674
JOHN ALROY
By equation (4), the conditional based on assuming extinction at the base of interval 4 is 2/3. The survival prior is 0.5 by definition, and the extinction prior is 0.5×0.25=0.125 (the 0.25 term reflects the assumption of a uniform distribution and the fact that there are four intervals). By Bayes’s theorem, the extinction posterior is therefore 1/3. In other words, the extinction hypothesis has a higher conditional (because it explains why both sightings are jammed into three intervals) but a lower poster- ior (because the prior is somuch lower). Note that a maximum likelihood inference
based on the conditionals would fail here completely, because hypothesizing extinction implies a conditional probability of the sam- pling pattern that is twice as high (namely, 2/3 vs. 1/3). Thus, extinction would be the clearly favored hypothesis. This style of inference has nothing better to go on. Likewise, in a classical frequentist framework we can neither reject the hypothesis of survival (because p=1/3) nor the hypothesis of extinction in interval 4 (p=2/3). Perversely, all such p-values would be tiny in a real-world case where the number of intervals was large—leading to the firm conclusion that a species had neither survived nor gone extinct! A frequentist would counter that one would instead only examine one of the two hypotheses. But which hypothesis? Based on such considerations, it should be clear that Bayesian methods are more powerful than the alternatives when it comes to inferring extinction. To get a better sense of how the new
equation actually works and how it differs from CSD, it may be helpful to look at a small exemplary data set. Figure 1 shows posterior probabilities based on the dodo sighting record reported by Roberts and Solow (2003). Three differences between the current and earlier methods are apparent. First, the new equation yields posteriors that
initially climb almost linearly before plateauing, whereas CSD suggests a sigmoidal shape. The former patternmight be more intuitive, because we expect that if a species is really extinct, then the chance it was extinct in a given year should not be particularly low at the start. Or it might not: I leave the matter of intuitiveness to the reader. Regardless, the pattern stems directly
FIGURE 1. Performance of the agnostic equation and of CSD (Alroy 2014) when applied to the dodo sighting record reported by Roberts and Solow (2003). Small boxes show the sighting years; black line shows the agnostic posteriors; gray line shows the CSD posteriors.
from the prior assumption of an exponentially distributed extinction probability (andwould be much the same on the assumption of a uniform distribution). Second, the new method is more conserva-
tive at high values. For example, it exceeds the 0.95 level in the year 1732, whereas CSD hits this point in 1715. The equation’s conservative tendency is put forth in this paper as its main advantage. Third, both methods yield a “best guess”
value of 0.5 at about the same time: the newone reaches this threshold in 1690 and the old one in 1692. Interestingly, Roberts and Solow (2003) also reported a best guess of 1690. Their figure is not truly comparable, because it represents the 0.5 confidence limit and is therefore a condi- tional probability instead of a posterior. The coincidence is nonetheless intriguing.
Simulations Although it is interesting to see how the
agnostic equation works, the real question is whether it works. We will therefore turn to a simulation algorithmclosely modeled on that of Alroy (2014), which demonstrates the new method’s accuracy with respect to computing
posteriors.Bydefault, there are 50 time intervals, the per-interval extinction probability E is 0.05, and the per-interval sampling probability ps is 0.50. The initial cohort is 1000 species, and it is assumed that all 1000were sampled at time 0.
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