## Info

Bayes is less well known to biologists, but is becoming more so. Bayes was a British minister born in 1702. His essay on "solving a problem in the doctrine of chances'' was published posthumously and is generally hard to get, so it was republished in 1958 by B/ometr/ka, with a nice historical introduction (Barnard and Bayes 1958). Frequentist and Bayesian statistics differ in both operational (how one does certain kinds of calculations) and philosophical (what exactly one is trying to accomplish) aspects. It is impossible to review the issues here in any comprehensive manner; the articles by Suter (1996), Ludwig et a/. (2001) and Ellison (1996, 2004) and the book by Taper and Lele (2004) point out the variety of issues and towards the primary literature. Here, we shall briefly focus on just one question that allows us to see the difference. When dealing with any kind of statistics, we have both data (D) and hypotheses (H). The approach of classical, frequentist statistics, is to ask questions about the probability of the data, given a hypothesis. That is, formally we compute Pr{D|H}. For example, a standard hypothesis test (with a 5% significance value) asks: what is the probability of obtaining these or more extreme data, given that the hypothesis is true? If this probability is less than 5%, then the hypothesis is rejected (the choice of 5% is arbitrary, but now more or less accepted). The alternative approach is to ask what kind of support the data provide for the hypothesis. Formally, we compute Pr{H|D}. The big problem is that in general Pr{D|H} = Pr{H|D} so that testing a statistical hypothesis often does not give us what we need for scientific understanding. This difficulty has been recognized for a long time (Yates 1951, Royall 1997) but it is only with modern computing that application of the Bayesian approach in general has become practicable. (Another way to think about the difference is that frequentists believe that unknown parameters are fixed and real and that the data are drawn from a distribution of possible observations while Bayesians believe that the data are real and that the unknown parameters are drawn from a distribution.) The second difference between frequentist and Bayesian statistics is how we deal with prior information. In many problems arising in ecology or evolutionary biology, we have such prior information. For example, when managing a fish stock, we may know life history information for similar stocks or the same species elsewhere; when computing an evolutionary tree, we may know something about the relationships between different species in the tree. Bayesian statistics provides a consistent means for dealing with this prior information, while frequentist statistics does not. A general introduction to the Bayesian approach is the book by DeGroot (1970); while old, is still a great read and one of the classic texts in the field. Efron (2005) offers a view for the twentyfirst century. Bayesian approaches are now used extensively in phylogeny and evolutionary biology (Huelsenbeck and Ronquist 2001, Huelsenbeck et a/. 2001) and in fishery management (McAllister et a/. 1994, McAllister and Kirkwood 1998a, b, 1999, McAllister et a/. 2001).

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