Self-Indication Assumption Doomsday argument rebuttal

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.

This objection to the Doomsday Argument (DA), originally by Dennis Dieks (1992), developed by Bartha & Hitchcock (1999), and expanded by Ken Olum (2001), is that the possibility of you existing at all depends on how many humans will ever exist (N). If N is big, then the chance of you existing is higher than if only a few humans will ever exist. Since you do indeed 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 current name for this attack within the (very active) DA community is the "Self-Indication Assumption" (SIA), proposed by one of its opponents, the DA-advocate Nick Bostrom. His (2000) definition reads:

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 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:

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:

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:

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