ON A CONSERVATIVE BAYESIAN METHOD OF INFERRING EXTINCTION
temporal range of a species is believed to fall within or beyond the sampling win- dow. If one thinks not only that the extinction process operated over this entire range but that it began before the first sampling event, then a reasonable approach based on the Copernican anthro- pic principle would be to halve the overall extinction prior and recompute the interval-by-interval priors accordingly.
2. When the data are historical instead of paleontological, there may be reason to believe that extinction rates were near- zero prior to some year (i.e., that there is a “critical range” starting at that year during which rates were nontrivial). If the date precedes the sampling win- dow, meaning there are no observations that old, then the exponential extinction rate μ can be fixed accordingly. For example, if the oldest sighting went back to 1951 but it is believed that extinction processes were operating from 1901, then the per-interval prior for an observation window ending at 2000 would be a function of log(0.5)/100 instead of log (0.5)/50.
3. Unlike the situation with CSD, here it is technically possible to analyze data for species sighted only once. I assume read- ers agree it would be meaningless to do so. In simulation, adding in species sighted once has no visible effect on the rate of inferring extinction with high certainty but does increase the average extinction probability so that the averages are no longer accurate. Meanwhile, a verbal case could be made for using a cutoff of three instead of two. However, imposing higher cutoffs has the opposite result, progressively downward biasing the results relative to the true extinction tallies. The reason is that ignoring rarely sighted taxa means preferentially excluding ones that go extinct quickly. Therefore, anyone carrying out a study of many species really should stick to a two- sighting lower limit.
4. We may have independent information that no sightings could have been made in some intervals in the first place. For
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example, in a paleontological context there may be no relevant rock record, and in a conservation context there may have been no collecting at the right time in the right place. Equations (4) and (5) can be modified appropriately by decreasing L and either i or N. The prior P(A) presumably should be left alone.
5. There may be multiple sighting opportu- nities within some intervals (e.g., because sightings are being tracked separately in different geographic regions; Alroy [2016]). In such cases, the counts can be increased without complicating the math any further. For example, if the sighting opportunity record is 1–0–2–1–1 and the sighting record is 0–0–2–0–0, then N=5, n=2, and i=5 or 6, depending on which conditional is being calculated.
The agnostic equation does, however, fail
to handle one potential bias: clustering of sightings that does not have to do with whether sampling opportunities seem to exist. The distinction is subtle. Clustering could reflect, for example, spatiotemporal concentra- tion of sampling effort that is not reflected in the data used to establish whether sampling is possible (e.g., sightings of related species in the same general region). In any case, this problem often results in runs of consecutive sightings or of consecutive nonsightings. CSD resolves the clustering problem by computing separate sampling probabilities that are conditional on whether a sighting has been made immediately before the interval at hand. This round goes to CSD, so if clustering is severe and cannot be related to an objective indication of sampling opportu- nity, then the agnostic method might not be the best choice.
Examples Working through a simple example based on
uniform priors (instead of exponentially distributed priors) might help to make the general methodology more understandable. Suppose the sighting history is 1–0–1–0. Here, N=4, L=3, and i=4. By equation (5), the conditional probability of obtaining this pattern based on the hypothesis of survival is 1/3.
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