ON A CONSERVATIVE BAYESIAN METHOD OF INFERRING EXTINCTION


term that is a simple frequency instead of an agnostically varied free parameter. Finally, the assumptions of the method may seem confus- ing, and readers may specifically object to such ideas as only using the information up to a certain interval to compute a posterior for that interval. Only the first point has any practical consequences, the others being philosophical quibbles, so in this paper I will focus on it. A simple, recently published method (Alroy


2015) sidesteps these concerns and could hardly be more different from CSD. The purpose of the present paper is to document this new method’s behavior using simulations. No claim is made that greater accuracy is obtained with this approach, but readers might consider it as a viable and elegant alternative for studying either individual sighting records or sets of records. Potential applications include inferring historical extinction dates from museum specimen records (Burgman et al. 1995; Alroy 2015), testing for simulta- neous extinctions in the fossil record (Marshall and Ward 1996), and possibly even fixing first appearance dates for clades (Marshall 1999).


Method Let D be a string of sighting and nonsighting


records of a single species that are binned into discrete intervals, let P(E) be the prior prob- ability of extinction, and suppose that P(E)is 0.5 (one could also assume it varies uniformly over a range from 0 to 1, but in this paper I sidestep the issue because simulations demon- strate that it makes no practical difference). By Bayes’s theorem, the posterior chance of extinction is:


PðEjDÞ= PðEÞPðDjEÞ +PðAÞPðDjAÞ PðEÞPðDjEÞ (1)


where P(E|D) = the posterior probability of extinction given the data, P(A) is the prior probability of survival (i.e., 1−P(E)), and P(D|E) and P(D|A) respectively equal the conditional probability of the data given extinction or survival. Alternatively,


PðEjDÞ=1PðAÞPðDjAÞ +PðEÞPðDjEÞ PðAÞPðDjAÞ (2)


671 For consistency with Alroy (2014), the P(E|D)


value for each interval is notated as ε later on. Suppose P(E) is uniformly distributed across


intervals, meaning that each one shares a “slice” of this probability that is the same, and suppose that each interval i has its own separate conditional probability P(D|Ei). The product P(E) P(D|Ei) must be zero for all intervals through the one including the last sighting (L) because we know the extinction hypothesis is false during this time interval. We can then reformat equation (2) (Alroy 2015):


PðEjDÞ =1 PðAÞPðDjAÞ


PðAÞPðDjAÞ + P N


i=L+ 1 PðEiÞPðDjEiÞ Now all we need to do is plug in the


conditionals. These could be computed in various ways, for example, by finding the binomial probability of the data given a sampling probability that could be estimated as the frequency of presences within the species’ known range of existence. However, a simple and clean method is to use combina- torics to find the chance that all of the n sightings are jammed into the range from 1 through L given that there areNintervals and that the species survived until i or (if it is alive) through N. Goodman (1952: 627) provided a good way to calculate such conditionals:


 PðDjEiÞ=


L1 n1





i1 n


(4) On a minor point, also note that we write


i-1 instead of i (as in Goodman [1952] and Höhle and Held [2006]) because extinction is assumed here to have occurred at the base of interval i, not the top. In other words, i is the first interval during which the species is extinct, not the last during which it is alive. Trivially, the conditional probability given that a species is alive is:


 PðDjAÞ=


L1 n1





N n


(5) (3)


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