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Doomsday argument
You will learn: Trace rank evidence through finite priors and explicit observer-selection assumptions.
Start with: Bayes' theorem
Can your position in a sequence tell you how long the sequence will ultimately be? If a ticket is sampled uniformly from a numbered urn, its label can provide evidence about the urn’s size. Applying an analogous argument to a person’s birth rank requires much stronger assumptions about what counts as the sampled observer and how that observer was selected.
The doomsday argument is controversial because those observation-selection assumptions are not supplied by Bayes’ theorem itself. We use a small finite example to expose their effect. It contains hypothetical completed populations and ranks, with no real birth counts, dates, or forecast of humanity’s future.
Compare two selection assumptions
Compare three hypothetical completed populations of 100, 1,000, or 10,000 people. Your observed integer rank is assumed uniformly sampled among the people in whichever population exists. No calendar dates or real population estimates enter this model.
| Total N | Base normalized prior | P(rank | N) | Rank-only posterior | Population-weighted prior | Weighted-prior posterior |
|---|---|---|---|---|---|
| 100 | 0.3333333 | 0.01000000 | 0.9009009 | 0.009009009 | 0.3333333 |
| 1000 | 0.3333333 | 0.001000000 | 0.09009009 | 0.09009009 | 0.3333333 |
| 10000 | 0.3333333 | 0.0001000000 | 0.009009009 | 0.9009009 | 0.3333333 |
Log10 totals 2, 3, and 4 mean 100, 1,000, and 10,000 people. Solid left marks: base prior updated by rank. Dashed right marks: first multiply base weights by population size, then update by rank. For compatible populations, that extra N exactly cancels the likelihood's 1/N. The result is the base prior conditioned only on N being at least the observed rank.
The three possible totals are one hundred, one thousand, and ten thousand people. The base weights specify a prior over these three hypotheses. They need not be equal, but at least one must be positive to define a prior. Within any selected population of size N, assume the observed rank r is uniformly sampled from the integers one through N.
Under this assumption the likelihood of observing that exact rank is 1/N when r is no larger than N, and zero otherwise. Larger populations offer more possible rank labels, so each particular compatible label has a smaller likelihood. This is a statement about the specified uniform sampling mechanism, not an automatic consequence of being a person.
Update the base prior by rank
Bayes’ rule multiplies each prior probability by its rank likelihood and normalizes. If r = 50 and the base weights are equal, the unnormalized weights are proportional to 1/100, 1/1000, and 1/10000. Multiplying through by ten thousand gives 100, 10, and 1.
The posterior probabilities are therefore 100/111, 10/111, and 1/111, approximately 0.900901, 0.090090, and 0.009009. All three populations can contain rank fifty; they receive different posterior weights because of the likelihood, not because the larger two are logically impossible.
Raise the rank to 101. The hundred-person hypothesis now becomes impossible. With equal base weights, the remaining posterior probabilities are 10/11 and 1/11. If the rank exceeds ten thousand, every displayed hypothesis is incompatible with the observation, so the posterior is undefined. The experiment reports that contradiction instead of manufacturing a distribution.
Add population weighting before observing rank
The alternative column first multiplies each base weight by N, then normalizes. This models an additional population weighting inspired by the self-indication assumption: hypotheses with more candidate observers receive more weight before the precise rank is used.
Updating these adjusted weights by the same 1/N rank likelihood cancels the population-size factor for every compatible hypothesis. The result is simply the original base prior restricted to totals at least as large as r. With equal base weights and r = 50, all three posterior probabilities are one third. With r = 101, the two compatible populations receive one half each.
This cancellation is an algebraic property of the displayed two-stage model. It does not prove that population weighting is required, or that uniform-rank sampling correctly models self-location. The two columns deliberately disagree about a premise that comes before the Bayesian update.
The reference class is doing work
Who belongs to the population from which an observer is considered sampled? All humans, people sharing some information, all observers, or observer-moments? How are possible worlds weighted? Is there any actual random selection process, or is a sampling principle being used to describe uncertainty about one’s own location?
Different answers can change the hypothesis space, likelihood, or prior weighting. The finite experiment holds most of these choices fixed so that the effect of one change is visible. Its three arbitrary population sizes are teaching inputs, not a data-driven prior. Changing to a continuous or infinite set of hypotheses would require a properly specified measure; an unspecified “uniform prior over all totals” is not available.
Make a prediction
If all prior weight is assigned to a population smaller than the observed rank, should the model move that weight to a larger population automatically?
Explore the answer
No. The observation has zero probability under the specified model. Bayes’ rule cannot normalize zero evidence. The model or prior must be reconsidered explicitly; replacing it silently would conceal a substantive assumption change.
Bostrom’s primer introduces the argument, while Bostrom and Ćirković’s reply concerning self-indication illustrates the dispute over observer weighting. The lesson’s conclusion is conditional: the rank update follows once the sampling and prior assumptions are supplied. Continue to Sleeping Beauty to compare trials with occasions of observation.