{"history":[],"article":{"id":"a4350d7e-040b-443c-8b20-c5e7b3fcbacd","title":"Doomsday argument","slug":"doomsday-argument","content":":::summary\n:::\n\n# \nThe **doomsday argument** (**DA**), or **Carter catastrophe**, is a [probabilistic argument](https://bliptext.com/articles/probability-theory) that claims to predict the future population of the human species based on an estimation of the number of humans born to date.\nThe doomsday argument was originally proposed by the [astrophysicist](https://bliptext.com/articles/astrophysics) [Brandon Carter](https://bliptext.com/articles/brandon-carter) in 1983, leading to the initial name of the Carter catastrophe.\nThe argument was subsequently championed by the [philosopher](https://bliptext.com/articles/philosopher) [John A. Leslie](https://bliptext.com/articles/john-a-leslie) and has since been independently conceived by [J. Richard Gott](https://bliptext.com/articles/j-richard-gott) and [Holger Bech Nielsen](https://bliptext.com/articles/holger-bech-nielsen).\nSimilar principles of [eschatology](https://bliptext.com/articles/eschatology) were proposed earlier by [Heinz von Foerster](https://bliptext.com/articles/heinz-von-foerster), among others.\nA more general form was given earlier in the [Lindy effect](https://bliptext.com/articles/lindy-effect), 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](https://bliptext.com/articles/mortality-rate) over time.\n# [World population from 10,000 BC to AD 2000 | upload.wikimedia.org/wikipedia/commons/b/b7/Population///curve.svg]\n\n# Summary\nThe premise of the argument is as follows: suppose that the total number of human beings that will ever exist is fixed.\nIf 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.\nGiven 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.\nIf the total number of humans who were born or will ever be born is denoted by N, then the [Copernican principle](https://bliptext.com/articles/copernican-principle) suggests that any one human is equally likely (along with the other N-1 humans) to find themselves in any position n of the total population Nso humans assume that our fractional position f=n/N is uniformly distributed on the [interval](https://bliptext.com/articles/interval-mathematics) [0,1] before learning our absolute position.\nf is uniformly distributed on (0,1) even after learning the absolute position n. For example, there is a 95% chance that f is in the interval (0.05,1), that is f > 0.05.\nIn 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.\nIf the absolute position n is known, this argument implies a 95% confidence upper bound for N obtained by rearranging n/N > 0.05 to give N < 20n.\nIf Leslie's figure 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 N will be less than 20\\times60 billion = 1.2 trillion.\nAssuming that the world population stabilizes at 10 billion and a life expectancy of [80 years](https://bliptext.com/articles/longevity), it can be estimated that the remaining 1140 billion humans will be born in 9120 years.\nDepending 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.\n\n# Aspects\nAssume, for simplicity, that the total number of humans who will ever be born is 60 billion (N1), or 6,000 billion (N2).\nIf 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.\nIt is possible to sum the [probabilities](https://bliptext.com/articles/probability) for each value of N and, therefore, to compute a statistical 'confidence limit' on N.\nFor example, taking the numbers above, it is 99% certain that N is smaller than 6 trillion.\nNote that as remarked above, this argument assumes that the prior probability for N is flat, or 50% for N1 and 50% for N2 in the absence of any information about X.\nOn the other hand, it is possible to conclude, given X, that N2 is more likely than N1 if a different prior is used for N.\nMore 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).\nTaking P(X) to be flat, we still have to assume the prior probability P(N) that the total number of humans is N.\nIf we conclude that N2 is much more likely than N1 (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.\nA further, more detailed discussion, as well as relevant distributions P(N), are given below in the [Rebuttals](https://bliptext.com/articles/) section.\nThe doomsday argument does not say that humanity cannot or will not exist indefinitely.\nIt 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](https://bliptext.com/articles/extinct).\nAn abbreviated form of the argument does make these claims, by confusing probability with certainty.\nHowever, 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.\n(The precise numbers vary among specific doomsday arguments.)\n\n# Variations\nThis argument has generated a philosophical debate, and no consensus has yet emerged on its solution.\nThe variants described below produce the DA by separate derivations.\n\n## Gott's formulation: \"vague prior\" total population\nGott specifically proposes the functional form for the [prior distribution](https://bliptext.com/articles/prior-probability) of the number of people who will ever be born (N).\nGott's DA used the [vague prior distribution](https://bliptext.com/articles/prior-probability):\nwhere\nSince Gott specifies the [prior](https://bliptext.com/articles/prior-probability) distribution of total humans, P(N), [Bayes' theorem](https://bliptext.com/articles/bayes-s-theorem) and the [principle of indifference](https://bliptext.com/articles/principle-of-indifference) alone give us P(N|n), the probability of N humans being born if n is a random draw from N:\nThis is [Bayes' theorem](https://bliptext.com/articles/bayes-theorem) for the [posterior probability](https://bliptext.com/articles/posterior-probability) of the total population ever born of N, [conditioned](https://bliptext.com/articles/conditioning-probability) on population born thus far of n.\nNow, using the indifference principle:\nThe unconditioned n distribution of the current population is identical to the vague prior N probability density function, so:\ngiving P (N | n) for each specific N (through a substitution into the posterior probability equation):\nThe easiest way to produce the doomsday estimate with a given [confidence](https://bliptext.com/articles/confidence-interval) (say 95%) is to pretend that N is a [continuous variable](https://bliptext.com/articles/continuous-random-variable) (since it is very large) and [integrate](https://bliptext.com/articles/integral) over the probability density from N = n to N = Z.\n(This will give a function for the probability that N ≤ Z):\nDefining Z = 20n gives:\nThis is the simplest [Bayesian](https://bliptext.com/articles/bayes-factor) derivation of the doomsday argument:\nThe use of a [vague prior](https://bliptext.com/articles/prior-probability) distribution seems well-motivated as it assumes as little knowledge as possible about N, given that some particular function must be chosen.\nIt 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).\nGott'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 [at 200,000](https://bliptext.com/articles/human-evolution).\nAlso, Gott tried to give a 95% confidence interval between a minimum survival time and a maximum.\nBecause of the 2.5% chance that he gives to underestimating the minimum, he has only a 2.5% chance of overestimating the maximum.\nThis 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 = 40n, and n = 200,000 years:\nThis is how Gott produces a 97.5% confidence of extinction within N ≤ 8,000,000 years.\nThe number he quoted was the likely time remaining, N − n = 7.8 million years.\nThis was much higher than the temporal confidence bound produced by counting births, because it applied the principle of indifference to time.\n(Producing different estimates by sampling different parameters in the same hypothesis is [Bertrand's paradox](https://bliptext.com/articles/bertrand-s-paradox-probability).) 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\nIn 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.\nGott has also tested this formulation against the [Berlin Wall](https://bliptext.com/articles/berlin-wall) and [Broadway](https://bliptext.com/articles/broadway-theatre) and off-[Broadway](https://bliptext.com/articles/broadway-theatre) plays.\nLeslie's argument differs from Gott's version in that he does not assume a vague prior probability distribution for N.\nInstead, 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.\nHe calls this the probability shift.\n[Heinz von Foerster](https://bliptext.com/articles/heinz-von-foerster) argued that humanity's abilities to construct societies, civilizations and technologies do not result in self-inhibition.\nRather, societies' success varies directly with population size.\nVon Foerster found that this model fits some 25 data points from the birth of [Jesus](https://bliptext.com/articles/jesus) to 1958, with only 7% of the [variance](https://bliptext.com/articles/variance) left unexplained.\nSeveral follow-up letters (1961, 1962, ...) were published in Science showing that von Foerster's equation was still on track.\nThe data continued to fit up until 1973.\nThe most remarkable thing about von Foerster's model was it predicted that the human population would reach infinity or a mathematical singularity, on Friday, November 13, 2026.\nIn fact, von Foerster did not imply that the world population on that day could actually become infinite.\nThe real implication was that the world population growth pattern followed for many centuries prior to 1960 was about to come to an end and be transformed into a radically different pattern.\nNote that this prediction began to be fulfilled just in a few years after the \"doomsday\" argument was published.\n\n# Reference classes\nThe [reference class](https://bliptext.com/articles/reference-class-problem) from which n is drawn, and of which N is the ultimate size, is a crucial point of contention in the doomsday argument argument.\nThe \"standard\" doomsday argument [hypothesis](https://bliptext.com/articles/hypothesis) skips over this point entirely, merely stating that the reference class is the number of \"people\".\nGiven 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 [practical](https://bliptext.com/articles/anthropology) and [philosophical](https://bliptext.com/articles/philosophy) reasons.\nAccording to [Nick Bostrom](https://bliptext.com/articles/nick-bostrom), [consciousness](https://bliptext.com/articles/consciousness) is (part of) the discriminator between what is in and what is out of the reference class, and therefore [extraterrestrial intelligence](https://bliptext.com/articles/extraterrestrial-intelligence) might have a significant impact on the calculation.\nThe following sub-sections relate to different suggested reference classes, each of which has had the standard doomsday argument applied to it.\n\n## SSSA: Sampling from observer-moments\n[Nick Bostrom](https://bliptext.com/articles/nick-bostrom), [considering observation selection effects](https://bliptext.com/articles/anthropic-principle), has produced a [Self-Sampling Assumption](https://bliptext.com/articles/self-sampling-assumption) (SSA): \"that you should think of yourself as if you were a random observer from a suitable reference class\".\nIf the \"reference class\" is the set of humans to ever be born, this gives N < 20n with 95% confidence (the standard doomsday argument).\nHowever, he has [refined](https://bliptext.com/articles/anthropic-bias) this idea to apply to observer-moments rather than just observers.\nHe has formalized this as:\nAn 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.\nIf the mean lifespan in the future is twice the historic mean lifespan, this implies 95% confidence that N < 10n (the average future human will account for twice the observer-moments of the average historic human).\nTherefore, the 95th percentile extinction-time estimate in this version is 4560 years.\n\n# Rebuttals\n\n## We are in the earliest 5%, a priori\nOne counterargument to the doomsday argument agrees with its statistical methods but disagrees with its extinction-time estimate.\nThis 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.\nBy disagreeing with the doomsday argument, it implies that the observer is within the first 5% of humans to be born.\nBy 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.\nHowever, if one's characteristics are typical of an [early adopter](https://bliptext.com/articles/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.\nFor instance, the mainstream of potential users will prefer to be involved when the project is nearly complete.\nHowever, 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.\nIf 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.\nThe analogy to the total-human-population form of the argument is that confidence in a prediction of the [distribution](https://bliptext.com/articles/probability-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.\nThis is an argument for changing the reference class.\nFor example, if one is certain that 99% of humans who will ever live will be [cyborgs](https://bliptext.com/articles/cyborg), but that only a negligible fraction of humans who have been born to date are [cyborgs](https://bliptext.com/articles/cyborg), one could be equally certain that at least one hundred times as many people remain to be born as have been.\n[Robin Hanson](https://bliptext.com/articles/robin-hanson)'s paper sums up these criticisms of the doomsday argument:\n\"All else is not equal; we have good reasons for thinking we are not randomly selected humans from all who will ever live.\"\n\n## Human extinction is distant, a posteriori\nThe [a posteriori](https://bliptext.com/articles/a-posteriori) observation that [extinction level events](https://bliptext.com/articles/extinction-level-event) are rare could be offered as evidence that the doomsday argument's predictions are implausible; typically, [extinctions](https://bliptext.com/articles/extinction) of dominant [species](https://bliptext.com/articles/species) happen less often than once in a million years.\nTherefore, it is argued that [human extinction](https://bliptext.com/articles/human-extinction) is unlikely within the next ten millennia.\n(Another [probabilistic argument](https://bliptext.com/articles/probability-theory), drawing a different conclusion than the doomsday argument.)\nIn 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.\nIf N is distributed uniformly from 1012 to 1013, for example, then the probability of N < 1,200 billion inferred from n = 60 billion will be extremely small.\nThis is an equally impeccable Bayesian calculation, rejecting the [Copernican principle](https://bliptext.com/articles/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.\nThis response is accused of overlooking the [technological threats to humanity's survival](https://bliptext.com/articles/human-extinction), to which earlier life was not subject, and is specifically rejected by most academic critics of the doomsday argument (arguably excepting [Robin Hanson](https://bliptext.com/articles/robin-hanson)).\n\n## The prior N distribution may make n very uninformative\n[Robin Hanson](https://bliptext.com/articles/robin-hanson) argues that N prior may be [exponentially distributed](https://bliptext.com/articles/exponential-distribution):\nHere, c and q are constants.\nIf q is large, then our 95% confidence upper bound is on the uniform draw, not the exponential value of N.\nThe 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).\nThis corresponds to the idea that humanity's growth may be exponential in time with doomsday having a vague prior [probability density function](https://bliptext.com/articles/probability-density-function) in time.\nThis would mean that N, the last birth, would have a distribution looking like the following:\nThis 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 \\alpha = 1 in this distribution):\nSubstituting into the posterior probability equation):\nIntegrating the probability of any N above xn:\nFor example, if x = 20, and \\alpha = 0.5, this becomes:\nTherefore, with this prior, the chance of a trillion births is well over 20%, rather than the 5% chance given by the standard DA.\nIf \\alpha is reduced further by assuming a flatter prior N distribution, then the limits on N given by n become weaker.\nAn \\alpha of one reproduces Gott's calculation with a birth reference class, and \\alpha around 0.5 could approximate his temporal confidence interval calculation (if the population were expanding exponentially).\nAs (gets smaller) n becomes less and less [informative](https://bliptext.com/articles/uninformative-prior) about N.\nIn the limit this distribution approaches an (unbounded) [uniform distribution](https://bliptext.com/articles/uniform-distribution-continuous), where all values of N are equally likely.\nThis is Page et al.'s \"Assumption 3\", which they find few reasons to reject, a priori.\n(Although all distributions with are improper priors, this applies to Gott's vague-prior distribution also, and they can all be converted to produce [proper integrals](https://bliptext.com/articles/improper-integral) by postulating a finite upper population limit.) Since the probability of reaching a population of size 2N is usually thought of as the chance of reaching N multiplied by the survival probability from N to 2N it follows that Pr(N) must be a [monotonically](https://bliptext.com/articles/monotonic-function) decreasing function of N, but this doesn't necessarily require an inverse proportionality.\n\n## Infinite expectation\nAnother objection to the doomsday argument is that the [expected](https://bliptext.com/articles/expected-value) total human population is actually [infinite](https://bliptext.com/articles/infinity).\nThe calculation is as follows:\nFor a similar example of counterintuitive infinite expectations, see the [St. Petersburg paradox](https://bliptext.com/articles/st-petersburg-paradox).\n\n## Self-indication assumption: The possibility of not existing at all\nOne objection is that the possibility of a human existing at all depends on how many humans will ever exist (N).\nIf this is a high number, then the possibility of their existing is higher than if only a few humans will ever exist.\nSince they do indeed exist, this is evidence that the number of humans that will ever exist is high.\nThis objection, originally by [Dennis Dieks](https://bliptext.com/articles/dennis-dieks) (1992), is now known by [Nick Bostrom](https://bliptext.com/articles/nick-bostrom)'s name for it: the \"[Self-Indication Assumption](https://bliptext.com/articles/self-indication-assumption) objection\".\nIt can be shown that some [SIAs](https://bliptext.com/articles/self-indication-assumption) prevent any inference of N from n (the current population).\n\n## Caves' rebuttal\nThe [Bayesian](https://bliptext.com/articles/bayesian-inference) argument by [Carlton M. Caves](https://bliptext.com/articles/carlton-m-caves) states that the uniform distribution assumption is incompatible with the [Copernican principle](https://bliptext.com/articles/copernican-principle), not a consequence of it.\nCaves gives a number of examples to argue that Gott's rule is implausible.\nFor instance, he says, imagine stumbling into a birthday party, about which you know nothing:\n\"Your friendly enquiry about the age of the celebrant elicits the reply that she is celebrating her (tp=) 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 below.)\"\nAlthough this example exposes a weakness in [J. Richard Gott](https://bliptext.com/articles/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 [modern DA](https://bliptext.com/articles/); [epistemological](https://bliptext.com/articles/epistemology) refinements of Gott's argument by [philosophers](https://bliptext.com/articles/philosopher) such as [Nick Bostrom](https://bliptext.com/articles/nick-bostrom) specify that:\nCareful 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.\nSince human age gives an estimate of survival time (via [actuarial](https://bliptext.com/articles/actuary) tables) Caves' Birthday party age-estimate could not fall into the class of DA problems defined with this proviso.\nTo 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).\nHowever, this would remove the modified example from everyday experience.\nTo keep it in the everyday realm the lady's age must be hidden prior to the survival estimate being made.\n(Although this is no longer exactly the DA, it is much more comparable to it.)\nWithout knowing the lady's age, the DA reasoning produces a rule to convert the birthday (n) into a maximum lifespan with 50% confidence (N).\nGott's [Copernicus method](https://bliptext.com/articles/copernicus-principle) rule is simply: Prob (N < 2n) = 50%.\nHow accurate would this estimate turn out to be?\nWestern [demographics](https://bliptext.com/articles/demographics) are now fairly [uniform](https://bliptext.com/articles/uniform) across ages, so a random birthday (n) could be (very roughly) approximated by a U(0,M ] 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 2n estimate of N will be under M, its true figure. The other half of the time 2n underestimates M, and in this case (the one Caves highlights in his example) the subject will die before the 2n estimate is reached. In this \"flat [demographics](https://bliptext.com/articles/demographics)\" model Gott's 50% confidence figure is proven right 50% of the time.\n\n## Self-referencing doomsday argument rebuttal\nSome philosophers have suggested that only people who have contemplated the doomsday argument (DA) belong in the reference class \"[human](https://bliptext.com/articles/human)\".\nIf that is the appropriate reference class, [Carter](https://bliptext.com/articles/brandon-carter) defied his own prediction when he first described the argument (to the [Royal Society](https://bliptext.com/articles/royal-society)).\nAn attendant could have argued thus:\n\"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.\"\nJeff Dewynne and Professor Peter Landsberg suggested that this line of reasoning will create a [paradox](https://bliptext.com/articles/paradox) for the doomsday argument:\nIf 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.\nAlso, 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.\n\n## Conflation of future duration with total duration\nVarious authors have argued that the doomsday argument rests on an incorrect conflation of future duration with total duration.\nThis 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.\nA rebuttal in Pisaturo (2009) argues that the doomsday argument relies on the equivalent of this equation:\nPisaturo then observes:\nPisaturo takes numerical examples based on two possible corrections to this equation: considering only future durations and considering only total durations.\nIn 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.\nThis argument is also echoed in O'Neill (2014).\nIn 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.\nAs 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.\nAccording to O'Neill:\n\n## Confusion over the meaning of confidence intervals\nGelman and Robert assert that the doomsday argument confuses frequentist [confidence intervals](https://bliptext.com/articles/confidence-intervals) with Bayesian [credible intervals](https://bliptext.com/articles/credible-intervals).\nSuppose that every individual knows their number n and uses it to estimate an upper bound on N.\nEvery 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.\nThis, say Gelman and Robert, is the defining property of a frequentist lower-tailed 95% confidence interval.\nBut, they say, \"this does not mean that there is a 95% chance that any particular interval will contain the true value.\"\nThat 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.\nThe latter is a different property and is the defining characteristic of a Bayesian credible interval.\nGelman and Robert conclude:\n\"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.\"","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T20:03:27.832Z","updated_at":"2025-01-25T20:03:27.832Z","search_vector":"'argument':2 'doomsday':1","search_content":" \n \n\n# \nThe **doomsday argument** (**DA**), or **Carter catastrophe**, is a [probabilistic argument](https://bliptext.com/articles/probability-theory) that claims to predict the future population of the human species based on an estimation of the number of humans born to date.\nThe doomsday argument was originally proposed by the [astrophysicist](https://bliptext.com/articles/astrophysics) [Brandon Carter](https://bliptext.com/articles/brandon-carter) in 1983, leading to the initial name of the Carter catastrophe.\nThe argument was subsequently championed by the [philosopher](https://bliptext.com/articles/philosopher) [John A. Leslie](https://bliptext.com/articles/john-a-leslie) and has since been independently conceived by [J. Richard Gott](https://bliptext.com/articles/j-richard-gott) and [Holger Bech Nielsen](https://bliptext.com/articles/holger-bech-nielsen).\nSimilar principles of [eschatology](https://bliptext.com/articles/eschatology) were proposed earlier by [Heinz von Foerster](https://bliptext.com/articles/heinz-von-foerster), among others.\nA more general form was given earlier in the [Lindy effect](https://bliptext.com/articles/lindy-effect), 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](https://bliptext.com/articles/mortality-rate) over time.\n# [World population from 10,000 BC to AD 2000 | upload.wikimedia.org/wikipedia/commons/b/b7/Population///curve.svg]\n\n# Summary\nThe premise of the argument is as follows: suppose that the total number of human beings that will ever exist is fixed.\nIf 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.\nGiven 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.\nIf the total number of humans who were born or will ever be born is denoted by N, then the [Copernican principle](https://bliptext.com/articles/copernican-principle) suggests that any one human is equally likely (along with the other N-1 humans) to find themselves in any position n of the total population Nso humans assume that our fractional position f=n/N is uniformly distributed on the [interval](https://bliptext.com/articles/interval-mathematics) [0,1] before learning our absolute position.\nf is uniformly distributed on (0,1) even after learning the absolute position n. For example, there is a 95% chance that f is in the interval (0.05,1), that is f > 0.05.\nIn 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.\nIf the absolute position n is known, this argument implies a 95% confidence upper bound for N obtained by rearranging n/N > 0.05 to give N < 20n.\nIf Leslie's figure 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 N will be less than 20\\times60 billion = 1.2 trillion.\nAssuming that the world population stabilizes at 10 billion and a life expectancy of [80 years](https://bliptext.com/articles/longevity), it can be estimated that the remaining 1140 billion humans will be born in 9120 years.\nDepending 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.\n\n# Aspects\nAssume, for simplicity, that the total number of humans who will ever be born is 60 billion (N1), or 6,000 billion (N2).\nIf 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.\nIt is possible to sum the [probabilities](https://bliptext.com/articles/probability) for each value of N and, therefore, to compute a statistical 'confidence limit' on N.\nFor example, taking the numbers above, it is 99% certain that N is smaller than 6 trillion.\nNote that as remarked above, this argument assumes that the prior probability for N is flat, or 50% for N1 and 50% for N2 in the absence of any information about X.\nOn the other hand, it is possible to conclude, given X, that N2 is more likely than N1 if a different prior is used for N.\nMore 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).\nTaking P(X) to be flat, we still have to assume the prior probability P(N) that the total number of humans is N.\nIf we conclude that N2 is much more likely than N1 (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.\nA further, more detailed discussion, as well as relevant distributions P(N), are given below in the [Rebuttals](https://bliptext.com/articles/) section.\nThe doomsday argument does not say that humanity cannot or will not exist indefinitely.\nIt 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](https://bliptext.com/articles/extinct).\nAn abbreviated form of the argument does make these claims, by confusing probability with certainty.\nHowever, 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.\n(The precise numbers vary among specific doomsday arguments.)\n\n# Variations\nThis argument has generated a philosophical debate, and no consensus has yet emerged on its solution.\nThe variants described below produce the DA by separate derivations.\n\n## Gott's formulation: \"vague prior\" total population\nGott specifically proposes the functional form for the [prior distribution](https://bliptext.com/articles/prior-probability) of the number of people who will ever be born (N).\nGott's DA used the [vague prior distribution](https://bliptext.com/articles/prior-probability):\nwhere\nSince Gott specifies the [prior](https://bliptext.com/articles/prior-probability) distribution of total humans, P(N), [Bayes' theorem](https://bliptext.com/articles/bayes-s-theorem) and the [principle of indifference](https://bliptext.com/articles/principle-of-indifference) alone give us P(N|n), the probability of N humans being born if n is a random draw from N:\nThis is [Bayes' theorem](https://bliptext.com/articles/bayes-theorem) for the [posterior probability](https://bliptext.com/articles/posterior-probability) of the total population ever born of N, [conditioned](https://bliptext.com/articles/conditioning-probability) on population born thus far of n.\nNow, using the indifference principle:\nThe unconditioned n distribution of the current population is identical to the vague prior N probability density function, so:\ngiving P (N | n) for each specific N (through a substitution into the posterior probability equation):\nThe easiest way to produce the doomsday estimate with a given [confidence](https://bliptext.com/articles/confidence-interval) (say 95%) is to pretend that N is a [continuous variable](https://bliptext.com/articles/continuous-random-variable) (since it is very large) and [integrate](https://bliptext.com/articles/integral) over the probability density from N = n to N = Z.\n(This will give a function for the probability that N ≤ Z):\nDefining Z = 20n gives:\nThis is the simplest [Bayesian](https://bliptext.com/articles/bayes-factor) derivation of the doomsday argument:\nThe use of a [vague prior](https://bliptext.com/articles/prior-probability) distribution seems well-motivated as it assumes as little knowledge as possible about N, given that some particular function must be chosen.\nIt 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).\nGott'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 [at 200,000](https://bliptext.com/articles/human-evolution).\nAlso, Gott tried to give a 95% confidence interval between a minimum survival time and a maximum.\nBecause of the 2.5% chance that he gives to underestimating the minimum, he has only a 2.5% chance of overestimating the maximum.\nThis 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 = 40n, and n = 200,000 years:\nThis is how Gott produces a 97.5% confidence of extinction within N ≤ 8,000,000 years.\nThe number he quoted was the likely time remaining, N − n = 7.8 million years.\nThis was much higher than the temporal confidence bound produced by counting births, because it applied the principle of indifference to time.\n(Producing different estimates by sampling different parameters in the same hypothesis is [Bertrand's paradox](https://bliptext.com/articles/bertrand-s-paradox-probability).) 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\nIn 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.\nGott has also tested this formulation against the [Berlin Wall](https://bliptext.com/articles/berlin-wall) and [Broadway](https://bliptext.com/articles/broadway-theatre) and off-[Broadway](https://bliptext.com/articles/broadway-theatre) plays.\nLeslie's argument differs from Gott's version in that he does not assume a vague prior probability distribution for N.\nInstead, 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.\nHe calls this the probability shift.\n[Heinz von Foerster](https://bliptext.com/articles/heinz-von-foerster) argued that humanity's abilities to construct societies, civilizations and technologies do not result in self-inhibition.\nRather, societies' success varies directly with population size.\nVon Foerster found that this model fits some 25 data points from the birth of [Jesus](https://bliptext.com/articles/jesus) to 1958, with only 7% of the [variance](https://bliptext.com/articles/variance) left unexplained.\nSeveral follow-up letters (1961, 1962, ...) were published in Science showing that von Foerster's equation was still on track.\nThe data continued to fit up until 1973.\nThe most remarkable thing about von Foerster's model was it predicted that the human population would reach infinity or a mathematical singularity, on Friday, November 13, 2026.\nIn fact, von Foerster did not imply that the world population on that day could actually become infinite.\nThe real implication was that the world population growth pattern followed for many centuries prior to 1960 was about to come to an end and be transformed into a radically different pattern.\nNote that this prediction began to be fulfilled just in a few years after the \"doomsday\" argument was published.\n\n# Reference classes\nThe [reference class](https://bliptext.com/articles/reference-class-problem) from which n is drawn, and of which N is the ultimate size, is a crucial point of contention in the doomsday argument argument.\nThe \"standard\" doomsday argument [hypothesis](https://bliptext.com/articles/hypothesis) skips over this point entirely, merely stating that the reference class is the number of \"people\".\nGiven 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 [practical](https://bliptext.com/articles/anthropology) and [philosophical](https://bliptext.com/articles/philosophy) reasons.\nAccording to [Nick Bostrom](https://bliptext.com/articles/nick-bostrom), [consciousness](https://bliptext.com/articles/consciousness) is (part of) the discriminator between what is in and what is out of the reference class, and therefore [extraterrestrial intelligence](https://bliptext.com/articles/extraterrestrial-intelligence) might have a significant impact on the calculation.\nThe following sub-sections relate to different suggested reference classes, each of which has had the standard doomsday argument applied to it.\n\n## SSSA: Sampling from observer-moments\n[Nick Bostrom](https://bliptext.com/articles/nick-bostrom), [considering observation selection effects](https://bliptext.com/articles/anthropic-principle), has produced a [Self-Sampling Assumption](https://bliptext.com/articles/self-sampling-assumption) (SSA): \"that you should think of yourself as if you were a random observer from a suitable reference class\".\nIf the \"reference class\" is the set of humans to ever be born, this gives N < 20n with 95% confidence (the standard doomsday argument).\nHowever, he has [refined](https://bliptext.com/articles/anthropic-bias) this idea to apply to observer-moments rather than just observers.\nHe has formalized this as:\nAn 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.\nIf the mean lifespan in the future is twice the historic mean lifespan, this implies 95% confidence that N < 10n (the average future human will account for twice the observer-moments of the average historic human).\nTherefore, the 95th percentile extinction-time estimate in this version is 4560 years.\n\n# Rebuttals\n\n## We are in the earliest 5%, a priori\nOne counterargument to the doomsday argument agrees with its statistical methods but disagrees with its extinction-time estimate.\nThis 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.\nBy disagreeing with the doomsday argument, it implies that the observer is within the first 5% of humans to be born.\nBy 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.\nHowever, if one's characteristics are typical of an [early adopter](https://bliptext.com/articles/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.\nFor instance, the mainstream of potential users will prefer to be involved when the project is nearly complete.\nHowever, 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.\nIf 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.\nThe analogy to the total-human-population form of the argument is that confidence in a prediction of the [distribution](https://bliptext.com/articles/probability-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.\nThis is an argument for changing the reference class.\nFor example, if one is certain that 99% of humans who will ever live will be [cyborgs](https://bliptext.com/articles/cyborg), but that only a negligible fraction of humans who have been born to date are [cyborgs](https://bliptext.com/articles/cyborg), one could be equally certain that at least one hundred times as many people remain to be born as have been.\n[Robin Hanson](https://bliptext.com/articles/robin-hanson)'s paper sums up these criticisms of the doomsday argument:\n\"All else is not equal; we have good reasons for thinking we are not randomly selected humans from all who will ever live.\"\n\n## Human extinction is distant, a posteriori\nThe [a posteriori](https://bliptext.com/articles/a-posteriori) observation that [extinction level events](https://bliptext.com/articles/extinction-level-event) are rare could be offered as evidence that the doomsday argument's predictions are implausible; typically, [extinctions](https://bliptext.com/articles/extinction) of dominant [species](https://bliptext.com/articles/species) happen less often than once in a million years.\nTherefore, it is argued that [human extinction](https://bliptext.com/articles/human-extinction) is unlikely within the next ten millennia.\n(Another [probabilistic argument](https://bliptext.com/articles/probability-theory), drawing a different conclusion than the doomsday argument.)\nIn 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.\nIf N is distributed uniformly from 1012 to 1013, for example, then the probability of N < 1,200 billion inferred from n = 60 billion will be extremely small.\nThis is an equally impeccable Bayesian calculation, rejecting the [Copernican principle](https://bliptext.com/articles/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.\nThis response is accused of overlooking the [technological threats to humanity's survival](https://bliptext.com/articles/human-extinction), to which earlier life was not subject, and is specifically rejected by most academic critics of the doomsday argument (arguably excepting [Robin Hanson](https://bliptext.com/articles/robin-hanson)).\n\n## The prior N distribution may make n very uninformative\n[Robin Hanson](https://bliptext.com/articles/robin-hanson) argues that N prior may be [exponentially distributed](https://bliptext.com/articles/exponential-distribution):\nHere, c and q are constants.\nIf q is large, then our 95% confidence upper bound is on the uniform draw, not the exponential value of N.\nThe 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).\nThis corresponds to the idea that humanity's growth may be exponential in time with doomsday having a vague prior [probability density function](https://bliptext.com/articles/probability-density-function) in time.\nThis would mean that N, the last birth, would have a distribution looking like the following:\nThis 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 \\alpha = 1 in this distribution):\nSubstituting into the posterior probability equation):\nIntegrating the probability of any N above xn:\nFor example, if x = 20, and \\alpha = 0.5, this becomes:\nTherefore, with this prior, the chance of a trillion births is well over 20%, rather than the 5% chance given by the standard DA.\nIf \\alpha is reduced further by assuming a flatter prior N distribution, then the limits on N given by n become weaker.\nAn \\alpha of one reproduces Gott's calculation with a birth reference class, and \\alpha around 0.5 could approximate his temporal confidence interval calculation (if the population were expanding exponentially).\nAs (gets smaller) n becomes less and less [informative](https://bliptext.com/articles/uninformative-prior) about N.\nIn the limit this distribution approaches an (unbounded) [uniform distribution](https://bliptext.com/articles/uniform-distribution-continuous), where all values of N are equally likely.\nThis is Page et al.'s \"Assumption 3\", which they find few reasons to reject, a priori.\n(Although all distributions with are improper priors, this applies to Gott's vague-prior distribution also, and they can all be converted to produce [proper integrals](https://bliptext.com/articles/improper-integral) by postulating a finite upper population limit.) Since the probability of reaching a population of size 2N is usually thought of as the chance of reaching N multiplied by the survival probability from N to 2N it follows that Pr(N) must be a [monotonically](https://bliptext.com/articles/monotonic-function) decreasing function of N, but this doesn't necessarily require an inverse proportionality.\n\n## Infinite expectation\nAnother objection to the doomsday argument is that the [expected](https://bliptext.com/articles/expected-value) total human population is actually [infinite](https://bliptext.com/articles/infinity).\nThe calculation is as follows:\nFor a similar example of counterintuitive infinite expectations, see the [St. Petersburg paradox](https://bliptext.com/articles/st-petersburg-paradox).\n\n## Self-indication assumption: The possibility of not existing at all\nOne objection is that the possibility of a human existing at all depends on how many humans will ever exist (N).\nIf this is a high number, then the possibility of their existing is higher than if only a few humans will ever exist.\nSince they do indeed exist, this is evidence that the number of humans that will ever exist is high.\nThis objection, originally by [Dennis Dieks](https://bliptext.com/articles/dennis-dieks) (1992), is now known by [Nick Bostrom](https://bliptext.com/articles/nick-bostrom)'s name for it: the \"[Self-Indication Assumption](https://bliptext.com/articles/self-indication-assumption) objection\".\nIt can be shown that some [SIAs](https://bliptext.com/articles/self-indication-assumption) prevent any inference of N from n (the current population).\n\n## Caves' rebuttal\nThe [Bayesian](https://bliptext.com/articles/bayesian-inference) argument by [Carlton M. Caves](https://bliptext.com/articles/carlton-m-caves) states that the uniform distribution assumption is incompatible with the [Copernican principle](https://bliptext.com/articles/copernican-principle), not a consequence of it.\nCaves gives a number of examples to argue that Gott's rule is implausible.\nFor instance, he says, imagine stumbling into a birthday party, about which you know nothing:\n\"Your friendly enquiry about the age of the celebrant elicits the reply that she is celebrating her (tp=) 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 below.)\"\nAlthough this example exposes a weakness in [J. Richard Gott](https://bliptext.com/articles/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 [modern DA](https://bliptext.com/articles/); [epistemological](https://bliptext.com/articles/epistemology) refinements of Gott's argument by [philosophers](https://bliptext.com/articles/philosopher) such as [Nick Bostrom](https://bliptext.com/articles/nick-bostrom) specify that:\nCareful 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.\nSince human age gives an estimate of survival time (via [actuarial](https://bliptext.com/articles/actuary) tables) Caves' Birthday party age-estimate could not fall into the class of DA problems defined with this proviso.\nTo 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).\nHowever, this would remove the modified example from everyday experience.\nTo keep it in the everyday realm the lady's age must be hidden prior to the survival estimate being made.\n(Although this is no longer exactly the DA, it is much more comparable to it.)\nWithout knowing the lady's age, the DA reasoning produces a rule to convert the birthday (n) into a maximum lifespan with 50% confidence (N).\nGott's [Copernicus method](https://bliptext.com/articles/copernicus-principle) rule is simply: Prob (N < 2n) = 50%.\nHow accurate would this estimate turn out to be?\nWestern [demographics](https://bliptext.com/articles/demographics) are now fairly [uniform](https://bliptext.com/articles/uniform) across ages, so a random birthday (n) could be (very roughly) approximated by a U(0,M ] 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 2n estimate of N will be under M, its true figure. The other half of the time 2n underestimates M, and in this case (the one Caves highlights in his example) the subject will die before the 2n estimate is reached. In this \"flat [demographics](https://bliptext.com/articles/demographics)\" model Gott's 50% confidence figure is proven right 50% of the time.\n\n## Self-referencing doomsday argument rebuttal\nSome philosophers have suggested that only people who have contemplated the doomsday argument (DA) belong in the reference class \"[human](https://bliptext.com/articles/human)\".\nIf that is the appropriate reference class, [Carter](https://bliptext.com/articles/brandon-carter) defied his own prediction when he first described the argument (to the [Royal Society](https://bliptext.com/articles/royal-society)).\nAn attendant could have argued thus:\n\"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.\"\nJeff Dewynne and Professor Peter Landsberg suggested that this line of reasoning will create a [paradox](https://bliptext.com/articles/paradox) for the doomsday argument:\nIf 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.\nAlso, 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.\n\n## Conflation of future duration with total duration\nVarious authors have argued that the doomsday argument rests on an incorrect conflation of future duration with total duration.\nThis 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.\nA rebuttal in Pisaturo (2009) argues that the doomsday argument relies on the equivalent of this equation:\nPisaturo then observes:\nPisaturo takes numerical examples based on two possible corrections to this equation: considering only future durations and considering only total durations.\nIn 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.\nThis argument is also echoed in O'Neill (2014).\nIn 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.\nAs 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.\nAccording to O'Neill:\n\n## Confusion over the meaning of confidence intervals\nGelman and Robert assert that the doomsday argument confuses frequentist [confidence intervals](https://bliptext.com/articles/confidence-intervals) with Bayesian [credible intervals](https://bliptext.com/articles/credible-intervals).\nSuppose that every individual knows their number n and uses it to estimate an upper bound on N.\nEvery 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.\nThis, say Gelman and Robert, is the defining property of a frequentist lower-tailed 95% confidence interval.\nBut, they say, \"this does not mean that there is a 95% chance that any particular interval will contain the true value.\"\nThat 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.\nThe latter is a different property and is the defining characteristic of a Bayesian credible interval.\nGelman and Robert conclude:\n\"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.\"","revisionCount":0},"hasMore":false,"nextCursor":null}

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