Marginal Revolution: Has the Doomsday Argument been refuted?

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Has the Doomsday Argument been refuted?

Randomness seems to confound us. For example, we have a tendency to infer non-randomness from apparent patterns in random events (witness the incorrigible optimists who spot trends in the spins of a roulette wheel or the ups and downs of the FT Share Index); at the same time, the history of statistics suggests that, when random samples are required, we often mistake the merely haphazard - or whatever happens to be near at hand - for the truly random. As I will show, the Doomsday Argument's fundamental mistake is to rely on the intuitive but misguided notion that we can in general take ourselves to be typical humans, and thus, in effect, random samples of the species.

Here is the paper. If you are familiar with the core argument, scroll down to p.9 "We are necessarily alive..." for the beginning of the bottom line. I have never been persuaded by the Doomsday Argument, if only because it does not specify the appropriate reference class of self-observing agents. I now see further reason to be skeptical. Comments are open, and thanks again to the what-would-I-do-without-it www.politicaltheory.info for the pointer.

Posted by Tyler Cowen on November 10, 2005 at 05:01 AM in Science | Permalink

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» Doomsday and Bayes from Statistical Modeling, Causal Inference, and Social Science
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Comments

I'm a late-comer to this one -- I've known of the doomsday argument, but never took the time to really think about it. (It seemed another of those annoying ideas that inadvertantly leak out of philosophy from time to time to bother the rest of us, and in such cases a little mental rock-kicking usually suffices).

With that proviso, I think the reference class argument is right--righter than you suggest. This, it seems to me, is fundamentally flawed:

"Finally, corresponding to finding you are in cubicle number 7 we have the fact that you find that your birth rank is about 60 billion"

But it is simply not a 'fact' that my birth rank is 'about 60 billion'. That's because there was no FIRST human--any line drawn in evolutionary history is completely arbitrary. Let us consider an ancestor from our evolutionary lineage that lived, say, 10 million years ago, or 20 if you like. By definition, no catastrophic event occured to wipe out his species--his direct descendants still live (and in enormous numbers). But his 'species' is extinct nonetheless. So is his situation consistent with 'doom' or 'survival'? If we say, 'doom', then the exercise is pointless--our descendents will ultimately evolve into creatures distinct from ourselves, and 'doom' will come gradually and imperceptibly even without a catastrophy. But if we say 'survival' then we must include all the individuals in the populations of our evolutionary ancestors in our reference class. And then our reference population becomes all living things on earth, and the birth rank of a randomly selected individual is many, many orders of magnitude greater than 60 billion.

Posted by: Slocum at Nov 10, 2005 8:40:13 AM

The "doomsday argument" is actually a classical frequentist confidence interval. Averaging over all members of the group under consideration, 95% of these confidence intervals will contain the true value. Thus, if we go back and apply the doomsday argument to thousands of past data sets, its 95% intervals should indeed have 95% coverage. If you look carefully at classical statistical theory, you'll see that it makes claims about averages, not about particular cases.

However, this does not mean that there is a 95% chance that any particular interval will contain the true value. Especially not in this situation, where we have additional subject-matter knowledge. That's where Bayesian statistics (or, short of that, some humility about applying frequentist inferences to particular cases) comes in.

Posted by: Andrew Gelman at Nov 10, 2005 9:28:02 AM

In all Bayesian reasoning, the first thing to consider is what the prior distribution is. There's no reason to believe that the "lifetime" of the human species will be uniform over a particular range, for example. We could look at the distribution of the lifetime of species that have lasted at least as long as humanity, but that's iffy as it would have lots of gaps in it. And then there's the issue stated above -- how can we tell when a species begins or ends?

But ignoring evolutionary reality, and thinking that there's a distinct begin and end to a particular species, who's to say that the lifetimes don't follow an exponential distribution? The exponential distribution is rather cool as a time of failure distribution, as the expected time to failure is constant, no matter where you stand. Suppose a lightbulb follows this lifetime distribution, and it has an expected lifetime of 1000 hours. If it makes it to 1000 hours, and you look at what's the expected time to failure, it's still 1000 hours. Under an exponential distribution as the prior lifetime distribution, knowing how long the population has survived tells you absolutely nothing.

I'm hoping the person who originally made the doomsday argument isn't a statistician or mathematician, because that's embarrassing.

Posted by: meep at Nov 10, 2005 9:33:28 AM

Just to be clear: I think the doomsday argument is pretty silly (and also, it's not Bayesian). Although maybe it's a good thing that Bayesian inference has such high prestige now that it's being misapplied in silly ways. That's a true sign of acceptance of a scientific method.

Posted by: Andrew Gelman at Nov 10, 2005 9:39:51 AM

"Improbable" things happen all the time, and the perception of improbability depends on perspective. Suppose I hit a golf ball off the tee box. Presuming it lands in the fairway ( not an especially reasonable presumption, but that's another matter) the odds that it will land on a particular blade of grass are millions to one. But it's certain to land on one of them.

So is the "outcome" remarkable? Depends on the "outcome" you're talking about. If someone had predicted which blade of grass the ball would land on, that'd be remarkable. But the after the fact remarking upon a multimillion to one shot? Not so much.

I think a similar thing is going on here... a forcing of perspective to make people go "wow."

Posted by: bk at Nov 10, 2005 10:09:55 AM

I don't have time to read the entire "Remarks on the Doomsday Argument" article, but doesn't the factor of time make a complete hash of the argument?

With the urns example, if you were putting the balls one-by-one into the urn in numerical order, and you stopped after ten balls had been added and picked one, then its number would tell you absolutely nothing about which urn you were picking from. Right? So how does checking your birth number after 60 billion people have lived (surely that is too high?) tell you anything about how many people would ever live?

Seems to me you'd have to randomly sample your person from the entire history (now and the future) of the human race to match the urn example. And we simply cannot do that, thanks to time.

Posted by: Sol at Nov 10, 2005 10:26:01 AM

I don't think this article adds anything to the existing literature. If you care about this, look at Nick Bostrom's book on the topic. If you don't care, just ignore it. It makes a prediction, but it doesn't predict an observation that you can ever make.
The main practical significance of the argument is to remind us that we systematically underestimate the risk of events that have never occurred because we would not be there if they had occurred. We should expect zero such events in our past even if the events in question are likely.

Posted by: michael vassar at Nov 10, 2005 12:14:14 PM

I'm probably missing something but:
The DA is apparently deciding between two possibilities of the total number of humans that there will ever be. However, there are far more than two possibilities. So even if the probability that the most recent birth is the last birth is greater than the probability that the 1 trillionth birth will be the last birth, that does not imply that the last birth has greater probability than there being more than a trillion births. Given how high the maximum number of human births that there could be, it seems reasonable that the probability that the true total number of human births will fall somewhere in a very large interval, much higher than the current count, will be greater than the probability that we have had our last birth.

Given that the DA question has been considered by others, the probability that the resolution is this simple seems small :-).

-Kevin

Posted by: kevin Postlewaite at Nov 10, 2005 1:02:28 PM

Refuting this seems completely straightforward. There are two urns being filled with balls at equal rates. One will stop being filled with balls at 100 billion, the other will never stop being filled. Both are still being filled when I grab a ball an urn and sample the number.

No conclusion can be drawn whatsoever regarding which urn I drew from.

Posted by: Mark Nau at Nov 10, 2005 1:23:34 PM

Michael Vassar mentioned Nick Bostrom, so it's worthwhile to point out that arguments against Doomsday are also effective against his Simulation Argument.
http://www.mwilliams.info/archives/006304.php

Posted by: Michael Williams at Nov 10, 2005 8:16:11 PM

The publishing of the DA article in SciAm was a major factor in my decision to discontinue my subscribtion. So did the article claiming we would run out of oil in 50 years. (Shame they weren't in the same issue.)

I'm fighting a cold now, so I don't have the ability to write out all of my objections to the DA now.

My first objection (to both arguments) is that any prediction which is constant over time is likely to be worthless for many purposes. We've been running out of oil in 50 years for 100 years. The DA argument has said that 50% of "people" have already lived since the first "people" were around, and will continue to do so even (if and) when the last "person" is born. I don't feel compelled to take such arguments seriously. The marginal utility to me of sitting down and detailing exactly where the argument fails is quite low, which is why I've not bothered to do so.

But if the first article is correct, and influential people are being convinced, then perhaps my marginal utility is being driven up.

There was a dart tournament. A latecomer managed to conive his way onto the list. His first dart was 2 feet to the left of center. His second, 3 to the right. His third was 1 to the left. The DA proponents declared him the winner on the basis that on average, he was spot on! (not real)

There was a lot of excitement at a voice recognition conference reported in the http://www.theinquirer.net a couple of years ago. Effective voice recognition was thought to be one year away. One of the participants explained the excitement, "For the last 15 years, we've been 2 years away!" (real--and the humorous irony was clear)

I our manager reported that the bridge chip for a processor we were testing was going to be later than expected (again)--9 months later. I replied that a year and a half ago, it was expected in 18 months, and that 6 months ago, it was expected in 12. I then confidently predicted that it would be available in 18 more months. He was not happy with my "joke". I was right. (real)

Posted by: Nathan Zook at Nov 11, 2005 10:00:20 AM

Every branch of science and philosophy points to the finite nature of life. Why should we expect statistics to disagree?

Posted by: will at Nov 11, 2005 5:32:16 PM

Mark Nau nailed it.

Posted by: Tom at Nov 14, 2005 11:50:04 AM

The DA is right to some extent but it has some flaws. Think about it this way: There's a sequence of N identical balls and you get to pick a ball n at random. Then you are told the sequence number n. You can certainly make a prediction about N. Now, suppose a small percentage of balls are red, and the ball you pick happens to be red. If you know the sequence number in the set of red balls, you can make a prediction about the total number of red balls. The original calculation about the total number of balls is still valid, unless you have additional information, such as 'only the first 5% of all balls are red'. The argument still works if you don't know where red starts and ends, i.e. if there are shades of red. You can just pick an arbitrary boundary.

So the DA can be used in any reference class you can imagine:

- Living organisms.
- Vertebrates.
- Mammals.
- Primates.
- Humans.
- Humans with some understanding of probability theory.
- Humans who know about the DA.

The narrower classes probably give a more accurate picture, which is a bit worrisome. It seems likely that what we think of as 'modern humans' will cease to exist relatively soon (perhaps this century).

Refutation # 1. Let's consider all observers who will ever think of the DA independently and let us assume that they are uniformly distributed among all humans who will ever live (perhaps an incorrect assumption but this is unknown). Let us further assume there will be more than 100 of them. We can conclude that the first such observer was wrong about the DA.

Refutation # 2. The act of making the observation can affect the result. Suppose you are hired at random to be employee number 5 of a startup company. You then tell the CEO, with 95% confidence, that there will never be more than 100 employees at the company. The CEO then goes and hires 96 employees in order to prove you wrong. Similarly, humans could implement counter-measures upon learning about the DA.

Posted by: Jose at Jan 16, 2006 10:41:55 AM

>Refuting this seems completely straightforward.
>There are two urns being filled with balls at
>equal rates. One will stop being filled with
>balls at 100 billion, the other will never stop
>being filled. Both are still being filled when
>I grab a ball an urn and sample the number.
>No conclusion can be drawn whatsoever regarding
>which urn I drew from.

One flaw with this refutation is that assuming N can be infinite is unrealistic. If N is finite and you have two urns with different Ns, to be analogous to 'being born' in either population, you would need to observe a random ball n from all those dropped. The time at which you make the observation needs to be a random time in the range that makes sense for each urn.

So let's say one urn is being filled with 100 billion balls, and the other with 100 trillion balls. One ball is dropped every second. You arrive at one of the two urns at a random time to observe the ball being dropped. Let's say a computer chooses an urn at random, and also chooses at random which time you should observe the ball being dropped. Clearly, if you observe ball number 60 billion being dropped, you can make a good guess as to which of the 2 urns the computer picked.

Posted by: Jose at Jan 16, 2006 11:27:29 AM

Jose said:

It does not really matter whether the second urn is finite or infinite. Let's suppose the second urn stops being filled after 100 trillion balls. Observing a ball with number 60 billion does not give you any information about which urn you are. The chances of you being at urn number 1 or number 2 are still 1/2.

By the way, being born in a population of x people is not analogous to drawing a random number from the set 1 to x. I was not born that way, and neither did you :).

Jose said:

Again, the point is not that you arrive at random at the urn. The point is that you observe ball number 60 billion being drawn, and you have to decide what to believe. Are you in urn number one or urn number 2? (to make it easier, imagine you are ball number 60 billion. This is even more analogous to you being born).

It does not really matter what you believe.

If after seeing ball number 60 billion, you believe you are watching urn number 1 and you are at urn number 1, you will be correct. But if you are at urn number 2, you will be wrong. Both urns have the same probability of having been selected, so you have a 1/2 probability of being correct and 1/2 of being wrong.

Put it in another way. If after seeing ball number 60 billion, you believe you are watching urn number 2 and you are at urn number 2, you will be correct. Otherwise, you will be wrong. Therefore you have the same chances of being correct if you believe you are at urn number 1 or at urn number 2. Thus, it does not really matter what you believe after watching ball number 60 billion.

Which is as it should be!.

Check also: here

Cheers,

Ricardo.

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