Of priors and prejudice

The methods of Maximum Entropy have been deployed for some years to address the problem of species abundance distributions. It is important to identify correctly weighting factors, or priors, to be applied before maximising the entropy function subject to constraints. The form of such priors depends not only on the exact problem but can also depend on the way it is set up; priors are determined by the underlying dynamics of the complex system under consideration. The problem is really one of statistical mechanics and the properties of the system yield the MaxEnt prior however the problem is framed. Here I calculate, in several different ways, the species abundance distribution resulting when individuals in a community are born and die independently. In the usual formulation the prior distribution for the number of species over the number of individuals is 1/n; the problem can be reformulated in terms of the distribution of individuals over species classes. Results are obtained using master equations for the dynamics and also through the combinatoric methods of elementary statistical mechanics; the priors then emerge a posteriori. The object is not only to establish the log series species abundance distribution as the outcome of per capita guild dynamics but also to illuminate the nature and origin of priors. The exposition is pedagogical.