Haldane’s Probability of Mutant Survival is Not the Probability of Allele Establishment
Bibliographic record
Abstract
ABSTRACT Haldane notably showed in 1927 that the probability of fixation for an advantageous allele is approximately 2 s , for selective advantage s . This widely known result is variously interpreted as either the fixation probability or the establishment probability, where the latter is considered the likelihood that an allele will survive long enough to have effectively escaped loss by drift. While Haldane was concerned with escape from loss by drift in the same paper, in this short note we point out that: 1) Haldane’s ‘probability of survival’ is analogous to the probability of fixation in a Wright-Fisher model (as also shown by others); and 2) This result is unrelated to Haldane’s consideration of how common an allele must be to ‘probably spread through the species’. We speculate that Haldane’s survival probability may have become misunderstood over time due to a conflation of terminology about surviving drift and ‘ultimately surviving’ ( i.e. , fixing). Indeed, we find that the probability of establishment remarkably appears to have been overlooked all these years, perhaps as a consequence of this misunderstanding. Using straightforward diffusion and Markov chain methods, we show that under Haldane’s assumptions, where establishment is defined by eventual fixation being more likely that extinction, the establishment probability is actually 4 s when the fixation probability is 2 s . Generalizing consideration to deleterious, neutral, and adaptive alleles in finite populations, if establishment is defined by the odds ratio between eventual fixation and extinction, k , the general establishment probability is (1 + k )/ k times the fixation probability. It is therefore 4 s when k = 1, or 3 s when k = 2 for beneficial alleles in large populations. As k is made large, establishment becomes indistinguishable from fixation, and ceases to be a useful concept. As a result, we recommend establishment be generally defined as when the odds of ultimate fixation are greater than for extinction ( k = 1, following Haldane), or when fixation is twice as likely as extinction ( k = 2).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.024 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".