Determining whether a species has gone extinct is a central problem in both paleobiology and conservation biology. Past literature has mostly employed equations that yield confidence intervals around the endpoints of temporal ranges. These frequentist methods calculate the chance of not having seen a species lately given that it is still alive (a conditional probability). However, any reasonable person would instead want to know the chance that a species is extinct given that it has not been seen (the posterior probability). Here, I present a simple Bayesian equation that estimates posteriors. It uninterestingly assumes that the sampling probability equals the frequency of sightings within the range. It interestingly sets the prior probability of going extinct during any one time interval (E) by assuming that extinction is an exponential decay process and there is a 50% chance a species has gone extinct by the end of its observed range. The range is first adjusted for undersampling by using a routine equation. Bayes' theorem is then used to compute the posterior for interval 1 (ϵ1), which becomes the prior for interval 2. The following posterior ϵ2 again incorporates E because extinction might have happened instead during interval 2. The posteriors are called “creeping-shadow-of-a-doubt values” to emphasize the uniquely iterative nature of the calculation. Simulations show that the method is highly accurate and precise given moderate to high sampling probabilities and otherwise conservative, robust to random variation in sampling, and able to detect extinction pulses after a short lag. Improving the method by having it consider clustering of sightings makes it highly resilient to trends in sampling. Example calculations involving recently extinct Costa Rican frogs and Maastrichtian ammonites show that the method helps to evaluate the status of critically endangered species and identify species likely to have gone extinct below some stratigraphic horizon.
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