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{{Short description|Doomsday scenario on human births}}
{{distinguish|text=the [[Heinz von Foerster#Doomsday equation|Doomsday equation]] or the [[Doomsday rule]]}}
{{Primary sources |date=August 2016}}

[[Image:Population curve.svg|thumb|350px|World population from 10,000 BC to AD 2000]]
The '''doomsday argument''' ('''DA'''), or '''Carter catastrophe''', is a [[probability theory|probabilistic argument]] that claims to predict the future population of the human species based on an estimation of the number of humans born to date. The doomsday argument was originally proposed by the [[Astrophysics|astrophysicist]] [[Brandon Carter]] in 1983,<ref>{{Cite journal |author = [[Brandon Carter]] |title = The anthropic principle and its implications for biological evolution |journal = [[Philosophical Transactions of the Royal Society of London]]
|volume = A310 |pages = 347–363 |year = 1983 |doi = 10.1098/rsta.1983.0096 |last2 = McCrea |first2 = W. H. |issue = 1512 |bibcode = 1983RSPTA.310..347C |s2cid = 92330878
 }}</ref> leading to the initial name of the Carter catastrophe. The argument was subsequently championed by the [[philosopher]] [[John A. Leslie]] and has since been independently conceived by [[J. Richard Gott]]<ref>{{Cite journal |author = J. Richard Gott, III |title = Implications of the Copernican principle for our future prospects |journal = [[Nature (journal)|Nature]] |volume = 363 |pages = 315–319 |year = 1993 |doi = 10.1038/363315a0 |issue = 6427 |bibcode = 1993Natur.363..315G |s2cid = 4252750
 }}</ref> and [[Holger Bech Nielsen]].<ref>{{Cite journal |author = [[Holger Bech Nielsen]] |title = Random dynamics and relations between the number of fermion generations and the fine structure constants
|journal = [[Acta Physica Polonica]] |volume = B20
|pages = 427–468 |year = 1989
}}</ref> Similar principles of [[eschatology]] were proposed earlier by [[Heinz von Foerster]], among others. A more general form was given earlier in the [[Lindy effect]],<ref>{{Cite web |url=http://www.amstat.org/meetings/jsm/2014/onlineprogram/AbstractDetails.cfm?abstractid=313738 |title=Predicting Future Lifespan: The Lindy Effect, Gott's Predictions and Caves' Corrections, and Confidence Intervals |year=2014 |url-status=dead |archiveurl=https://web.archive.org/web/20160328183642/http://www.amstat.org/meetings/jsm/2014/onlineprogram/AbstractDetails.cfm?abstractid=313738 |archivedate=2016-03-28 |first=Colman |last=Humphrey}}</ref> which proposes that for certain phenomena, the future life expectancy is proportional to (though not necessarily equal to) the current age and is based on a decreasing [[mortality rate]] over time.
==Summary==
The premise of the argument is as follows: suppose that the total number of human beings who will ever exist is fixed. If so, the likelihood of a randomly selected person existing at a particular time in history would be proportional to the total population at that time. Given this, the argument posits that a person alive today should adjust their expectations about the future of the human race because their existence provides information about the total number of humans that will ever live.

If the total number of humans who were born or will ever be born is denoted by <math display="inline">N</math>, then the [[Copernican principle]] suggests that any one human is equally likely (along with the other <math display="inline">N-1</math> humans) to find themselves in any position <math display="inline">n</math> of the total population <math display="inline">N</math> so humans assume that our fractional position <math display="inline">f=n/N</math> is uniformly distributed on the [[Interval (mathematics)|interval]] [0,1] before learning our absolute position.

<math display="inline">f</math> is uniformly distributed on (0,1) even after learning the absolute position <math display="inline">n</math>. For example, there is a 95% chance that <math display="inline">f</math> is in the interval (0.05,1), that is <math display="inline">f > 0.05</math>. In other words, one can assume with 95% certainty that any individual human would be within the last 95% of all the humans ever to be born. If the absolute position <math display="inline">n</math> is known, this argument implies a 95% confidence upper bound for <math display="inline">N</math> obtained by rearranging <math display="inline">n/N > 0.05</math> to give <math display="inline">N < 20n </math>.

If Leslie's figure<ref>{{Cite journal |last1=Oliver |first1=Jonathan |last2=Korb |first2=Kevin |year=1998 |title=A Bayesian Analysis of the Doomsday Argument |journal=Philosophy |citeseerx=10.1.1.49.5899}}</ref> is used, then approximately 60 billion humans have been born so far, so it can be estimated that there is a 95% chance that the total number of humans <math display="inline">N</math> will be less than 20<math display="inline">\times</math>60 billion = 1.2 trillion. Assuming that the world population stabilizes at 10 billion and a life expectancy of [[Longevity#Change over time|80 years]], it can be estimated that the remaining 1140 billion humans will be born in 9120 years. Depending on the projection of the world population in the forthcoming centuries, estimates may vary, but the argument states that it is unlikely that more than 1.2 trillion humans will ever live.

==Aspects==
Assume, for simplicity, that the total number of humans who will ever be born is 60 billion (''N''<sub>1</sub>), or 6,000 billion (''N''<sub>2</sub>).<ref>{{Cite journal |doi = 10.1093/mind/107.426.403|title = A refutation of the doomsday argument|journal = Mind|volume = 107|issue = 426|pages = 403–410|year = 1998|last1 = Korb|first1 = K.}}</ref> If there is no prior knowledge of the position that a currently living individual, ''X'', has in the history of humanity, one may instead compute how many humans were born before ''X'', and arrive at say 59,854,795,447, which would necessarily place ''X'' among the first 60 billion humans who have ever lived.

It is possible to sum the [[Probability|probabilities]] for each value of ''N'' and, therefore, to compute a statistical 'confidence limit' on ''N''. For example, taking the numbers above, it is 99% certain that ''N'' is smaller than 6 trillion.

Note that as remarked above, this argument assumes that the prior probability for ''N'' is flat, or 50% for ''N''<sub>1</sub> and 50% for ''N''<sub>2</sub> in the absence of any information about ''X''. On the other hand, it is possible to conclude, given ''X'', that ''N''<sub>2</sub> is more likely than ''N''<sub>1</sub> if a different prior is used for ''N''. More precisely, Bayes' theorem tells us that P(''N''|''X'') = P(''X''|''N'')P(''N'')/P(''X''), and the conservative application of the Copernican principle tells us only how to calculate P(''X''|''N''). Taking P(''X'') to be flat, we still have to assume the prior probability P(''N'') that the total number of humans is ''N''. If we conclude that ''N''<sub>2</sub> is much more likely than ''N''<sub>1</sub> (for example, because producing a larger population takes more time, increasing the chance that a low probability but cataclysmic natural event will take place in that time), then P(''X''|''N'') can become more heavily weighted towards the bigger value of ''N''. A further, more detailed discussion, as well as relevant distributions P(''N''), are given below in the [[#Rebuttals|Rebuttals]] section.

The doomsday argument does ''not'' say that humanity cannot or will not exist indefinitely. It does not put any upper limit on the number of humans that will ever exist nor provide a date for when humanity will become [[extinct]]. An abbreviated form of the argument ''does'' make these claims, by confusing probability with certainty. However, the actual conclusion for the version used above is that there is a 95% ''chance'' of extinction within 9,120 years and a 5% chance that some humans will still be alive at the end of that period. (The precise numbers vary among specific doomsday arguments.)

==Variations==
This argument has generated a philosophical debate, and no consensus has yet emerged on its solution. The variants described below produce the DA by separate derivations.

===Heinz von Foerster's prediction of humanity's disappearance on 13 November 2026===
A 1960 issue of ''[[Science (magazine)|Science]]'' magazine included an article by [[Heinz von Foerster]] and his colleagues, P. M. Mora and L. W. Amiot, proposing an equation representing the best fit to the historical data on the Earth's population available in 1958:
<blockquote>
Fifty years ago, ''Science'' published a study with the provocative title “[https://www.researchgate.net/publication/233822850_Doomsday_friday_13_November_AD_2026 Doomsday: Friday, 13 November, A.D. 2026]”. It fitted world population during the previous two millennia with ''P'' = 179 × 10<sup>9</sup>/(2026.9 − ''t'')<sup>0.99</sup>. This “quasi-hyperbolic” equation (hyperbolic having exponent 1.00 in the denominator) projected to infinite population in 2026—and to an imaginary one thereafter.
:—Taagepera, Rein. [https://www.sciencedirect.com/science/article/abs/pii/S0040162513001613 A world population growth model: Interaction with Earth's carrying capacity and technology in limited space] ''Technological Forecasting and Social Change'', vol. 82, February 2014, pp. 34–41
</blockquote>
[[File:World population since 10,000 BCE (OurWorldInData series), OWID.svg|thumb|The global population is equal to <math>\tfrac{179000000000}{2026.9 - t}</math> and [[Hyperbolic growth|hyperbolically grows]] as ''t'' approaches 2026.9. When ''t'' surpasses 2026.9, the number of people living on the planet Earth suddenly becomes negative—on 13 November 2026 AD, all humans instantaneously disappear.]]
In 1975, [[Sebastian von Hoerner|von Hoerner]] suggested that von Foerster's doomsday equation can be written, without a significant loss of accuracy, in a simplified [[Hyperbolic growth|hyperbolic]] form (''i.e.'' with the exponent in the denominator assumed to be 1.00):
:<math>\text{Global population}=\frac{179000000000}{2026.9 - t},</math>
where
* 2026.9 is 13 November 2026 AD—the date of the so-called "demographic singularity"<ref>Korotayev, Andrey. [https://jbh.journals.villanova.edu/article/view/2329/2251 The 21st Century Singularity and its Big History Implications: A re-analysis] ''Journal of Big History'', II(3), June 2018, pp. 73–119</ref> and von Foerster's 115th anniversary;
* ''t'' is the number of a year of the [[Gregorian calendar]].<ref>Korotayev, Andrey. [https://jbh.journals.villanova.edu/article/view/2329/2251 The 21st Century Singularity and its Big History Implications: A re-analysis] ''Journal of Big History'', II(3), June 2018, pp. 73–119. "We have already mentioned that, as was the case with equations (8) and (9) above, in von Foerster’s Eq. (13) the denominator’s exponent (0.99) turns out to be only negligibly different from 1, and as was already suggested by von Hoerner (1975) and Kapitza (1992, 1999), it can be written more succinctly as ''N<sub>t</sub>'' = ''C''/(''t<sup>*</sup>'' − ''t'')."</ref>

Despite its simplicity, von Foerster's equation is very accurate in the range from 4,000,000 BP<ref>Korotayev, Andrey. [https://jbh.journals.villanova.edu/article/view/2329/2251 The 21st Century Singularity and its Big History Implications: A re-analysis] ''Journal of Big History'', II(3), June 2018, pp. 73–119. "Note that von Foerster and his colleagues detected the hyperbolic pattern of world population growth for 1 CE –1958 CE; later it was shown that this pattern continued for a few years after 1958, and also that it can be traced for many millennia BCE (Kapitza 1996a, 1996b, 1999; Kremer 1993; Tsirel 2004; Podlazov 2000, 2001, 2002; Korotayev, Malkov, Khaltourina 2006a, 2006b). In fact Kremer (1993) claims that this pattern is traced since 1,000,000 BP, whereas Kapitza (1996a, 1996b, 2003, 2006, 2010) even insists that it can be found since 4,000,000 BP."</ref> to 1997 AD. For example, the doomsday equation (developed in 1958, when the Earth's population was 2,911,249,671<ref name=Worldometer>[https://www.worldometers.info/world-population/world-population-by-year/ World Population by Year] Worldometer</ref>) predicts a population of 5,986,622,074 for the beginning of the year 1997:
:<math>\frac{179000000000}{2026.9 - 1997}=5986622074.</math>

The actual figure was 5,924,787,816.<ref name=Worldometer/>

The doomsday equation is called so because it predicts that the number of people living on the planet Earth will become maximally ''positive'' by 13 November 2026, and on the next moment will become ''negative''.  Said otherwise, the equation predicts that on 13 November 2026 all humans will instantaneously disappear.

===Gott's formulation: "vague prior" total population===
Gott specifically proposes the functional form for the [[Prior probability|prior distribution]] of the number of people who will ever be born (''N''). Gott's DA used the [[Prior probability#Uninformative priors|vague prior distribution]]:
:<math>P(N) = \frac{k}{N}</math>.
where
* P(N) is the probability prior to discovering ''n'', the total number of humans who have ''yet'' been born.
* The constant, ''k'', is chosen to [[Normalizing constant|normalize]] the sum of P(''N''). The value chosen is not important here, just the functional form (this is an [[improper prior]], so no value of ''k'' gives a valid distribution, but [[Bayesian inference]] is still possible using it.)

Since Gott specifies the [[Prior probability|prior]] distribution of total humans, ''P(N)'', [[Bayes's theorem|Bayes' theorem]] and the [[principle of indifference]] alone give us ''P(N|n)'', the probability of ''N'' humans being born if ''n'' is a random draw from ''N'':

:<math>P(N\mid n) = \frac{P(n\mid N) P(N)}{P(n)}.</math>

This is [[Bayes' theorem]] for the [[posterior probability]] of the total population ever born of ''N'', [[conditioning (probability)|conditioned]] on population born thus far of ''n''. Now, using the indifference principle:

:<math>P(n\mid N) = \frac{1}{N}</math>.

The unconditioned ''n'' distribution of the current population is identical to the vague prior ''N'' probability density function,<ref group=note>The only [[probability density function]]s that must be specified ''[[A priori and a posteriori|a priori]]'' are:
* Pr(''N'') - the ultimate number of people that will be born, assumed by J. Richard Gott to have a vague prior distribution, Pr(''N'') = ''k''/''N''
* Pr(''n''|''N'') - the chance of being born in any position based on a total population ''N'' - all DA forms assume the [[Copernican principle]], making Pr(''n''|''N'') = 1/''N''

From these two distributions, the doomsday argument proceeds to create a Bayesian inference on the distribution of ''N'' from ''n'', through [[Bayes' theorem#For continuous random variables|Bayes' rule]], which requires P(''n''); to produce this, integrate over all the possible values of ''N'' which might contain an individual born ''n''th (that is, wherever ''N'' > ''n''):

:<math> P(n) = \int_{N=n}^{N=\infty} P(n\mid N) P(N) \,dN = \int_{n}^{\infty}\frac{k}{N^2} \,dN </math> <math>= \frac{k}{n}.</math>

This is why the marginal distribution of n and N are identical in the case of P(''N'') = ''k''/''N''
</ref> so:

:<math>P(n) = \frac{k}{n}</math>,

giving P (''N'' | ''n'') for each specific ''N'' (through a substitution into the posterior probability equation):

:<math>P(N\mid n) = \frac{n}{N^2}</math>.

The easiest way to produce the doomsday estimate with a given [[Confidence interval|confidence]] (say 95%) is to pretend that ''N'' is a [[Continuous random variable|continuous variable]] (since it is very large) and [[Integral|integrate]] over the probability density from ''N'' = ''n'' to ''N'' = ''Z''. (This will give a function for the probability that ''N'' ≤ ''Z''):

:<math>P(N \leq Z) = \int_{N=n}^{N=Z} P(N|n)\,dN</math> <math> = \frac{Z-n}{Z}</math>

Defining ''Z'' = 20''n'' gives:

:<math>P(N \leq 20n) = \frac{19}{20}</math>.

This is the simplest [[Bayes factor|Bayesian]] derivation of the doomsday argument:
:The chance that the total number of humans that will ever be born (''N'') is greater than twenty times the total that have been is below 5%

The use of a [[Prior probability#Uninformative priors|vague prior]] distribution seems well-motivated as it assumes as little knowledge as possible about ''N'', given that some particular function must be chosen. It is equivalent to the assumption that the probability density of one's fractional position remains uniformly distributed even after learning of one's absolute position (''n'').

Gott's "reference class" in his original 1993 paper was not the number of births, but the number of years "humans" had existed as a species, which he put [[Human evolution#H. sapiens|at 200,000]]. Also, Gott tried to give a 95% confidence interval between a ''minimum'' survival time and a maximum. Because of the 2.5% chance that he gives to underestimating the minimum, he has only a 2.5% chance of overestimating the maximum. This equates to 97.5% confidence that extinction occurs before the upper boundary of his confidence interval, which can be used in the integral above with ''Z'' = 40''n'', and ''n'' = 200,000 years:

:<math>P(N \leq 40[200000]) = \frac{39}{40}</math>

This is how Gott produces a 97.5% confidence of extinction within ''N'' ≤ 8,000,000 years. The number he quoted was the likely time remaining, ''N''&nbsp;−&nbsp;''n'' = 7.8 million years. This was much higher than the temporal confidence bound produced by counting births, because it applied the principle of indifference to time. (Producing different estimates by sampling different parameters in the same hypothesis is [[Bertrand's paradox (probability)|Bertrand's paradox]].) Similarly, there is a 97.5% chance that the present lies in the first 97.5% of human history, so there is a 97.5% chance that the total lifespan of humanity will be at least

:<math>N \geq 200000 \times \frac{40}{39} \approx 205100~\text{years}</math>;

In other words, Gott's argument gives a 95% confidence that humans will go extinct between 5,100 and 7.8 million years in the future.

Gott has also tested this formulation against the [[Berlin Wall]] and [[Broadway theatre|Broadway]] and off-Broadway plays.<ref>{{cite magazine |url=http://www.newyorker.com/archive/1999/07/12/1999_07_12_035_TNY_LIBRY_000018591 |title=How to Predict Everything|magazine=The New Yorker| author=Timothy Ferris |date=July 12, 1999 |access-date=September 3, 2010}}</ref>

Leslie's argument differs from Gott's version in that he does not assume a'' vague prior'' probability distribution for ''N''. Instead, he argues that the force of the doomsday argument resides purely in the increased probability of an early doomsday once you take into account your birth position, regardless of your prior probability distribution for ''N''. He calls this the ''probability shift''.

==Reference classes==
The [[reference class problem|reference class]] from which ''n'' is drawn, and of which ''N'' is the ultimate size, is a crucial point of contention in the doomsday argument. The "standard" doomsday argument [[hypothesis]] skips over this point entirely, merely stating that the reference class is the number of "people". Given that you are human, the Copernican principle might be used to determine if you were born exceptionally early, however the term "human" has been heavily contested on [[Anthropology|practical]] and [[Philosophy|philosophical]] reasons. According to [[Nick Bostrom]], [[consciousness]] is (part of) the discriminator between what is in and what is out of the reference class, and therefore [[extraterrestrial intelligence]] might have a significant impact on the calculation.{{Citation needed|date=May 2023}}

The following sub-sections relate to different suggested reference classes, each of which has had the standard doomsday argument applied to it.

===SSSA: Sampling from observer-moments===
[[Nick Bostrom]], [[Anthropic principle#Character of anthropic reasoning|considering observation selection effects]], has produced a [[Self-Sampling Assumption]] (SSA): "that you should think of yourself as if you were a random observer from a suitable reference class". If the "reference class" is the set of humans to ever be born, this gives ''N'' < 20''n'' with 95% confidence (the standard doomsday argument). However, he has [[Anthropic bias|refined]] this idea to apply to ''observer-moments'' rather than just observers. He has formalized this as:<ref>{{Cite web |last=Bostrom |first=Nick |date=2005 |title=Self-Location and Observation Selection Theory |url=https://anthropic-principle.com/preprints/self-location |access-date=2023-07-02 |website=anthropic-principle.com}}</ref>

:The strong self-sampling assumption (SSSA): Each observer-moment should reason as if it were randomly selected from the class of all observer-moments in its reference class.

An application of the principle underlying SSSA (though this application is nowhere expressly articulated by Bostrom), is: If the minute in which you read this article is randomly selected from every minute in every human's lifespan, then (with 95% confidence) this event has occurred after the first 5% of human observer-moments. If the mean lifespan in the future is twice the historic mean lifespan, this implies 95% confidence that ''N'' < 10''n'' (the average future human will account for twice the observer-moments of the average historic human). Therefore, the 95th percentile extinction-time estimate in this version is 4560 years.

== Counterarguments ==
{{Tone|date=November 2010|section}}

===We are in the earliest 5%, ''a priori''===
One counterargument to the doomsday argument agrees with its statistical methods but disagrees with its extinction-time estimate. This position requires justifying why the observer cannot be assumed to be randomly selected from the set of all humans ever to be born, which implies that this set is not an appropriate reference class. By disagreeing with the doomsday argument, it implies that the observer is within the first 5% of humans to be born.

By analogy, if one is a member of 50,000 people in a collaborative project, the reasoning of the doomsday argument implies that there will never be more than a million members of that project, within a 95% confidence interval. However, if one's characteristics are typical of an [[early adopter]], rather than typical of an average member over the project's lifespan, then it may not be reasonable to assume one has joined the project at a random point in its life. For instance, the mainstream of potential users will prefer to be involved when the project is nearly complete. However, if one were to enjoy the project's incompleteness, it is already known that he or she is unusual, before the discovery of his or her early involvement.

If one has measurable attributes that set one apart from the typical long-run user, the project doomsday argument can be refuted based on the fact that one could expect to be within the first 5% of members, ''a priori''. The analogy to the total-human-population form of the argument is that confidence in a prediction of the [[probability distribution|distribution]] of human characteristics that places modern and historic humans outside the mainstream implies that it is already known, before examining ''n'', that it is likely to be very early in ''N''. This is an argument for changing the reference class.

For example, if one is certain that 99% of humans who will ever live will be [[cyborg]]s, but that only a negligible fraction of humans who have been born to date are cyborgs, one could be equally certain that at least one hundred times as many people remain to be born as have been.

[[Robin Hanson]]'s paper sums up these criticisms of the doomsday argument:<ref name=":0">{{Cite web |title=Critiquing the Doomsday Argument |url=http://mason.gmu.edu/~rhanson/Nodoom.html |access-date=2023-06-17 |website=mason.gmu.edu}}</ref>
{{blockquote|All else is not equal; we have good reasons for thinking we are not randomly selected humans from all who will ever live.}}

===Human extinction is distant, ''a posteriori''===
The [[a posteriori]] observation that [[extinction level event]]s are rare could be offered as evidence that the doomsday argument's predictions are implausible; typically, [[extinction]]s of dominant [[species]] happen less often than once in a million years. Therefore, it is argued that [[human extinction]] is unlikely within the next ten millennia. (Another [[probability theory|probabilistic argument]], drawing a different conclusion than the doomsday argument.)

In Bayesian terms, this response to the doomsday argument says that our knowledge of history (or ability to prevent disaster) produces a prior marginal for ''N'' with a minimum value in the trillions. If ''N'' is distributed uniformly from 10<sup>12</sup> to 10<sup>13</sup>, for example, then the probability of ''N'' < 1,200 billion inferred from ''n'' = 60 billion will be extremely small. This is an equally impeccable Bayesian calculation, rejecting the [[Copernican principle]] because we must be 'special observers' since there is no likely mechanism for humanity to go extinct within the next hundred thousand years.

This response is accused of overlooking the [[Human extinction#Natural vs. anthropogenic|technological threats to humanity's survival]], to which earlier life was not subject, and is specifically rejected by most{{By whom|date=April 2023|section=Critique:_Human_extinction_is_distant,_a_posteriori}} academic critics of the doomsday argument (arguably excepting [[Robin Hanson]]).

===The prior ''N'' distribution may make ''n'' very uninformative===
[[Robin Hanson]] argues that ''N''{{'s}} prior may be [[exponential distribution|exponentially distributed]]:<ref name=":0" />
:<math>N = \frac{e^{U(0, q]}}{c}</math>

Here, ''c'' and'' q'' are constants. If ''q'' is large, then our 95% confidence upper bound is on the uniform draw, not the exponential value of ''N''.

The simplest way to compare this with Gott's Bayesian argument is to flatten the distribution from the vague prior by having the probability fall off more slowly with ''N'' (than inverse proportionally). This corresponds to the idea that humanity's growth may be exponential in time with doomsday having a vague prior [[probability density function]] in ''time''. This would mean that ''N'', the last birth, would have a distribution looking like the following:

:<math>\Pr(N) = \frac{k}{N^\alpha}, 0 < \alpha < 1.
</math>

This prior ''N'' distribution is all that is required (with the principle of indifference) to produce the inference of ''N'' from ''n'', and this is done in an identical way to the standard case, as described by Gott (equivalent to <math>\alpha</math> = 1 in this distribution):

:<math> \Pr(n) = \int_{N=n}^{N=\infty} \Pr(n\mid N) \Pr(N) \,dN = \int_{n}^{\infty} \frac{k}{N^{(\alpha+1)}} \,dN = \frac{k}{{\alpha}n^{\alpha}}</math>

Substituting into the posterior probability equation):

:<math>\Pr(N\mid n) = \frac{{\alpha}n^{\alpha}}{N^{(1+\alpha)}}.</math>

Integrating the probability of any ''N'' above ''xn'':

:<math>\Pr(N > xn) = \int_{N=xn}^{N=\infty} \Pr(N\mid n)\,dN = \frac{1}{x^{\alpha}}.</math>

For example, if ''x'' = 20, and <math>\alpha</math> = 0.5, this becomes:

:<math>\Pr(N > 20n) = \frac{1}{\sqrt{20}} \simeq 22.3\%. </math>

Therefore, with this prior, the chance of a trillion births is well over 20%, rather than the 5% chance given by the standard DA. If <math>\alpha</math> is reduced further by assuming a flatter prior ''N'' distribution, then the limits on'' N'' given by ''n'' become weaker. An <math>\alpha</math> of one reproduces Gott's calculation with a birth reference class, and <math>\alpha</math> around 0.5 could approximate his temporal confidence interval calculation (if the population were expanding exponentially). As <math>\alpha \to 0</math> (gets smaller) ''n'' becomes less and less [[uninformative prior|informative]] about ''N''. In the limit this distribution approaches an (unbounded) [[uniform distribution (continuous)|uniform distribution]], where all values of ''N'' are equally likely. This is Page et al.'s "Assumption 3", which they find few reasons to reject, ''a priori''. (Although all distributions with <math>\alpha \leq 1</math> are improper priors, this applies to Gott's vague-prior distribution also, and they can all be converted to produce [[improper integral|proper integrals]] by postulating a finite upper population limit.) Since the probability of reaching a population of size 2''N'' is usually thought of as the chance of reaching ''N'' multiplied by the survival probability from ''N'' to 2''N'' it follows that Pr(''N'') must be a [[monotonic function|monotonically]] decreasing function of ''N'', but this doesn't necessarily require an inverse proportionality.<ref name=":0" />

===Infinite expectation===
Another objection to the doomsday argument is that the [[Expected value|expected]] total human population is actually [[Infinity|infinite]].<ref name=":1">{{Cite web |last1=Monton |first1=Bradley |last2=Roush |first2=Sherri |date=2001-11-20 |title=Gott's Doomsday Argument |url=http://philsci-archive.pitt.edu/1205/ |access-date=2023-06-17 |website=philsci-archive.pitt.edu |language=en}}</ref> The calculation is as follows:

:The total human population <var>N</var> = <var>n</var>/<var>f</var>, where <var>n</var> is the human population to date and <var>f</var> is our fractional position in the total.
:We assume that <var>f</var> is uniformly distributed on <nowiki>(0,1]</nowiki>.
: The expectation of <var>N</var> is <math display="block"> E(N) = \int_{0}^{1} {n \over f} \, df = n [\ln (f) ]_{0}^{1}= n \ln (1) - n \ln (0) = + \infty .</math>

For a similar example of counterintuitive infinite expectations, see the [[St. Petersburg paradox]].

===Self-indication assumption: The possibility of not existing at all===
{{main article|Self-indication assumption doomsday argument rebuttal}}
One objection is that the possibility of a human existing at all depends on how many humans will ever exist (''N''). If this is a high number, then the possibility of their existing is higher than if only a few humans will ever exist. Since they do indeed exist, this is evidence that the number of humans that will ever exist is high.<ref>{{Cite journal |last=Olum |first=Ken D. |year=2002 |title=The doomsday argument and the number of possible observers |journal= The Philosophical Quarterly|volume=52 |issue=207 |page=164 |doi=10.1111/1467-9213.00260 |arxiv=gr-qc/0009081 |s2cid=14707647 }}</ref>

This objection, originally by [[Dennis Dieks]] (1992),<ref>{{Cite web |last=Dieks |first=Dennis |date=2005-01-13 |title=Reasoning About the Future: Doom and Beauty. |url=https://philsci-archive.pitt.edu/2144/ |access-date=2023-06-17 |website=philsci-archive.pitt.edu |language=en}}</ref> is now known by [[Nick Bostrom]]'s name for it: the "[[Self-Indication Assumption]] objection". It can be shown that some [[Self-Indication Assumption|SIAs]] prevent any inference of ''N'' from ''n'' (the current population).<ref>{{Cite book |last=Bostrom |first=Nick |title=Anthropic Bias: Observational Selection Effects in Science and Philosophy |publisher=Routledge |year=2002 |isbn=0-415-93858-9 |location=New York & London |pages=124–126}}</ref>

===Caves' rebuttal===
The [[Bayesian inference|Bayesian]] argument by [[Carlton M. Caves]] states that the uniform distribution assumption is incompatible with the [[Copernican principle]], not a consequence of it.<ref>{{cite arXiv |last=Caves |first=Carlton M. |date=2008 |title=Predicting future duration from present age: Revisiting a critical assessment of Gott's rule |class=astro-ph |eprint=0806.3538 }}</ref>

Caves gives a number of examples to argue that Gott's rule is implausible. For instance, he says, imagine stumbling into a birthday party, about which you know nothing:

<blockquote>Your friendly enquiry about the age of the celebrant elicits the reply that she is celebrating her (''t''<sub>''p''</sub>=) 50th birthday. According to Gott, you can predict with 95% confidence that the woman will survive between [50]/39 = 1.28 years and 39[×50] = 1,950 years into the future. Since the wide range encompasses reasonable expectations regarding the woman's survival, it might not seem so bad, till one realizes that [Gott's rule] predicts that with probability 1/2 the woman will survive beyond 100 years old and with probability 1/3 beyond 150. Few of us would want to bet on the woman's survival using Gott's rule. ''(See Caves' online paper [[#External links|below]].)''</blockquote>

Although this example exposes a weakness in [[J. Richard Gott]]'s "Copernicus method" DA (that he does not specify when the "Copernicus method" can be applied) it is not precisely analogous with the [[#Numerical estimates of the doomsday argument|modern DA]]{{Clarify|date=September 2023 |reason=This link doesn't have a destination.}}; [[Epistemology|epistemological]] refinements of Gott's argument by [[philosopher]]s such as [[Nick Bostrom]] specify that:
: Knowing the absolute birth rank (''n'') must give no information on the total population (''N'').

Careful DA variants specified with this rule aren't shown implausible by Caves' "Old Lady" example above, because the woman's age is given prior to the estimate of her lifespan. Since human age gives an estimate of survival time (via [[actuary|actuarial]] tables) Caves' Birthday party age-estimate could not fall into the class of DA problems defined with this proviso.

To produce a comparable "Birthday Party Example" of the carefully specified Bayesian DA, we would need to completely exclude all prior knowledge of likely human life spans; in principle this could be done (e.g.: hypothetical Amnesia chamber). However, this would remove the modified example from everyday experience. To keep it in the everyday realm the lady's age must be ''hidden'' prior to the survival estimate being made. (Although this is no longer exactly the DA, it is much more comparable to it.)

Without knowing the lady's age, the DA reasoning produces a ''rule'' to convert the birthday (''n'') into a maximum lifespan with 50% confidence (''N''). Gott's [[Copernicus principle|Copernicus method]] rule is simply: Prob (''N'' < 2''n'') = 50%. How accurate would this estimate turn out to be? Western [[demographics]] are now fairly [[uniform]] across ages, so a random birthday (''n'') could be (very roughly) approximated by a U(0,''M''<nowiki>]</nowiki> draw where ''M'' is the maximum lifespan in the census. In this 'flat' model, everyone shares the same lifespan so ''N'' = ''M''. If ''n'' happens to be less than (''M'')/2 then Gott's 2''n'' estimate of ''N'' will be under ''M'', its true figure. The other half of the time 2''n'' underestimates ''M'', and in this case (the one Caves highlights in his example) the subject will die before the 2''n'' estimate is reached. In this "flat demographics" model Gott's 50% confidence figure is proven right 50% of the time.

===Self-referencing doomsday argument rebuttal===
{{Main article|Self-referencing doomsday argument rebuttal}}
Some philosophers have suggested that only people who have contemplated the doomsday argument (DA) belong in the reference class "[[human]]". If that is the appropriate reference class, [[Brandon Carter|Carter]] defied his own prediction when he first described the argument (to the [[Royal Society]]). An attendant could have argued thus:

<blockquote>Presently, only one person in the world understands the Doomsday argument, so by its own logic there is a 95% chance that it is a minor problem which will only ever interest twenty people, and I should ignore it.</blockquote>

Jeff Dewynne and Professor Peter Landsberg suggested that this line of reasoning will create a [[paradox]] for the doomsday argument:<ref name=":1" />

If a member of the Royal Society did pass such a comment, it would indicate that they understood the DA sufficiently well that in fact 2 people could be considered to understand it, and thus there would be a 5% chance that 40 or more people would actually be interested. Also, of course, ignoring something because you only expect a small number of people to be interested in it is extremely short sighted—if this approach were to be taken, nothing new would ever be explored, if we assume no ''a priori'' knowledge of the nature of interest and attentional mechanisms.

===Conflation of future duration with total duration===
Various authors have argued that the doomsday argument rests on an incorrect conflation of future duration with total duration. This occurs in the specification of the two time periods as "doom soon" and "doom deferred" which means that both periods are selected to occur ''after'' the observed value of the birth order. A rebuttal in Pisaturo (2009)<ref>{{Cite journal
 | author = Ronald Pisaturo
 | title = Past Longevity as Evidence for the Future
 | journal = [[Philosophy of Science (journal)|Philosophy of Science]]
 | volume = 76
 | pages = 73–100
 | year = 2009
 | doi = 10.1086/599273
| s2cid = 122207511
 }}</ref> argues that the doomsday argument relies on the equivalent of this equation:

:<math> P(H_{TS}|D_pX)/P(H_{TL}|D_pX) = [P(H_{FS}|X)/P(H_{FL}|X)] \cdot [P(D_p|H_{TS}X)/P(D_p|H_{TL}X)] </math>,
:where:
:''X'' = the prior information;
:''D<sub>p</sub>'' = the data that past duration is ''t<sub>p</sub>'';
:''H<sub>FS</sub>'' = the hypothesis that the future duration of the phenomenon will be short;
:''H<sub>FL</sub>'' = the hypothesis that the future duration of the phenomenon will be long;
:''H<sub>TS</sub>'' = the hypothesis that the ''total'' duration of the phenomenon will be short—i.e., that ''t<sub>t</sub>'', the phenomenon's ''total'' longevity, = ''t<sub>TS</sub>'';
: ''H<sub>TL</sub>'' = the hypothesis that the ''total'' duration of the phenomenon will be long—i.e., that ''t<sub>t</sub>'', the phenomenon's ''total'' longevity, = ''t<sub>TL</sub>'', with ''t<sub>TL</sub>'' > ''t<sub>TS</sub>''.
Pisaturo then observes:

:Clearly, this is an invalid application of Bayes' theorem, as it conflates future duration and total duration.

Pisaturo takes numerical examples based on two possible corrections to this equation: considering only future durations and considering only total durations. In both cases, he concludes that the doomsday argument's claim, that there is a "Bayesian shift" in favor of the shorter future duration, is fallacious.

This argument is also echoed in O'Neill (2014).<ref>{{Cite journal
 | author = Ben O'Neill
 | title = Assessing the 'Bayesian Shift' in the Doomsday Argument
 | journal = [[Journal of Philosophy]]
 | volume = 111 | issue = 4
 | pages = 198–218
 | year = 2014 | doi = 10.5840/jphil2014111412
 }}</ref>  In this work O'Neill argues that a unidirectional "Bayesian Shift" is an impossibility within the standard formulation of probability theory and is contradictory to the rules of probability.  As with Pisaturo, he argues that the doomsday argument conflates future duration with total duration by specification of doom times that occur after the observed birth order. According to O'Neill:

:The reason for the hostility to the doomsday argument and its assertion of a "Bayesian shift" is that many people who are familiar with probability theory are implicitly aware of the absurdity of the claim that one can have an automatic unidirectional shift in beliefs regardless of the actual outcome that is observed. This is an example of the "reasoning to a foregone conclusion" that arises in certain kinds of failures of an underlying inferential mechanism. An examination of the inference problem used in the argument shows that this suspicion is indeed correct, and the doomsday argument is invalid.  (pp. 216-217)

===Confusion over the meaning of confidence intervals===
Gelman and Robert<ref>{{Cite journal
 | author1 = Andrew Gelman
 | author2 = Christian P. Robert
 | title = 'Not Only Defended But Also Applied': The Perceived Absurdity of Bayesian Inference
 | journal = [[The American Statistician]]
 | volume = 67
 | issue = 4
 | year = 2013
 | pages = 1–5
 | doi = 10.1080/00031305.2013.760987
 | arxiv = 1006.5366
 | s2cid = 10833752
 }}</ref> assert that the doomsday argument confuses frequentist [[confidence intervals]] with Bayesian [[credible intervals]].  Suppose that every individual knows their number ''n'' and uses it to estimate an upper bound on ''N''.  Every individual has a different estimate, and these estimates are constructed so that 95% of them contain the true value of ''N'' and the other 5% do not. This, say Gelman and Robert, is the defining property of a frequentist lower-tailed 95% confidence interval. But, they say, "this does not mean that there is a 95% chance that any particular interval will contain the true value." That is, while 95% of the confidence intervals will contain the true value of ''N'', this is not the same as ''N'' being contained in the confidence interval with 95% probability. The latter is a different property and is the defining characteristic of a Bayesian credible interval. Gelman and Robert conclude:

{{Blockquote|text=the Doomsday argument is the ultimate triumph of the idea, beloved among Bayesian educators, that our students and clients do not really understand Neyman–Pearson confidence intervals and inevitably give them the intuitive Bayesian interpretation.}}

==See also==
* [[Anthropic principle]]
* [[Human overpopulation]]
*[[German tank problem]]
* [[Global catastrophic risk]]
* [[Doomsday event]]
* [[Fermi paradox]]
* [[Measure problem (cosmology)]]
* [[Mediocrity principle]]
* [[Quantum suicide and immortality]]
* [[Simulated reality]]
* [[Survival analysis]]
* [[Survivalism]]
* [[Technological singularity]]

==Notes==
{{Reflist|group=note}}

==References==
{{Reflist}}

==Further reading==
* [[John A. Leslie]], ''The End of the World: The Science and Ethics of Human Extinction'', Routledge, 1998, {{ISBN|0-41518447-9}}.
* [[John Richard Gott III|J. R. Gott III]], ''Future Prospects Discussed'', Nature, vol. 368, p.&nbsp;108, 1994.
* This argument plays a central role in [[Stephen Baxter (author)|Stephen Baxter]]'s science fiction book, ''[[Manifold: Time]]'', Del Rey Books, 2000, {{ISBN|0-345-43076-X}}.
* The same principle plays a major role in the [[Dan Brown]] novel, ''[[Inferno (Brown novel)|Inferno]]'', Corgy Books, {{ISBN|978-0-552-16959-2}}
* [[William Poundstone|Poundstone, William]], ''The Doomsday Calculation: How an Equation that Predicts the Future Is Transforming Everything We Know About Life and the Universe''. 2019 Little, Brown Spark. [https://books.google.com/books?id=Q55yDwAAQBAJ Description] & [https://books.google.com/books?id=Q55yDwAAQBAJ arrow/scrollable preview.] Also summarised in Poundstone's essay, [https://www.wsj.com/articles/doomsday-math-says-humanity-may-have-just-760-years-left-11561655839 "Math Says Humanity May Have Just 760 Years Left"]. ''The Wall Street Journal'', updated June 27, 2019. {{ISBN|9783164440707}}

==External links==
{{External links|date=November 2023}}
* [http://philpapers.org/browse/doomsday-argument The Doomsday argument category on PhilPapers]
* [http://flatrock.org.nz/topics/environment/doom_soon.htm A non-mathematical, unpartisan introduction to the DA]
* [http://www.anthropic-principle.com/preprints/ali/alive.html Nick Bostrom's response to Korb and Oliver]
* [http://www.anthropic-principle.com/preprints.html#doomsday Nick Bostrom's annotated collection of references]
* [https://arxiv.org/abs/gr-qc/9407002 Kopf, Krtouš & Page's early (1994) refutation] based on the [[Self-Indication Assumption|SIA]], which they called "Assumption 2".
* [https://arxiv.org/abs/gr-qc/0009081 The Doomsday argument and the number of possible observers by Ken Olum] In 1993 [[J. Richard Gott]] used his "Copernicus method" to predict the lifetime of Broadway shows. One part of this paper uses the same reference class as an empirical counter-example to Gott's method.
* [https://web.archive.org/web/20040217141525/http://hanson.gmu.edu/nodoom.html A Critique of the Doomsday Argument by Robin Hanson]
* [http://cogprints.org/7044/ A Third Route to the Doomsday Argument by Paul Franceschi], ''Journal of Philosophical Research'', 2009, vol. 34, pp.&nbsp;263–278
* [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=82931 Chambers' Ussherian Corollary Objection]
* [https://web.archive.org/web/20041205094143/http://info.phys.unm.edu/papers/2000/Caves2000a.pdf Caves' Bayesian critique of Gott's argument. C. M. Caves, "Predicting future duration from present age: A critical assessment", Contemporary Physics 41, 143-153 (2000).]
* [https://arxiv.org/abs/0806.3538 C.M. Caves, "Predicting future duration from present age: Revisiting a critical assessment of Gott's rule.]
* [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2400044 "Infinitely Long Afterlives and the Doomsday Argument" by John Leslie] shows that Leslie has recently modified his analysis and conclusion (Philosophy 83 (4) 2008 pp.&nbsp;519–524): Abstract—A recent book of mine defends three distinct varieties of immortality. One of them is an infinitely lengthy afterlife; however, any hopes of it might seem destroyed by something like Brandon Carter's 'doomsday argument' against viewing ourselves as extremely early humans. The apparent difficulty might be overcome in two ways. First, if the world is non-deterministic then anything on the lines of the doomsday argument may prove unable to deliver a strongly pessimistic conclusion. Secondly, anything on those lines may break down when an infinite sequence of experiences is in question.
* [https://www.lrb.co.uk/the-paper/v21/n13/mark-greenberg/apocalypse-not-just-now Mark Greenberg, "Apocalypse Not Just Now" in London Review of Books]
* [http://pthbb.org/manual/services/grim/laster.html Laster]: A simple webpage applet giving the min & max survival times of anything with 50% and 95% confidence requiring only that you input how old it is. It is designed to use the same mathematics as [[J. Richard Gott]]'s form of the DA, and was programmed by [[sustainable development]] researcher Jerrad Pierce.
* [https://www.youtube.com/watch?v=dSvgw9ZOK3I PBS Space Time The Doomsday Argument]

{{Doomsday}}

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[[Category:Probabilistic arguments]]
[[Category:1983 introductions]]
[[Category:Doomsday scenarios|*]]