> short peak -> collapse) predicts more or less exactly what we are seeing now.

You convinced me that your reference class is a good one, but I'm not convinced about that population model and so I'm having a hard time with the idea that this "prediction" is good evidence for the model (it seems like there must be a very large number of population models that would predict what we see right now).

The very simplest population model is an exponential growth pattern, which flattens out at a maximum when the population overshoots its planet's resources, and then drops vertically downward. That fits our current observations, since almost all observations will be made at or near the maximum. (Notice that human population is no longer growing exponentially, since percentage birth rates are falling dramatically almost everywhere. Recently, our growth is quite linear, with roughly equal periods going from 4-5 then 5-6 and 6-7 billion, and by a number of measures we are now in overshoot).

To make this model generic, assume that a generic planet supporting observers has a mixture of renewable and non-renewable resources. At some stage, the observers work out how to exploit the non-renewable resources and their population explodes. Use of the non-renewables allows the death-rate to fall and the population to grow far beyond a point where it can be sustained by the renewables alone; then as the non-renewable resources become exhausted, population plummets down again.

These dynamics arise out of a really simple population model, such as the Lotka-Volterra equation (a predator-prey model); the application to non-renewable resources is to treat them as the "prey" but then set the growth rate of the prey to zero. There are also plenty of real-life examples, such as yeast growing in a vat of sugar, where the population crashes as a result both of exhausting the non-renewable sugar in the vat, and the yeast polluting themselves with the waste product, alcohol. (This seems disturbingly like human behaviour to be honest: compare fossil fuels = sugar; co2 emissions = alcohol.)

Now I agree that other population models would fit as well. A demographic transition model whereby birth-rate falls below death-rate everywhere will lead to exponential behaviour on either side of the peak (exponential up, peak, exponential down) and a concentration of observations at the peak. One thing that's suspicious about this model though is understanding why it would apply generically across civilisations of observers, or even generically to all parts of human civilisation. If only a few sub-populations *don't* transition, but keep growing, then they quickly arrest the exponential decline, and push numbers up again. So I don't see this model as being very plausible to be honest.

It's also worth noting that a number of population models really *don't* fit under the reference class of all observations, assuming our current observations are random with respecr to that class. Here are a few which *don't* fit:

Populations of civilisations keep growing exponentially beyond planetary limits, as a result of really advanced technology (ultimately space travel). Population goes up into the trillions, and ultimately trillions of trillions. Our current observations are then very atypical.

Most civilisations follow a growth -> peak -> collapse model, but a small minority escape their planetary bounds and keep growing. The difficulty here is that almost all observations will be in the "big" civilisations which manage to expand hugely beyond their planet, whereas ours are not, so they are still atypical observations. Ken Olum made this point first (I think).

Long peak/plateau. Civilisations generally stabilise after the exponential growth phase, and maintain a "high" population for multiple generations. For instance ~10 billion for more than ~1000 years. Here the problem is that most observations will be made on the long plateau, well after the growth phase has ended, which makes our own observations atypical.

Decline arrested; long plateau. Here we imagine population dropping down somewhat, and then stabilising say at ~1 billion for more than ~10000 years. Again the difficulty is that with a long plateau, most observations are made on the plateau, rather than near the peak. Finally, it's a bit difficult to see how population could stabilise for so long; you'd have to somehow rule-out the civilisation ever creating space settlements while it's on the plateau (since these could then expand in numbers again). Perhaps it is just impossible to get the first settlements going at that stage in a civilisation's history (can't do it after the non-renewable resources have all gone).

Upvote for Breakfast of Champions reference.

Like everybody else in the cocktail lounge, he was softening his brain with alcohol. This was a substance produced by a tiny creature called yeast. Yeast organisms ate sugar and excreted alcohol. They killed themselves by destroying their own environment with yeast shit.

Kilgore Trout once wrote a short story which was a dialogue between two pieces of yeast. They were discussing the possible purposes of life as they ate sugar and suffocated in their own excrement. Because of their limited intelligence, they never came close to guessing that they were making champagne.

Also an illustration of the inherent silliness of seeking a transcendent meaning to life, I guess.

Maybe we should use not adult observers but observers who know and could understand probability theory. Probably there were several millions of them before now. Most of them lived in the 20 century. So it makes DA prediction even sharper.

You could also count your position between all people who understand Doomsday argument. Maybe only 10 000 people did it since 1983 when it was first time proposed. And this number is also growing exponentially. This means that only 10 years is left before Doom.

Also I could count my position between all who understand that Doomsday argument reference class is all people who understand DA. Probably only a few did it. I knows it last 3 years. And it means sooner Doom. Or that all this line of reasoning is false.

Another issue... Yes restricting the reference class to people who are discussing the DA is possible, which would imply that humans will stop discussing the DA soon... not necessarily that we will die out soon. This is one of the ways of escaping from a doomish conclusion.

Except when you then think "Hmm, but then why does the DA class disappear soon?" If the human race survives for even a medium-long period, then people will return to it from time to time over the next centuries/millennia (e.g. it could be part of a background course on how not to apply Bayes's theorem) in which case we look like atypically early members of the DA class right now.. Or even if humanity struggles on a few more decades, then collapses this century, we look like atypically early members of the DA class right now (I'd expect a lot of attention to the DA when it becomes clear to the world that we're about to collapse).

Finally, the DA reference class is more complicated than the wider reference class of all observations, since there is more built into its definition. Since, it is more complex and has less predictive power (it doesn't predict we'd be this early in the class) it looks like the incorrect reference class for us to use right now.

So there are 3 possibility:

1 We will die off very soon, in next 10 years perhaps. It is possible because of «success« in bioengineering and AI.

2 In next 10 years DA will be rebutted in very spectacular and obvious way. Everyone since will know this rebuttal.

3 DA is wrong.

My opinion is that very soon dieoff is inevitable and only something really crazy could save as. It could be quantum immortality, or AI crash project, or extraterrestrial intelligence or owners of our simulation.

I suppose a "really fast, really soon" decline is possible ... something so quick that essentially no-one notices, and hence there isn't a lot of discussion about why DA seems to have been right when the decline happens.

However, one problem is making this model generic across multiple civilisations of observers (not just humans). Is it really plausible that essentially every civilisation that arises crashes almost immediately after someone first postulates the DA (so the total class of DA-aware observers is really tiny in every civilisation)? If some civilisations are more drawn-out than others, and have a huge number of observers thinking about DA before collapse, then we are - again - atypical members of the DA class.

It is really interesting point - to see all DA aware observers in all civilizations. So maybe technologies are the main reason why all civilizations crash. And understanding of DA typically appear tougher with science. So this explain why understanding of DA is coincedent with global catastrophes.

But more strong could be idea that understanding of DA has casual relation with catastrophes. Something like strong anthropic principle. Now I think that it is good idea for science fiction, because it is not clear how DA understanding could destroy the world, but may be it worth more thinking about it.

Maybe we should use not adult observers but observers who know and could understand probability theory.

You can't just set the observer class to whatever you want. You get different answers. You have to use the class of every possible observer. I can explain this mathematically if you wish, but I don't have time right now.

No, I can. But my answers give only time of existence of this referent class, not general Doom. For example, someone knows that he is a student in the University for 2 years. So he could conclude that he will be a student 2 more years with probability 50 per cent.

This is the answer to so called problem of referent class in DA. Each referent class has his own time of existence, its own end.

Yes, you can't. At least, not if you do it right.

P(There are m total humans | You are human number n) = P(You are human number n | There are m total humans) * P(You are human number n) / P(There are m total humans)

If P(You are human number n | There are m total humans) came out to equal n/m, it would work fine. It doesn't.

P(You are human number n | There are m total humans) = P(You are human number n | There are m total humans & You are human) * P(You are human | There are m total humans)

= n/m * P(You are human | There are m total humans)

If P(You are human | There are m total humans) was constant, it would still work. The problem is, it's not. It only works out that way if the number of observers is proportional to the number of humans. For example, if almost all life is either human, or wildlife on planets humans terraformed.

Interesting point... However my post was not favouring *adult* observations as such; just counting all observations, and noting that people who live longer will make more of them. There is no need to exclude observations by children and infants from the reference class.

**[deleted]**· 2012-01-21T00:35:46.785Z · score: 2 (2 votes) · LW(p) · GW(p)

First off, the *mean* of our expected position in civilization is halfway thru, but because of how the civilization size changes depending on whether we are in the last 95% or first 5% (these are equally likely right?) I don't think you can go on to say we are half way thru using up our total population. If someone could do the math for this, that would be cool. Like think what's the expected value for total length of human civilization.

Second, saying that there were lots of baby observations in history might not mean much, because our observation are not baby observations. Then again, our observations are not transhuman future observations or medieval peasant observations either. Can someone who know this better point out where I'm confused here?

Anyways, I'm pretty sure this doomsday thing is totally bunk, but that could be because I don't understand it.

Correct - we can't predict that we are exactly in the middle of human observational history, only roughly in the middle. This is why the prediction from the random observation model is ~200 years, with the ~ representing a range of variation around the central estimate.

Giving formal confidence intervals (like Gott does) seems a bit of a stretch in my view, since the bounds of these then become acutely sensitive to the prior distribution. Under the "vague" prior over total person-years of observations, and with a 90% confidence interval, we could predict between 100 billion and 40 trillion of person-years to come.

**[deleted]**· 2012-01-21T21:03:35.220Z · score: 0 (0 votes) · LW(p) · GW(p)

No I'm not disputing taking the mean to be the middle of the probability distribution, that's elementary. I mean because the population numbers are on such different scales if we are at the end or the beginning, the mean of the *total* distribution may be far in the future. I don't actually know if this is true or not because I don't understand what the probability distribution looks like.

If we looked at just two cases instead of one, it might help things. The two cases are: we are in the last 95% or we are in the first 5%. In the 5% case, we estimate another 2 trillion people are coming. in the 95% case, we estimate another 5 billion. To calculate expected value of future people, we have to know the relative likelyhoods of cases like these. If we consider anthropic issues in a naive way (just assume all observations are equally likely) I don't know what the distribution looks like. Maybe someone else can help. I do know that we shouldn't just considr anthropic bias naively, we should also consider what we currently observe about the world. We are not transhuman and we are not prehistoric peasants, so that is some evidence. My intuition tells me that once you consider what you actually see as evidence, it screens off all the anthropic stuff and expected populations and such.

Following the reasoning behind the Doomsday Argument, this particular thought is likely to be in the middle along the timeline of all thoughts experienced. This observation reduces the chances that in the future we will create AI that will experience many orders of magnitude more thoughts than those of all humans put together.

That's an interesting point.

The whole doomsday argument seems to me to be based on a vaguely frequentist approach, where you can define anything as the class of observations. You raise a great point here, changing the reference class from "people" to "experiences." The fact that the predicted end of the world date varies according to the reference class chosen sounds a lot like accusations of subjectivity in frequentism.

Actually, I think it is more like the charge of subjectivism in Bayesian theory, and for similar reasons.

If we take a Bayesian model, then we have to consider a wide range of hypotheses for what our reference class could be (all observers, all observations, all human observers, all human observations, all observations by observers aware of Bayes's theorem, all observers aware of the DA). We should then apply a prior probability to each reference class (on general grounds of simplicity, economy, overall reasonableness or whatever) as well a a prior probability to each hypothesis about population models (how observer numbers change over time; total number of observers or total integrated observer experiences over time). Then we churn through Bayes' theorem using our actual observations, and see what drops out.

My point in the post is that a pretty simple reference class (all observations) combined with a pretty simple population model (exponential growth -> short peak -> collapse) seems to do the job. It predicts what we observe now very well.

We should then apply a prior probability to each reference class (on general grounds of simplicity, economy, overall reasonableness or whatever) as well a a prior probability to each hypothesis

What is applying a prior probability to a reference class? As opposed to applying a prior probability to a hypothesis?

The hypothesis is "this reference class is right for my observations" (in the sense that the observations are typical for that class). There might be several different reference classes which are "right", so such hypotheses are non-exclusive.

I suspect they all are, weighted by "general grounds of simplicity, economy, overall reasonableness or whatever)" i.e. Kolmogorov complexity.

Therefore, asking if "wildlife" or "humans" or whichever *simple* reference class is the right one is a wrong approach.

To me, this is inside view vs. outside view. If an alien randomly selected human-experience-seconds after humanity went extinct in a sudden cataclysm, most of the selected experiences would be within a human lifetime of the cataclysm, for the reasons you specified. But our experience of our own life is NOT random in the same way that the alien's selection is random. So there's no reason to say that our observation of the shape of the population curve is evidence that there is a sudden cataclysm in the near future.

If there's some glaring math error in my post, I'd appreciate it if someone could explain it to me.

I think the issue is that we need *some* sort of insider model, otherwise we cannot extract any predictions about what we are supposed to be observing.

Nick Bostrom gives some examples to illustrate the point. Suppose physicists come up with a universe model which is infinite or really-really large. The consequence of the model is that every observation which could be made actually *is* made by someone, somewhere. Now suppose that the background radiation in this model is uniform and has a temperature of 1 Kelvin; we would generally believe that the model is falsified because we observe a temperature of around 3K. However, we know that at least *some* observations in the model (albeit a tiny minority) wil be exactly like ours, so how is the model falsified? If we don't make any assumptions about how our observations are sampled from the whole (in the "insider" model we are atypical; so what?) then we have no way of testing or falsifying such models.

There is also a statistical point justifying insiders arguing they are randomly selected (*as if* they had been selected by an outsider). If *all* the insiders in the class were to reason in this way, and predict - for each of their observations - that there will be about as many observations still to come in the class as there have been so far in the class then *most* of them will be right, most of the time. Statistically, it seems very reasonable to use a method which works for most of the people most of the time.

I think the issue is that we need some sort of insider model, otherwise we cannot extract any predictions about what we are supposed to be observing.

So you agree that the predictions your are making are not well founded if using the insider model is not justified in these circumstances?

There is also a statistical point justifying insiders arguing they are randomly selected (as if they had been selected by an outsider). If all the insiders in the class were to reason in this way, and predict - for each of their observations - that there will be about as many observations still to come in the class as there have been so far in the class then most of them will be right, most of the time. Statistically, it seems very reasonable to use a method which works for most of the people most of the time.

But there are patterns in the reasons why an insider would be wrong to pretend to be an outsider. Those patterns should be readily apparent to an outsider. Imagine if the population curve we were discussing were of Europe instead of the whole world. A non-European who randomly selects European experience moments might be justified in making the argument you present, but a European doesn't seem able to justify pretending that their our experience moment is randomly selected (for reasons that are really obvious to the non-European). Or am I misunderstanding what statistics means by random selection?

Nick Bostrom gives some examples to illustrate the point. Suppose physicists come up with a universe model which is infinite or really-really large. The consequence of the model is that every observation which could be made actually is made by someone, somewhere. Now suppose that the background radiation in this model is uniform and has a temperature of 1 Kelvin; we would generally believe that the model is falsified because we observe a temperature of around 3K. However, we know that at least some observations in the model (albeit a tiny minority) wil be exactly like ours, so how is the model falsified? If we don't make any assumptions about how our observations are sampled from the whole (in the "insider" model we are atypical; so what?) then we have no way of testing or falsifying such models.

I don't understand the point of this example. Most importantly, the question assumes that the correct answer is already known. I agree it is quite confusing for every measurement to be wrong when the right answer is independently known. But that's not anywhere close to the state of science, and we don't expect scientific observation to ever be in the position of trying to confirm what we already known through non-scientific means.

Even granting the assumption that the true theory is known before observation, there's an ambiguity about what is meant by "observations that the background is 3K." Do you mean that every human observation in our light-cone shows 3K? If so, doesn't Ockam's Razor suggest that there is something unusual about local space, not that all our measurements (and theories about the background temp) are wrong? In a certain sense, we'll never know that our observations are caused by strangeness in local space rather than being uniform everywhere, but the difficulty of understanding the universe beyond our light-cone is a fact about the physical universe, NOT a fact about statistics. And if it's only one erroneous measurement, why are we worried that we can't show it was in error?

No, background radiation is uniformly 1K, but it just happens that every observation we've ever made of it has suggested it's 3K. The point is that in a really big 1K universe, there are *some* observers who are in that unfortunate position of thinking it's 3K. Not very many of them, but how do we know we're not part of that unfortunate minority?

Well it's crashingly obvious really that if the universe did have 1K background, then we'd expect to be part of the overwhelming majority who observe 1K rather than the teeny-tiny minority who appear to observe 3K. However, that "crashingly obvious" conceals an observer selection hypothesis, which is Bostrom's point.

Let's back up for a moment. There are two propositions at issue here. Following is the first:

If an alien observer took a random sample of human experiences and came up with me and my dorm mates, the observer would be justified in certain belief about the shape of the curve of human population over time. Specifically, the observation would be evidence that a cataclysm occurred within our lifetimes.

I agree with this point. Second proposition:

The process that led me and my dorm mates to be placed together and able to share our experiences is sufficiently random that I am justified in reaching the same conclusion as the hypothetical alien observer. That is, by looking

onlyat the shape of the population curve and my place in it, I can make predictions about the future shape of the curve.

This proposition seems mathematically unjustified. (The true of my assertion is what we are debating, right?). I don't understand what the background radiation hypothetical does to support the mathematical position I'm rejecting. I agree that we would expect to be part of the 1K-measurement population rather than the 3K-measurement population. But the hypothesis is that the universe is so large that someone "wins" this unlikely lottery.

So it unexpectedly turns out to be us. We'll NEVER know that we are the victims of a galactic improbability. And we'll NEVER know the true theory of background radiation since we don't have access to the data necessary to justify the theory. From our point of view, the background temperature really IS 3K. And we're wrong. I'm not trained on the technical issues, so I don't understand why this helps the proposition above that I labelled as mathematically unjustified.

PS Of course the assumption is "mathematically unjustified". Every core assumption in every scientific theory is "mathematically unjustified". Science is not mathematics.

When I say "mathematically unjustified," I mean contrary to mathematical rules. Physics relies on induction, which is not mathematically justified in some sense (but Hume doesn't suggest we should abandon the scientific method). But physics never says anything like "assume 1 + 5 = 7 when talking about quarks."

Your argument looks at two items (1) the shape of the population curve, and (2) our place in the population curve. From that, you infer something about the shape of the curve in the future. I say that making inferences from those two items requires certain technical definitions of random selection be satisfied. And point (2) does not appear to satisfy those requirements. By contrast, the observations you describe in your parallel comment do appear to satisfy those technical requirements.

I'm not a trained statistician, so my understanding of the precise contours of the technical requirements could be wrong. But saying that relaxing the requirements usually doesn't contaminate the results is insufficient because it seems to me that the reasons for the strict requirements are highly relevant to your argument, but not all statistical arguments. In short, there's no way to infer the future shape of the population chart from the chart up to this date. Why should I believe that my experience of the chart is additional evidence, independent of the chart, that justifies additional inferences that the chart could not justify on its own?

Edit: Imagine you are trying to make inferences about the skills of professional baseball players, and you start by assuming that talent for playing baseball is normally distributed. This assumption is almost certainly false because professional baseball players are a selected off subset of all people with the capacity to play baseball. That is, we expect the shape of talent *among players who are selected for their skill at playing baseball* to resemble one tail of a normal curve, not the entire curve. Applying statistical tests that assume normal distribution will almost certainly lead to incorrect conclusions.

So, by "mathematically unjustified" you meant something like "mathematically inconsistent" in the same way that "1 + 5 = 7" is inconsistent. However, now I'm puzzled, since why is it "mathematically inconsistent" for an insider to model his observation as a random sample from some population?

Provided the sampling model follows the Kolmogorov probability axioms, it is mathematically consistent. And this is true even if it is a totally weird and implausible sampling model (like being a random sample from the population {me now, Joan of Arc at the stake, Obama's left shoe, the number 27} ... ).

In the world of the unlucky physicists, they are assuming that their data is randomly selected. If I understand the thought experiment, this assumption is correct. If this were a computer game, we'd say the physicists have been cursed by the random number generator to receive extremely unlikely results given the true state of the universe. But that doesn't mean the sample isn't still random - unlikely occurrences can happen randomly.

Likewise, the doomsday argument assumes that the sample of human experiences is randomly selected. Yet there is no reason to think this is so. You are using your experiences as the sample because it is the only one truly available to you. To me, this looks like convenience sampling, with all the limitations on drawing conclusions that this implies. And if your assumption that your sample is random is wrong, then the whole doomsday argument falls apart.

In short, cursed by the random number generator != nonrandom sample.

What I'm trying to understand is the difference between these two arguments:

Model A predicts that the vast majority of observations of the universe will conclude it has a background radiation with a temperature of 1K, whereas a tiny minority of observations will conclude it has a temperature of 3K. Model B predicts that the vast majority of observations of the universe will conclude a background radiation temperature of 3K. Our current observations conclude a temperature of 3K. This is evidence against model A and in favour of model B.

Model 1 predicts that the vast majority of observations of the universe will be in civilisations which have expanded away from their planet of origin and have made many trillion trillion person-years of observations so far; a tiny minority will be in civilisations which are still on their planet of origin and have made less than 10 trillion person-years of observations so far. Model 2 predicts that the vast majority of observations will be in civilisations which are still on their planet of origin and have made less than 10 trillion person-years of observations so far. Our current observations are in a civilisation which is still on its planet of origin, and has made less than 10 trillion person-years of observations so far. This is evidence in favour of Model 2.

Formally, these look identical, but it seems you accept the first argument yet reject the second. And the difference is... ?

In both cases, the inferences being drawn rely on the fact that the observation was randomly selected.

In the physics example, the physicist started with no observation, made a random observation, and made inferences from the random observation.

In the population example, we start with an observation (our own lives). You treat this observation as a random sample, but you have no reason to think that "random sample" is a real property of your observation. Certainly, you *didn't* random select the observation. Instead, you are using your own experience essentially because it is the only one available.

But then why do you assume that the physicist made a "random observation"? The model A description just says that there are lots of observations, and only a tiny minority are such as to conclude 3K. If both model A and model B were of deterministic universes, so that there are strictly no "random" observations in either of them (because there are no random processes at all) then would you reverse your conclusion?

Is your basic objection to the application of probability theory when it concerns processes other than physically random processes?

If the physicists are not receiving random samples of the population of possible observations, then their inferences are also unjustified. And if random processes are impossible because the universe is deterministic . . . my head hurts, but I think raising that problem is changing the subject. I don't really want to talk about whether counter-factuals (like scientists proposing a different theory than the one actually proposed) are a coherent concept in a deterministic universe.

Is your basic objection to the application of probability theory when it concerns processes other than physically random processes?

That could be, but I'm not familiar with the technical vocabulary you are using. What's an example of a non-physical random process?

Maybe take a look at the Wikipedia entry http://en.wikipedia.org/wiki/Randomness

This discusses lots of different interpretations of "random". The general sense seems to be that a random process is unpredictable in detail, but has some predictable properties such that the process can be modelled mathematically by a random variable (or sequence of random variables).

Here, the notion of "modelling by a random variable" means that if we take the actual outcome and apply statistical tests to check whether the outcome is drawn from the distribution defined by the random variable, then the actual outcome passes those tests. This doesn't mean of course that it *is* in an objective sense a random process with that distribution, but it does mean that the model "fits".

Hope that helps...

P.S. For the avoidance of doubt, you can assume that models A and B involve pseudo-random processes, and these obey the usual frequency statistics of true random processes.

OK, let me ask you a question. Suppose that physicists have produced these two models of the universe.

Model A has a uniform background radiation temperature of 1K. Model B has a uniform background radiation temperature of 3K.

Both models are extremely large (infinite, if you prefer), so both models will contain some observers whose observations suggest a temperature of 3K, as well as observers whose observations suggest a temperature of 1K.

Our own observations suggest a temperature of 3K.

In your view, does that observation give us *any reason at all* to favour Model B over Model A as a description of the universe? If so, why? If not, how *can* we do science when some scientific models
imply a very large (or infinite) universe?