Paleobiology, 42(4), 2016, pp. 670–679 DOI: 10.1017/pab.2016.12


On a conservative Bayesian method of inferring extinction John Alroy


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 aggres- sive, generatingmanynear-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.


John Alroy. Department of Biological Sciences, Macquarie University, New South Wales 2109, Australia. E-mail: john.alroy@mq.edu.au.


Accepted: 11 February 2016 Published online: 5 May 2016


Introduction The scientific literature on the problem of


inferring extinction given a sighting history is littered with dead ends. Most methods are frequentist, as with the pioneering equation of Strauss and Sadler (1989). They have been reviewed elsewhere (Rivadeneira et al. 2009; Marshall 2010; Bradshaw et al. 2012), and because they are not Bayesian they are categori- cally unsuited for setting a coherent extinction probability. Furthermore, various tests of these methods against historicaldata (e.g.,Collen et al. 2010) suggest that they tend to be too aggressive. At thesametime,most olderBayesianmethods make problematic assumptions, such as positing that an extinction could have happened anywhere within a stratigraphic section but not beyond the end of the section (Strauss and Sadler 1989). Meanwhile, a recently proposed Bayesian method (Caley and Barry 2014) requires highly complex calculations. However, a computational algorithm called the “creeping-shadow-of-a-doubt” method


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(CSD) has been shown to be highly accurate under a wide array of circumstances (Alroy 2014). This method uses an iterative Bayesian calculation. At each step from the last sighting it computes a posterior probability as a function of the last interval’s posterior, a fixed per-interval prior extinction probability, and a chance of sampling failure. CSD requires one to assume that extinction is an exponential process and that there is a 50% chance a species would have gone extinct by the time of its last appearance. There are three problems with CSD. First,


although it is very accurate, it tends to produce a relatively narrow range of extinction probabilities and therefore creates a mix of false-negative and false-positive values (Alroy 2014: Fig. 2B). Some might prefer a method that would tend to only rarely produce high extinction probabilities and, therefore, few false positives. Second, CSD could be argued to be a hybrid frequentist–Bayesian method, because it employs a sampling probability


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