of P(b|N) across the [1, ] range of possible N: This range starts at n rather than 1, because n can’t be greater than N. It uses the calculation above for n’s distribution given N, and implies: Substituting these marginals into the conditional equation (assuming N below ) gives: The probabilistic bounds on N with the SIA The chance of doomsday before an arbitrary factor of the current population, x, is born can be inferred, by integrating the chance of N having any value above xn. (Normally x = 20.) Therefore, given the posterior information that we have been born and that we are nth in line: For any factor, x << ( / n), of the current population:Conclusion''': n provides no information about N, in an unbounded vague prior SIA universe. Significance of Omega The finite is essential to this solution in order to produce finite integrals. In a bounded universe, actually must be finite, although this is not usually an argument used by those proposing the SIA rebuttal. However, other proponents of indefinite survival of human (and posthuman) intelligence have postulated a finite endpoint, as the (extremely high) “Omega”. Specifying any finite upper limit, , was not a part of Dieks's argument, and critics of the SIA have argued that an infinite upper bound on N creates an Improper integral (or summation) in the bayesian inference on N, which is a challenge to the logic of the critique. (For example Eastmond, and Bostrom, who argues that if the SIA cannot rule out an infinite number of potential humans, it is fatally flawed.) The unbounded vague prior is scale invariant, in that the mean is arbitrary. Therefore no finite value can be selected with more than a 50% chance of being above N (the marginal distribution of N). Olum's critique depends on such a limit existing; without this his critique is technically not applicable. Therefore it must be cautioned that the simplification here (to bound Ns distribution at ) omits a significant hurdle to the credibility of the Self-Indication Assumption Doomsday argument rebuttal. Remarks Many people, (such as Bostrom) believe the leading candidate for Doomsday argument refutation is a Self-Indication Assumption of some kind. It is popular partly because it is a purely Bayesian argument which accepts some of the DA's premises (such as the Indifference and Copernican principles). Other observations: The joint prior distribution, P(n|N), can be manipulated