Doomsday argument - AbsoluteAstronomy.com

The Doomsday argument is a probabilistic argument

Probabilistic argument

Probabilistic argument can refer to the following:* In some contexts, probabilistic argument means any argument involving probability theory...

that claims to predict the number of future members of the human species given only an estimate of the total number of humans born so far. Simply put, it says that supposing the humans alive today are in a random place in the whole human history timeline, chances are we are about halfway through it.

It was first proposed in an explicit way by the astrophysicist Brandon Carter

Brandon Carter

Brandon Carter, FRS is an Australian theoretical physicist, best known for his work on the properties of black holes and for being the first to name and employ the anthropic principle in its contemporary form. He is a researcher at the Meudon campus of the Laboratoire Univers et Théories, part of...

in 1983, from which it is sometimes called the Carter catastrophe; the argument was subsequently championed by the philosopher John A. Leslie

John A. Leslie

John Andrew Leslie is a Canadian philosopher. He was educated at Wadham College, Oxford, earning his B.A. in English Literature in 1962 and his M.Litt. in Classics in 1968...

and has since been independently discovered by J. Richard Gott

J. Richard Gott

John Richard Gott III is a professor of astrophysical sciences at Princeton University. He is known for developing and advocating two cosmological theories with the flavor of science fiction: Time travel and the Doomsday argument.- Exotic matter time travel theories :Paul Davies's bestseller How...

and Holger Bech Nielsen

Holger Bech Nielsen

Holger Bech Nielsen is a Danish theoretical physicist, professor at the Niels Bohr Institute, at the University of Copenhagen, where he started studying physics in 1961....

. Similar principles of eschatology

Eschatology

Eschatology is a part of theology, philosophy, and futurology concerned with what are believed to be the final events in history, or the ultimate destiny of humanity, commonly referred to as the end of the world or the World to Come...

were proposed earlier by Heinz von Foerster

Heinz von Foerster

Heinz von Foerster was an Austrian American scientist combining physics and philosophy. Together with Warren McCulloch, Norbert Wiener, John von Neumann, Lawrence J. Fogel, and others, Heinz von Foerster was an architect of cybernetics.-Biography:Von Foerster was born in 1911 in Vienna, Austria,...

, among others.

Denoting by N the total number of humans who were ever or will ever be born, the Copernican principle

Copernican principle

In physical cosmology, the Copernican principle, named after Nicolaus Copernicus, states that the Earth is not in a central, specially favored position. More recently, the principle has been generalized to the relativistic concept that humans are not privileged observers of the universe...

suggests that humans are equally likely (along with the other N − 1 humans) to find themselves at any position n, so humans assume that our fractional position f = n/N is uniformly distributed

Uniform distribution (continuous)

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...

on the interval

Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

[0, 1] prior

Prior probability

In Bayesian statistical inference, a prior probability distribution, often called simply the prior, of an uncertain quantity p is the probability distribution that would express one's uncertainty about p before the "data"...

to learning our absolute position.

f is uniformly distributed on (0, 1] even after learning of the absolute position n. That is, for example, there is 95% chance that f is in the interval (0.05, 1], that is f > 0.05. In other words we could assume that we could be 95% certain that we would be within the last 95% of all the humans ever to be born. If we know our absolute position n, this implies an upper bound for N obtained by rearranging n/N > 0.05 to give N < 20n.

If Leslie's Figure is used, then 60 billion humans have been born so far, therefore it can be estimated that there is a 95% chance that the total number of humans N will be less than 20 × 60 billion = 1.2 trillion. Assuming that the world population

World population

The world population is the total number of living humans on the planet Earth. As of today, it is estimated to be billion by the United States Census Bureau...

stabilizes at 10 billion and a life expectancy

Life expectancy

Life expectancy is the expected number of years of life remaining at a given age. It is denoted by ex, which means the average number of subsequent years of life for someone now aged x, according to a particular mortality experience...

of 80 years, it can be estimated that the remaining 1140 billion humans will be born in 9120 years. Depending on the projection of world population in the forthcoming centuries, estimates may vary, but the main point of the argument is that it is unlikely that more than 1.2 trillion humans will ever live. This problem is similar to the famous German tank problem

German tank problem

In the statistical theory of estimation, estimating the maximum of a uniform distribution is a common illustration of differences between estimation methods...

Remarks

The step that converts N into an extinction time depends upon a finite human lifespan. If immortality
Immortality
Immortality is the ability to live forever. It is unknown whether human physical immortality is an achievable condition. Biological forms have inherent limitations which may or may not be able to be overcome through medical interventions or engineering...

becomes common, and the birth rate drops to zero, then the human race could continue forever even if the total number of humans N is finite.
A precise formulation of the Doomsday Argument requires the Bayesian
Bayesian probability
Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...

interpretation of probability
Even among Bayesians some of the assumptions of the argument's logic would not be acceptable; for instance, the fact that it is applied to a temporal phenomenon (how long something lasts) means that Ns distribution simultaneously represents an "aleatory probability" (as a future event), and an "epistemic probability" (as a decided value about which we are uncertain).
The U (0,1] f distribution is derived from two choices, which whilst being the default are also arbitrary:
- The principle of indifference
  Principle of indifference
  The principle of indifference is a rule for assigning epistemic probabilities.Suppose that there are n > 1 mutually exclusive and collectively exhaustive possibilities....
  
  , so that it is as likely for any other randomly selected person to be born after you as before you.
- The assumption of no 'prior' knowledge on the distribution of N.

Simplification: two possible total number of humans

Assume for simplicity that the total number of humans who will ever be born is 60 billion (N₁), or 6,000 billion (N₂). If there is no prior knowledge of the position that a currently living individual, X, has in the history of humanity, we may instead compute how many humans were born before X, and arrive at (say) 59,854,795,447, which would roughly place X amongst the first 60 billion humans who have ever lived.

Now, if we assume that the number of humans who will ever be born equals N₁, the probability that X is amongst the first 60 billion humans who have ever lived is of course 100%. However, if the number of humans who will ever be born equals N₂, then the probability that X is amongst the first 60 billion humans who have ever lived is only 1%. Since X is in fact amongst the first 60 billion humans who have ever lived, this means that the total number of humans who will ever be born is more likely to be much closer to 60 billion than to 6,000 billion. In essence the DA therefore suggests that human extinction

Human extinction

Human extinction is the end of the human species. Various scenarios have been discussed in science, popular culture, and religion . The scope of this article is existential risks. Humans are very widespread on the Earth, and live in communities which are capable of some kind of basic survival in...

is more likely to occur sooner rather than later.

It is possible to sum the 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 6,000 billion.

Note 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. On 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. 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 make an assumption about the prior probability P(N) that the total number of humans is N. If 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. A further, more detailed discussion, as well as relevant distributions P(N), are given below in the Rebuttals section.

What the argument is not

The Doomsday argument (DA) 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 DA's conclusion is:

There is a 95% chance of extinction within 9120 years.

The DA gives a 5% chance that some humans will still be alive at the end of that period. (These dates are based on the assumptions above; the precise numbers vary among specific Doomsday arguments.)

The DA does not predict the extinction of all intelligent life on Earth, only the extinction of humans.

Variations

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

Gott's formulation: 'vague prior' total population

Gott specifically proposes the functional form for the prior distribution

Prior probability

of the number of people who will ever be born (N). Gott's DA used the vague prior distribution:

.
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 normalize
Normalizing constant
The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics.-Definition and examples:In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g.,...

the sum of P(N). The value chosen isn't important here, just the functional form (this is an improper prior, so no value of k gives a valid distribution, but Bayesian inference
Bayesian inference
In statistics, Bayesian inference is a method of statistical inference. It is often used in science and engineering to determine model parameters, make predictions about unknown variables, and to perform model selection...

is still possible using it.)

Since Gott specifies the prior

Prior probability

distribution of total humans, P(N), Bayes's theorem and the principle of indifference

Principle of indifference

The principle of indifference is a rule for assigning epistemic probabilities.Suppose that there are n > 1 mutually exclusive and collectively exhaustive possibilities....

alone give us P(N|n), the probability of N humans being born if n is a random draw from N:

This is Bayes's theorem for the posterior probability

Posterior probability

In Bayesian statistics, the posterior probability of a random event or an uncertain proposition is the conditional probability that is assigned after the relevant evidence is taken into account...

of total population exactly N, conditioned

Conditioning (probability)

Beliefs depend on the available information. This idea is formalized in probability theory by conditioning. Conditional probabilities, conditional expectations and conditional distributions are treated on three levels: discrete probabilities, probability density functions, and measure theory...

on current population exactly n. Now, using the indifference principle:

.

The unconditioned n distribution of the current population is identical to the vague prior N probability density function, so:

,

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

.

The easiest way to produce the doomsday estimate with a given confidence (say 95%) is to pretend that N is a continuous variable (since it is very large) and integrate

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

over the probability density from N = n to N = Z. (This will give a function for the probability that N ≤ Z):

Defining Z = 20n gives:

.

This is the simplest Bayesian

Bayes factor

In statistics, the use of Bayes factors is a Bayesian alternative to classical hypothesis testing. Bayesian model comparison is a method of model selection based on Bayes factors.-Definition:...

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 vague prior distribution seems well-motivated as it assumes as little knowledge as possible about N, given that any 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 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.

97.5% is one chance in forty, which can be used in the integral above with Z = 40n, and n = 200,000 years:

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 − 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

Bertrand's paradox (probability)

The Bertrand paradox is a problem within the classical interpretation of probability theory. Joseph Bertrand introduced it in his work Calcul des probabilités as an example to show that probabilities may not be well defined if the mechanism or method that produces the random variable is not...

.)

His choice of 95% confidence bounds (rather than 80% or 99.9%, say) matched the scientifically accepted limit of statistical significance

Statistical significance

In statistics, a result is called statistically significant if it is unlikely to have occurred by chance. The phrase test of significance was coined by Ronald Fisher....

for hypothesis rejection. Therefore, he argued that the hypothesis

Hypothesis

A hypothesis is a proposed explanation for a phenomenon. The term derives from the Greek, ὑποτιθέναι – hypotithenai meaning "to put under" or "to suppose". For a hypothesis to be put forward as a scientific hypothesis, the scientific method requires that one can test it...

: “humanity will cease to exist before 5,100 years or thrive beyond 7.8 million years” can be rejected.

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.

Heinz von Foerster

Heinz von Foerster

argued that humanity's abilities to construct societies, civilizations and technologies do not result in self inhibition. Rather, societies' success varies directly with population size. Von Foerster found that this model fit some 25 data points from the birth of Jesus

Jesus

Jesus of Nazareth , commonly referred to as Jesus Christ or simply as Jesus or Christ, is the central figure of Christianity...

to 1958, with only 7% of the variance

Variance

In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

left unexplained. Several follow-up letters (1961, 1962, …) were published in Science showing that von Foerster's equation was still on track. The data continued to fit up until 1973. The 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. In fact, von Foerster did not imply that the world population on that day could actually become infinite. The 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. Note that this prediction began to be fulfilled just in a few years after the "Doomsday" was published.

Reference classes

One of the major areas of Doomsday Argument debate is the reference class

Reference class problem

In statistics, the reference class problem is the problem of deciding what class to use when calculating the probability applicable to a particular case...

from which n is drawn, and of which N is the ultimate size. The 'standard' Doomsday Argument hypothesis

Hypothesis

doesn't spend very much time on this point, and simply says that the reference class is the number of 'humans'. Given that you are human, the Copernican principle could be applied to ask if you were born unusually early, but the grouping of 'human' has been widely challenged on practical

Anthropology

Anthropology is the study of humanity. It has origins in the humanities, the natural sciences, and the social sciences. The term "anthropology" is from the Greek anthrōpos , "man", understood to mean mankind or humanity, and -logia , "discourse" or "study", and was first used in 1501 by German...

and philosophical

Philosophy

Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

grounds. Nick Bostrom

Nick Bostrom

Nick Bostrom is a Swedish philosopher at the University of Oxford known for his work on existential risk and the anthropic principle. He holds a PhD from the London School of Economics...

has argued that consciousness

Consciousness

Consciousness is a term that refers to the relationship between the mind and the world with which it interacts. It has been defined as: subjectivity, awareness, the ability to experience or to feel, wakefulness, having a sense of selfhood, and the executive control system of the mind...

is (part of) the discriminator between what is in and what is out of the reference class, and that extraterrestrial intelligences might affect the calculation dramatically.

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

Sampling only WMD-era humans

The Doomsday clock

Doomsday Clock

The Doomsday Clock is a symbolic clock face, maintained since 1947 by the board of directors of the Bulletin of the Atomic Scientists at the University of Chicago. The closer the clock is to midnight, the closer the world is estimated to be to global disaster. , the Doomsday Clock now stands at six...

shows the expected time to nuclear doomsday

Doomsday event

A doomsday event is a specific, plausibly verifiable or hypothetical occurrence which has an exceptionally destructive effect on the human race...

by the judgment of an expert board

Bulletin of the Atomic Scientists

The Bulletin of the Atomic Scientists is a nontechnical online magazine that covers global security and public policy issues, especially related to the dangers posed by nuclear and other weapons of mass destruction...

, rather than a Bayesian model. If the twelve hours of the clock symbolize the lifespan of the human species, its current time of 11:54 implies that we are among the last 1% of people who will ever be born (i.e. that n > 0.99N). J. Richard Gott

J. Richard Gott

's temporal version of the Doomsday argument (DA) would require very strong prior evidence to overcome the improbability of being born in such a special

Copernican principle

time.

If the clock's doomsday estimate is correct, there is less than 1 chance in 100 of seeing it show such a late time in human history, if observed at a random time within that history.

The scientists'

Bulletin of the Atomic Scientists

warning can be reconciled with the DA, however: The Doomsday clock specifically estimates the proximity of atomic

Nuclear weapon

A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either fission or a combination of fission and fusion. Both reactions release vast quantities of energy from relatively small amounts of matter. The first fission bomb test released the same amount...

self-destruction—which has only been possible for sixty years.
If doomsday requires nuclear weaponry then the Doomsday Argument 'reference class' is: people contemporaneous with nuclear weapons. In this model, the number of people living through, or born after Hiroshima

Atomic bombings of Hiroshima and Nagasaki

During the final stages of World War II in 1945, the United States conducted two atomic bombings against the cities of Hiroshima and Nagasaki in Japan, the first on August 6, 1945, and the second on August 9, 1945. These two events are the only use of nuclear weapons in war to date.For six months...

is n, and the number of people who ever will is N. Applying Gott's

J. Richard Gott

DA to these variable definitions gives a 50% chance of doomsday within 50 years.

In this model, the clock's hands are so close to midnight because a condition

Conditional probability

In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...

of doomsday is living post-1945, a condition which applies now but not to the earlier 11 hours and 53 minutes of the clock's metaphorical human 'day'.

If your life is randomly selected from all lives lived under the shadow of the bomb, this simple model gives a 95% chance of doomsday within 1000 years.

The scientists' recent use of moving the clock forward to warn of the dangers posed by global warming

Global warming

Global warming refers to the rising average temperature of Earth's atmosphere and oceans and its projected continuation. In the last 100 years, Earth's average surface temperature increased by about with about two thirds of the increase occurring over just the last three decades...

muddles this reasoning, however.

SSSA: Sampling from observer-moments

Nick Bostrom

Nick Bostrom is a Swedish philosopher at the University of Oxford known for his work on existential risk and the anthropic principle. He holds a PhD from the London School of Economics...

, 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 < 20n with 95% confidence (the standard Doomsday argument). However, he has refined this idea to apply to observer-moments rather than just observers. He has formalized this (http://anthropic-principle.com/preprints/self-location.html as:

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.

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 < 10n (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.

We are in the earliest 5%, a priori

If you agree with the statistical methods, still disagreeing with the Doomsday argument (DA) implies that:

We are within the first 5% of humans to be born.
This is not purely a coincidence.

Therefore, these rebuttals try to give reasons for believing that we are some of the earliest humans.

For instance, you are member 50,000 in a collaborative project, the Doomsday Argument implies a 95% chance that there will never be more than a million members of that project. This can be refuted if your other characteristics are typical of the early adopter. The mainstream of potential users will prefer to be involved when the project is nearly complete. If you enjoy the project's incompleteness, we already know that you are unusual, prior to the discovery of your early involvement.

If you have measurable attributes that set you apart from the typical long run user, the project DA can be refuted based on the fact that you would expect to be within the first 5% of members, a priori

A priori and a posteriori (philosophy)

The terms a priori and a posteriori are used in philosophy to distinguish two types of knowledge, justifications or arguments...

. The analogy to the total-human-population form of the argument is: Confidence in a prediction of the distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

of human characteristics that places modern & historic humans outside the mainstream, implies that we already know, before examining n that it is likely to be very early in N.

For example, if you are certain that 99% of humans who will ever live will be cyborg

Cyborg

A cyborg is a being with both biological and artificial parts. The term was coined in 1960 when Manfred Clynes and Nathan S. Kline used it in an article about the advantages of self-regulating human-machine systems in outer space. D. S...

s, but that only a negligible fraction of humans who have been born to date are cyborgs, you could be equally certain that at least one hundred times as many people remain to be born as have been.

Robin Hanson

Robin Hanson

Robin D. Hanson is an associate professor of economics at George Mason University and a research associate at the Future of Humanity Institute of Oxford University. He is known as an expert on idea futures, markets and was involved in the creation of the Foresight Exchange and DARPA's FutureMAP...

's paper sums up these criticisms of the DA:

"All else is not equal; we have good reasons for thinking we are not randomly selected humans from all who will ever live."

Drawbacks of this rebuttal:

The question of how the confident prediction is derived. We need an uncannily prescient picture of humanity's statistical distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

through all time, before we can pronounce ourselves extreme members of that population
Population
A population is all the organisms that both belong to the same group or species and live in the same geographical area. The area that is used to define a sexual population is such that inter-breeding is possible between any pair within the area and more probable than cross-breeding with individuals...

. (In contrast, project pioneers have clearly distinct psychology from the mainstream.)
If the majority of humans have characteristics we do not share, some would argue that this is equivalent to the Doomsday argument, since people like us will become extinct. (Friedrich Nietzsche
Friedrich Nietzsche
Friedrich Wilhelm Nietzsche was a 19th-century German philosopher, poet, composer and classical philologist...

outlines this pseudoextinction
Pseudoextinction
Pseudoextinction of a species occurs where there are no more living members of that species, but members of a daughter species or subspecies remain alive. As all species must have an ancestor of a previous species, much of evolution is believed to occur through pseudoextinction...

point of view in his 1885 book Thus Spoke Zarathustra
Thus Spoke Zarathustra
Thus Spoke Zarathustra: A Book for All and None is a philosophical novel by German philosopher Friedrich Nietzsche, composed in four parts between 1883 and 1885...

.)

Critique: Human extinction is distant, a posteriori

The a posteriori

A Posteriori

Apart from the album, some additional remixes were released exclusively through the iTunes Store. They are:*"Eppur si muove" – 6:39*"Dreaming of Andromeda" Apart from the album, some additional remixes were released exclusively through the iTunes Store. They are:*"Eppur si muove" (Tocadisco...

observation that extinction level events are rare could be offered as evidence that the DA's predictions are implausible; typically, extinction

Extinction

In biology and ecology, extinction is the end of an organism or of a group of organisms , normally a species. The moment of extinction is generally considered to be the death of the last individual of the species, although the capacity to breed and recover may have been lost before this point...

s of a dominant species

Species

In biology, a species is one of the basic units of biological classification and a taxonomic rank. A species is often defined as a group of organisms capable of interbreeding and producing fertile offspring. While in many cases this definition is adequate, more precise or differing measures are...

happens less often than once in a million years. Therefore, it is argued that Human extinction

Human extinction

is unlikely within the next ten millennia. (Another probabilistic argument

Probabilistic argument

Probabilistic argument can refer to the following:* In some contexts, probabilistic argument means any argument involving probability theory...

, drawing a different conclusion from the DA.)

In Bayesian terms, this response to the DA 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 10¹² to 10¹³, 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

Copernican principle

on the grounds that 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 technological threats to humanity's survival, to which earlier life was not subject, and is specifically rejected by most of the DA's academic critics (arguably excepting Robin Hanson

Robin Hanson

).

In fact, many futurologists believe the empirical situation is worse than Gott's DA estimate. For instance, Sir Martin Rees believes that the technological dangers give an estimated human survival duration of ninety-five years (with 50% confidence

Our Final Hour

Our Final Hour is a 2003 book by the British Astronomer Royal Sir Martin Rees. The full title of the book is Our Final Hour: A Scientist's Warning: How Terror, Error, and Environmental Disaster Threaten Humankind's Future In This Century - On Earth and Beyond...

.) Earlier prophets made similar predictions and were 'proven' wrong (e.g. on surviving the nuclear arms race). It is possible that their estimates were accurate, and that their common image as alarmists is a survivorship bias

Survivorship bias

Survivorship bias is the logical error of concentrating on the people or things that "survived" some process and inadvertently overlooking those that didn't because of their lack of visibility. This can lead to false conclusions in several different ways...

The prior N distribution may make n very uninformative

Robin Hanson

argues that Ns prior may be exponentially distributed

Exponential distribution

In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...

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 best 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 pdf

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

in time. This would mean than N, the last birth, would have a distribution looking like the following:

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

= 1 in this distribution):

Substituting into the posterior probability equation):

Integrating the probability of any N above xn:

For example, if x = 20, and

= 0.5, this becomes:

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

is reduced further by assuming a flatter prior N distribution, then the limits on N given by n become weaker. An

of one reproduces Gott's calculation with a birth reference class, and

around 0.5 could approximate his temporal confidence interval calculation (if the population were expanding exponentially). As

(gets smaller) n becomes less and less informative about N. In the limit this distribution approaches an (unbounded) uniform distribution

Uniform distribution

-Probability theory:* Discrete uniform distribution* Continuous uniform distribution-Other:* "Uniform distribution modulo 1", see Equidistributed sequence*Uniform distribution , a type of species distribution* Distribution of military uniforms...

, 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

are improper priors, this applies to Gott's vague-prior distribution also, and they can all be converted to produce proper integrals

Improper integral

In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits....

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 seems that Pr(N) must be a monotonically

Monotonic function

In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

decreasing function of N, but this doesn't necessarily require an inverse proportionality.

A prior distribution with a very low

parameter

Parameter

Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

makes the DA's ability to constrain the ultimate size of humanity very weak.

Infinite Expectation

Another objection to the Doomsday Argument is that the expected

Expected value

In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

total human population is actually infinite. The calculation is as follows:

The total human population N = n/f, where n is the human population to date and f is our fractional position in the total.

We assume that f is uniformly distributed on (0,1].

The expectation of N is

This infinite expectation shows that, under the framework of the DA, humanity still has some chance of surviving an arbitrarily long time.

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

St. Petersburg paradox

In economics, the St. Petersburg paradox is a paradox related to probability theory and decision theory. It is based on a particular lottery game that leads to a random variable with infinite expected value, i.e., infinite expected payoff, but would nevertheless be considered to be worth only a...

SIA: The possibility of not existing at all

One objection is that the possibility of your existing at all depends on how many humans will ever exist (N). If this is a high number, then the possibility of your existing is higher than if only a few humans will ever exist. Since you do indeed exist, this is evidence that the number of humans that will ever exist is high.

This objection, originally by Dennis Dieks

Dennis Dieks

Dennis Dieks is a Dutch physicist and philosopher of physics. In 1982 he proved the no-cloning theorem . In 1989 he proposed a new interpretation of quantum mechanics, later known as a version of the modal interpretation of quantum mechanics...

(1992), is now known by Nick Bostrom

Nick Bostrom

Nick Bostrom is a Swedish philosopher at the University of Oxford known for his work on existential risk and the anthropic principle. He holds a PhD from the London School of Economics...

's name for it: the "Self-Indication Assumption

Self-Indication Assumption

The Self Indication Assumption Nick Bostrom originally used the term SIA in a slightly different way. What is here referred to as SIA, he referred to as the combined SSA+SIA, a philosophical principle defined by Nick Bostrom, one of the two major schools of anthropic probability , states that:Note...

objection". It can be shown that some SIAs

Self-Indication Assumption

prevent any inference of N from n (the current population); for details of this argument from the Bayesian inference perspective see: Self-Indication Assumption Doomsday argument rebuttal

Self-Indication Assumption Doomsday argument rebuttal

The Self-Indication Assumption Doomsday argument rebuttal is an objection to the Doomsday argument by arguing that the chance of being born is not one, but is an increasing function of the number of people who will be born.- History :This objection to the Doomsday Argument ,...

Caves' rebuttal

The Bayesian

Bayesian inference

In statistics, Bayesian inference is a method of statistical inference. It is often used in science and engineering to determine model parameters, make predictions about unknown variables, and to perform model selection...

argument by Carlton M. Caves

Carlton M. Caves

Carlton Morris Caves holds the position of Distinguished Professor in physicsat the University of New Mexico. He is notable for his work in the areas of...

says that the uniform distribution assumption is incompatible with the Copernican principle

Copernican principle

, not a consequence of it.

He 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:

Your friendly enquiry about the age of the celebrant elicits the reply that she is celebrating her (t_p = ) 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.)

Although this example exposes a weakness in J. Richard Gott

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; epistemological refinements of Gott's argument by philosophers such as Nick Bostrom

Nick Bostrom

Nick Bostrom is a Swedish philosopher at the University of Oxford known for his work on existential risk and the anthropic principle. He holds a PhD from the London School of Economics...

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 actuarial

Actuary

An actuary is a business professional who deals with the financial impact of risk and uncertainty. Actuaries provide expert assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms ....

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 method rule is simply: Prob (N < 2n) = 50%. How accurate would this estimate turn out to be? Western demographics

Demographics

Demographics are the most recent statistical characteristics of a population. These types of data are used widely in sociology , public policy, and marketing. Commonly examined demographics include gender, race, age, disabilities, mobility, home ownership, employment status, and even location...

are now fairly uniform

Uniform

A uniform is a set of standard clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are worn by armed forces and paramilitary organizations such as police, emergency services, security guards, in some workplaces and schools and by inmates...

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' model Gott's 50% confidence figure is proven right 50% of the time.

Self-referencing doomsday argument rebuttal

Some philosophers have been bold enough to suggest that only people who have contemplated the Doomsday argument (DA) belong in the reference class 'human

Human

Humans are the only living species in the Homo genus...

'. If that is the appropriate reference class, Carter

Brandon Carter

defied his own prediction when he first described the argument (to the Royal Society

Royal Society

The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...

). A member present could have argued thus:

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.

Jeff Dewynne and Professor Peter Landsberg suggested that this line of reasoning will create a paradox

Paradox

Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...

for the Doomsday argument:

If a member 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 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.

Additionally, it should be considered that because Carter

Brandon Carter

did present and describe his argument, in which case the people to whom he explained it did contemplate the DA, as it was inevitable, the conclusion could then be drawn that in the moment of explanation Carter

Brandon Carter

created the basis for his own prediction.

Conflation of future duration with total duration

A rebuttal by Ronald Pisaturo in 2009 argues that the Doomsday Argument conflates future duration and total duration.

According to Pisaturo, the Doomsday Argument relies on the equivalent of this equation:

where:

X = the prior information;

D_p = the data that past duration is t_p;

H_FS = the hypothesis that the future duration of the phenomenon will be short;

H_FL = the hypothesis that the future duration of the phenomenon will be long;

H_TS = the hypothesis that the total duration of the phenomenon will be short—i.e., that t_t, the phenomenon’s total longevity, = t_TS;

H_TL = the hypothesis that the total duration of the phenomenon will be long—i.e., that t_t, the phenomenon’s total longevity, = t_TL, with t_TL > t_TS.

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.

Mathematics-free explanation by analogy

Assume the human species is a car driver. The driver has encountered some bumps but no catastrophes, and the car (Earth

Earth

Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

) is still road-worthy. However, insurance is required. The cosmic insurer has not dealt with humanity before, and needs some basis on which to calculate the premium. According to the Doomsday Argument, the insurer merely need ask how long the car and driver have been on the road—currently at least 40,000 years without an "accident"—and use the response to calculate insurance based on a 50% chance that a fatal "accident" will occur inside that time period.

Consider a hypothetical insurance company that tries to attract drivers with long accident-free histories not because they necessarily drive more safely than newly qualified drivers, but for statistical reasons: the hypothetical insurer estimates that each driver looks for insurance quotes every year, so that the time since the last accident

Accident

An accident or mishap is an unforeseen and unplanned event or circumstance, often with lack of intention or necessity. It implies a generally negative outcome which may have been avoided or prevented had circumstances leading up to the accident been recognized, and acted upon, prior to its...

is an evenly distributed random sample between accidents. The chance of being more than halfway through an evenly distributed random sample is one-half, and (ignoring old-age effects) if the driver is more than half way between accidents then they are closer to their next accident than their previous one. A driver who was accident-free for 10 years would be quoted a very low premium for this reason, but someone should not expect cheap insurance if they only passed their test two hours ago (equivalent to the accident-free record of the human species in relation to 40,000 years of geological time.)

Analogy to the estimated final score of a cricket batsman

A random in-progress cricket

Cricket

Cricket is a bat-and-ball game played between two teams of 11 players on an oval-shaped field, at the centre of which is a rectangular 22-yard long pitch. One team bats, trying to score as many runs as possible while the other team bowls and fields, trying to dismiss the batsmen and thus limit the...

test match

Test cricket

Test cricket is the longest form of the sport of cricket. Test matches are played between national representative teams with "Test status", as determined by the International Cricket Council , with four innings played between two teams of 11 players over a period of up to a maximum five days...

is sampled for a single piece of information: the current batsman's run tally so far. If the batsman is dismissed (rather than declaring), what is the chance that he will end up with a score more than double his current total?

A rough empirical

Empirical

The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....

result is that the chance is half (on average).

The Doomsday argument (DA) is that even if we were completely ignorant of the game we could make the same prediction, or profit by offering a bet paying odds

Odds

The odds in favor of an event or a proposition are expressed as the ratio of a pair of integers, which is the ratio of the probability that an event will happen to the probability that it will not happen...

of 2-to-3 on the batsmen doubling his current score.

Importantly, we can only offer the bet before the current score is given (this is necessary because the absolute value of the current score would give a cricket expert a lot of information about the chance of that tally doubling). It is necessary to be ignorant of the absolute run tally before making the prediction because this is linked to the likely total, but if the likely total and absolute value are not linked the survival prediction can be made after discovering the batter's current score. Analogously, the DA says that if the absolute number of humans born gives no information on the number that will be, we can predict the species’ total number of births after discovering that 60 billion people have ever been born: with 50% confidence it is 120 billion people, so that there is better-chance-than-not that the last human birth will occur before the 23rd century.

It is not true that the chance is half, whatever is the number of runs currently scored; batting

Batting

Batting may refer to:*Batting , the act of attempting to hit a ball thrown by the pitcher with a baseball bat, in order to score runs*Batting , the act of defending one's wicket with the cricket bat while attempting to score runs...

records give an empirical correlation

Correlation

In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

between reaching a given score (50 say) and reaching any other, higher score (say 100). On the average, the chance of doubling the current tally may be half, but the chance of reaching a century having scored fifty is much lower than reaching ten from five. Thus, the absolute value of the score gives information about the likely final total the batsman will reach, beyond the “scale invariant”.

An analogous Bayesian critique of the DA is that is somehow possessed prior

Prior probability

knowledge of the all-time human population distribution (total runs scored), and that this is more significant than the finding of a low number of births until now (a low current run count).

There are two alternative methods of making uniform

Uniform

draws from the current score (n):

Put the runs actually scored by dismissed player in order, say 200, and randomly choose between these scoring increments by U(0, 200].
Select a time randomly from the beginning of the match to the final dismissal.

The second sampling-scheme will include those lengthy periods of a game where a dismissed player is replaced, during which the ‘current batsman’ is preparing to take the field and has no runs. If people sample based on time-of-day rather than running-score they will often find that a new batsman has a score of zero when the total score that day was low, but humans will rarely sample a zero if one batsman stayed at the crease, piling on runs all day long. Therefore, the fact that sampling a non-zero score would tell us something about the likely final score that the current batsman will achieve.

Choosing sampling method 2 rather than method 1 would give a different statistical link between current and final score: any non-zero score would imply that the batsman reached a high final total, especially if the time to replace batsman is very long. This is analogous to the SIA

Self-Indication Assumption

-DA-refutation that Ns distribution should include N = 0 states, which leads to the DA having reduced predictive power

Predictive power

The predictive power of a scientific theory refers to its ability to generate testable predictions. Theories with strong predictive power are highly valued, because the predictions can often encourage the falsification of the theory...

(in the extreme, no power to predict N from n at all).

External links

A non-mathematical, unpartisan introduction to the DA
A compelling lecture from the University of Colorado-Boulder
Nick Bostrom's response to Korb and Oliver
Nick Bostrom's summary version of the argument
Nick Bostrom's annotated collection of references
Kopf, Krtouš & Page's early (1994) refutation based on the SIA
Sia
-People:*Sia Berkeley , British actress*Sia Figiel , Samoan novelist, poet and painter*Sia Furler , Australian singer...

, which they called "Assumption 2".
The Doomsday argument and the number of possible observers by Ken Olum In 1993 J. Richard Gott
J. Richard Gott
John Richard Gott III is a professor of astrophysical sciences at Princeton University. He is known for developing and advocating two cosmological theories with the flavor of science fiction: Time travel and the Doomsday argument.- Exotic matter time travel theories :Paul Davies's bestseller How...

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.
A Critique of the Doomsday Argument by Robin Hanson
A Third Route to the Doomsday Argument by Paul Franceschi, Journal of Philosophical Research, 2009, vol. 34, pp. 263–278
Chambers' Ussherian Corollary Objection
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).
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. 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.]
Mark Greenberg, "Apocalypse Not Just Now" in London Review of Books
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
J. Richard Gott
John Richard Gott III is a professor of astrophysical sciences at Princeton University. He is known for developing and advocating two cosmological theories with the flavor of science fiction: Time travel and the Doomsday argument.- Exotic matter time travel theories :Paul Davies's bestseller How...

's form of the DA, and was programmed by sustainable development
Sustainable development
Sustainable development is a pattern of resource use, that aims to meet human needs while preserving the environment so that these needs can be met not only in the present, but also for generations to come...

researcher Jerrad Pierce.

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Remarks

Simplification: two possible total number of humans

What the argument is not

Variations

Gott's formulation: 'vague prior' total population

Reference classes

Sampling only WMD-era humans

SSSA: Sampling from observer-moments

We are in the earliest 5%, a priori

Critique: Human extinction is distant, a posteriori

The prior N distribution may make n very uninformative

Infinite Expectation

SIA: The possibility of not existing at all

Caves' rebuttal

Self-referencing doomsday argument rebuttal

Conflation of future duration with total duration

Mathematics-free explanation by analogy

Analogy to the estimated final score of a cricket batsman

See also

External links