# The Doomsday Argument and Self-Sampling Assumption are wrong, but induction is alive and well.

Since the Doomsday Argument still is discussed often on _Less Wrong_, I would like to call attention to my new, short, self-published e-book, _The Longevity Argument_, which is a much-revised and much-expanded work that began with my paper, “[Past Longevity as Evidence for the Future](http://www.jstor.org/pss/10.1086/599273),” in the January 2009 issue of _[Philosophy of Science](http://www.jstor.org/action/showPublication?journalCode=philscie)_. In my judgment, my work provides a definitive refutation of the Doomsday Argument, identifying two elementary errors in the argument.

The first elementary error is that the Doomsday Argument conflates total duration and future duration. Although the Doomsday Argument’s Bayesian formalism is stated in terms of total duration, all attempted real-life applications of the argument—with one exception, a derivation by Gott (1994, 108) of his delta _t_ argument introduced in Gott 1993—actually plug in prior probabilities for future duration.

For example, Leslie (1996, 198–200) presents a Bayesian equation stated in terms of prior probabilities of total instances. But then Leslie (1996, 201–203) plugs into this equation prior probabilities for _future_ instances: humans being born for the next 150 years vs. humans being born for the next many thousands of centuries. Bostrom (2002, 94–96) recounts Leslie’s general argument in terms of births instead of durations of time, using 200 billion total births vs. 200 trillion total births. (A closer parallel to Leslie 1996 would be 80 billion total births vs. 80 trillion total births.) But the error persists: the actual prior probabilities that are plugged in to Leslie’s Bayesian equation, based on all of the real-life risks actually considered by Leslie (1996, 1–153) and Bostrom (2002, 95), are of future births, not total births.

In other words, Leslie supposes a prior probability of doom within the next 150 years or roughly 20 billion births. (The prior probabilities supposed in the Doomsday Argument are prior to knowledge of one’s birth rank.) Leslie then assumes that—since there have already been, say, 60 billion births—this prior probability is equal to the prior probability that the total number of births will have been 80 billion births. However, in the absence of knowledge of one’s birth rank, this assumption is absurd.

The second elementary error is the Doomsday Argument’s use of the Self-Sampling Assumption, which is contradicted by the prior information in all attempts at real-life applications in the literature.

For example, many risks to the human race—including most if not all the real-life risks discussed by Leslie and Bostrom—can reasonably be described mathematically as Poisson processes. Then the Self-Sampling Assumption implies that the risk per birth—the ‘lambda’ in the Poisson formula—is constant throughout the duration of the human race. But Leslie (1996, 202) also supposes that if mankind survives past the next century and a half, then the risk per birth will drop dramatically, because mankind will begin spreading throughout the galaxy. (The Doomsday Argument implicitly relies on such a drop in lambda—and the resultant bifurcation of risk into ‘doom soon’ and ‘doom very much later’—for the argument’s significant claims.) In other words, Leslie’s prior probabilities of doom are mathematical contradictions of the Self-Sampling Assumption that Leslie and Bostrom invoke in the Doomsday Argument.

In my book, I perform Bayesian analyses that correct these errors. These analyses demonstrate that gaining more knowledge of the past can indeed update one’s assessment of the future; but this updating is consistent with common sense instead of with the Doomsday Argument. In short, while refuting the Doomsday Argument, I vindicate induction.

 The price of my e-book is $4. However, professional scholars and educators are invited to email me to request a complimentary evaluation copy (not for further distribution, of course). I extend the same offer to the first ten _Less Wrong_ members with a Karma Score of 100 or greater who email me. (I may send to more than ten, or to some with lower Karma Scores, but I don’t want to make an open-ended commitment.)

For an abstract of the e-book, see [this entry on PhilPapers](http://philpapers.org/rec/PISTLA). For a non-technical introduction, see [here on my blog](http://ronpisaturo.com/blog/2011/06/29/longevity-argument/).

The e-book covers much more than the Doomsday Argument; here is a one-sentence summary: The Doomsday Argument, Self-Sampling Assumption, and Self-Indication Assumption are wrong; Gott’s delta _t_ argument (Gott 1993, 315–316; 1994) underestimates longevity, providing lower bounds on probabilities of longevity, and is equivalent to Laplace’s Rule of Succession (Laplace 1812, xii–xiii; \[1825\] 1995, 10–11); but Non-Parametric Predictive Inference based on the work of Hill (1968, 1988, 1993) and Coolen (1998, 2006) forms the basis of a calculus of induction.

References

Bostrom, Nick (2002), _Anthropic Bias: Observation Selection Effects in Science and Philosophy_. New York & London: Routledge.

Coolen, Frank P.A. (1998), “Low Structure Imprecise Predictive Inference For Bayes' Problem”, _Statistics & Probability Letters_ 36: 349–357.

——— (2006), On Probabilistic Safety Assessment in the Case of Zero Failures. _Journal of Risk and Reliability_ 220 (_Proceedings of the Institute of Mechanical Engineers_ O): 105–114.

Gott, J. Richard III (1993), “Implications of the Copernican Principle for our Future Prospects”, _Nature_ 363: 315–319.

 ——— (1994), “Future Prospects Discussed”, _Nature_ 368: 108.

Hill, Bruce M. (1968), “Posterior Distribution of Percentiles: Bayes' Theorem for Sampling from a Population”, _Journal of the American Statistical_  _Association_ 63: 677–691.

——— (1988), “De Finetti’s Theorem, Induction, and A(n) or Bayesian Nonparametric Predictive Inference”, _Bayesian Statistics_ 3, Edited by Bernardo J.M., DeGroot, M.H., Lindley, D.V. & Smith A.F.M. Oxford: Oxford University Press: 211–241.

——— (1993), “Parametric Models for An: Splitting Processes and Mixtures”, _Journal of the Royal Statistical Society_ B 55: 423–433.

Laplace, Pierre-Simon (1812), _Theorie Analytique des Probabilités_. Paris: Courcier.

——— (\[1825\] 1995), _Philosophical Essay on Probabilities_. Translated by Andrew I. Dale. Originally published as _Essai philosophique sur les probabilite´s_ (Paris: Bachelier). New York: Springer-Verlag.

Leslie, John (1996), _The End of the World: The Science and Ethics of Human_  _Extinction_. London: Routledge.

Here is and **Addendum** addressing the question by Manfred to elaborate on my statement, "the Self-Sampling Assumption implies that the risk per birth—the ‘lambda’ in the Poisson formula—is constant throughout the duration of the human race."

To avoid integrals, let me discuss a binomial process, which is a discrete version of a Poisson process.

Suppose you are studying a species from another planet. Suppose the only main risk to the species is an asteroid hitting the planet. Suppose the risk of an asteroid hit in a year is q. Given that the present moment is within a window (from the past through to the future) of N years without an asteroid hit, what is the probability P(Y) that the present moment is within year Y of that window?

P(Y) = \[q(1 – q)^Y^(1 – q)^N–Y^q\]/B, where B is the probability that the window is N years.

P(Y) = \[q^2^(1 – q)^N^\]/B.

Since Y does not appear in this formula, it is clear that P(Y) is constant for all Y. That is, since q is constant, P(Y) is uniform in \[1, N\], and P(Y) = 1/N. This result is equivalent to the Self-Sampling Assumption with units of time (years) as the reference class.

But suppose that the risk of an asteroid hit in the past was q, but the species has just built an asteroid destroyer, and the risk in the future is r where r << q. Then

P(Y) = \[q(1 – q)^Y^(1 – r)^N–Y^r\]/B.

\[8/16/2011: Corrected the final 'r' in the above equation from a 'q'.\] Y does appear in this formula. Clearly, the greater the value of Y, the smaller the value of P(Y). That is, contrary to the Self-Sampling Assumption, it is very likely that the present moment is in the early part of the window of N years.

The above argument demonstrates why the choice of ‘reference class’ matters. If the risk is constant per unit time, then the correct reference class is units of time. If the risk is constant per birth, then the correct reference class is births. Suppose birth rates increase exponentially. Then constant risk per unit time precludes constant risk per birth, and vice versa. The two reference classes cannot both be right. More generally, if the prior information stipulates that risk per birth is not constant, then the Self-Sampling Assumption using a reference class of births does not apply.

This passage is from my book (p. 59):

Here is a more philosophical and less mathematical perspective on the same point. SSA \[the Self-Sampling Assumption\] rests on the premise that all indexical information has been removed from the prior information. One's birth rank, which applies only to oneself, is such indexical information that is removed from the prior information before SSA is invoked. But even in the absence of birth rank, the prior information may—and usually does—include information that is indexical. For example, if the prior information states that _λ_~past~ is large and _λ_~future~ is small, then the prior information is stating something that is true only of the present—namely, that the present is when _λ_ changes abruptly from a large value to a small value. It turns out that this indexical information contradicts the mathematical conclusion of SSA. Moreover, this indexical information cannot be removed without consequence from the prior information, because the prior probabilities rest on it.

Perhaps the statement that Manfred quotes would have been clearer if I had instead written the following: The Self-Sampling Assumption implies that the risk per birth—the ‘lambda’ in the Poisson formula—is constant throughout the past and present.