672


JOHN ALROY Höhle and Held (2006) went on from


Goodman’s equation to present a formula that yields each interval’s posterior extinction prob- ability given that the species is extinct, period (because they envisioned examining an infinite series of observations). Equation (3) (Alroy 2015) additionally incorporates a term that captures the hypothesis of survival up to the end of the observation period, that is, the product P(A)P(D|A) that includes the 0.5 prior. So, it builds in complete uncertainty about whether extinction has happened in the first place. I therefore suggest calling it an agnostic extinction equation. The method as described to this point


assumes a uniform distribution of extinction probabilities through time, which is simply not realistic. Instead, if all other things are equal, then it makes sense to assume that extinction is an exponential process (as also assumed by Alroy [2014]). For example, if half of the population dies each year, then we expect one-quarter to survive 2 years and therefore one-quarter to have died in the second year; one-eighth to survive 3 years and therefore one-eighth to die in the third year; and so on. Switching to an exponential model only requires treating the P(Ei) term in equation (3) as a changing variable that depends on the time interval number i. Specifically, the function is:


PðEiÞ=exp½μði1ÞμexpðμiÞ (6)


where μ=the per-interval instantaneous extinction rate, which is log(0.5)/N on the aforementioned assumption that there is a 50% chance of surviving to the end of the observa- tion period. Note that if some other prior is used, then μ=log[P(A)]/N, not log(0.5)/N. Another concern is the fact that when real


extinction rates are minuscule, employing a 0.5 value for the overall prior P(E) creates a strong upward bias in the P(E|D) posteriors. There is no real solution to this problem if a single species is being analyzed. However, if the data set consists of a substantial popula- tion, then one possibility would be to compute a mean posterior for the population and then recycle the mean as a new prior and recompute all the individual posteriors using it


(Alroy 2015). As shown later, when the rate is very close to zero, recycling produces much better results than using a 0.5 prior. There is also a noticeable improvement when extinction rates are moderate, which is the case inmost of the simulations. Solow (2016) also developed an equation


that eliminates sampling probabilities by conditioning on the number of sightings. His method and mine (Alroy 2015), which were developed independently, both stem from the work of Goodman (1952) and Höhle and Held (2006). However, Solow’s method is biased and unrealistic because of the way its prior is formulated (Alroy 2016), and so it does not merit detailed discussion. Because the agnostic method and CSD are


both Bayesian and both simple, there is some chance of assuming that they are somehow related on a mathematical level. It must be emphasized that they are not. First, obviously, the new equation involves combinatorics instead of sampling probabilities. Second, the computa- tion is not iterative: equation (3) is applied separately to each interval, and no attention is paid to the values for the other intervals. Third and last, the extinction prior is 0.5 for the entire time series of N intervals, instead of being 0.5 at interval L. In practice, because intervals before L have zero priors, the actual overall prior is (N − L)/2N. For example, if L=10 and N=10, then the overall prior is zero for every interval based on this newmethod, but 0.5 based onCSD (despite the fact that the posterior is zero according to both methods). Calculations for cases where N > L are highly complicated, but regardless, the values produced by the two methods still come out differently. In sum, there is no connection whatsoever


between the agnostic and CSD methods except that they are Bayesian and highly accurate. The method of Solow (2016) is actually more related on a conceptual level. That said, CSD has a number of advantages over all other methods with respect to the way it handles complexities in the data (Alroy 2014). Most of these advantages are also shared by the new method.


1. When CSD is used, the prior extinction rate depends on whether the entire


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