Self-Indication Assumption Doomsday Argument Rebuttal - The Bayesian Inference of N From n Under The SIA

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:

1. Being born: With marginal probability P(b).
2. 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 doesn’t create Leslie’s “immaterial souls” is the possibility of being born into any of a large number of universes within a multiverse. You 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