The self-indication assumption doomsday argument rebuttal is an objection to the doomsday argument (that there is only a 5% chance of more than twenty times the historic number of humans ever being born) 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 (DA), originally by Dennis Dieks (1992), developed by Bartha and Hitchcock (1999), and expanded by Ken Olum (2001), is that the possibility of an individual existing all depends on how many humans will ever exist (N). If N is big, then the chance of said individual existing is higher than if only a few humans will ever exist. If the individual does exist, this is evidence that N is high. The argument is sometimes expressed in an alternative way by having the posterior marginal distribution of n based on N without explicitly invoking a non-zero chance of existing. The Bayesian inference mathematics are identical.
The name for this attack within the DA community is the "self-indication assumption" (SIA), proposed by one of its opponents, the DA-advocate Nick Bostrom. His (2000) definition reads:
SIA: Given the fact that people exist, people should (other things equal) favor hypotheses according to which many observers exist over hypotheses on which few observers exist.
A development of Dieks's original paper by Kopf, Krtous and Page (1994), showed that the SIA precisely cancels out the effect of the doomsday argument, and therefore, one's birth position (n) gives no information about the total number of humans that will exist (N). This conclusion of SIA is uncontroversial with modern DA-proponents, who instead question the validity of the assumption itself, not the conclusion which would follow, if the SIA were true.
The Bayesian inference of N from n under the SIA
The SIA-mathematics considers the chance of being the nth human as being conditioned on the joint probability of two separate events, both of which must be true: Being born: With marginal probability P(b). Being nth in line: With marginal probability (1/N), under the Principle of indifference. This means that the pdf for n, is concentrated at P(n = 0) = 1 - P(b), and that for P(n > 0) the marginal distribution can be calculated from the conditional:
Where n > 0
J. Richard Gott's DA could be formulated similarly up to this point, where it has P(b | N) = P(b) = 1, producing Gott's inference of n from N. However, Dennis Dieks argues that P(b) < 1, and that P(b | N) rises proportionally in N (which is a SIA). This can be expressed mathematically:
Where c is a constant.
The SIA's effect was expressed by Page et al. as Assumption 2 for the prior probability distribution, P(N): "The probability for the observer to exist somewhere in a history of length N is proportional to the probability for that history and to the number of people in that history." (1994 - Emphasis added to: )
They note that similar assumptions had been dismissed by Leslie on the grounds that: "it seems wrong to treat ourselves as if we were once immaterial souls harbouring hopes of becoming embodied, hopes that would have been greater, the greater the number of bodies to be created." (1992)
One argument given for P(b | N) rising in N that does not create Leslie's "immaterial souls" is the possibility of being born into any of a large number of universes within a multiverse. A person can only be born into one, so the indifference principle within this (humans-across-universes) reference class would mean that the chance of being born into a particular universe is proportional to its weight in humans, N. (Echoing the weak anthropic principle.) In this framework, the chance of 'not being born' is zero, but the chance of 'not being born into this universe is non-zero.
Whatever the reasoning, the essential idea of the self-indication assumption is that the prior probability of birth into this universe is rising in N, and is generally considered to be proportional to N. (The following discussion assumes they are proportional so P(b | N) = 2 P(b | 2N), since other functions increasing in N produce similar results.) Therefore:
Where n > 0
Effect of the "unborn" on the Bayesian inference
To clarify the exposition, Gott's vague prior N distribution is 'capped' at some "universal carrying capacity", . (This prevents N distribution being an improper prior.)
is the largest possible value for N if all living space in the 'universe' is consumed. The limit has no specified upper bounds (to habitable planets in the Galaxy, say) but makes N posterior distribution more tractable: