On a conservative Bayesian method of inferring extinction

Abstract

Few methods exist that put a posterior probability on the hypothesis that a thing is gone forever given its sighting history. A recently proposed Bayesian method is highly accurate but aggressive, generating many near-zero or near-one probabilities of extinction. Here I explore a Bayesian method called the agnostic equation that makes radically different assumptions and is much more conservative. The method assumes that the overall prior probability of extinction is 50% and that slices of the prior are exponentially distributed across the time series. The conditional probability of the data given survival is based on a simple, long-known combinatorial expression that captures the chance all presences would fall at random before or within the last sighting’s interval (L) given that they could fall anywhere. The same equation is used to compute the conditional probability given extinction at the start of each interval i, that is, the chance that all sightings would fall before or within L given that none could equal or postdate i. The conditional probability is zero for all hypothesized extinction intervals through L. Bayes’s theorem is then used to compute the posterior extinction probability. It is noted that recycling the mean posterior for a population as a prior improves the method’s accuracy. The agnostic equation differs from an earlier, related one, because it explicitly includes a term to represent the hypothesis of survival, and it therefore does not assume that the species has necessarily gone extinct within the sampling window. Simulations demonstrate that the posterior extinction probabilities are highly accurate when considered as a suite but individually indecisive, rarely approaching one. This property is advantageous whenever inferring extinction would have dangerous consequences. The agnostic method is therefore advocated in cases where conservative estimates are desired.