Ecological Archives A013-011-A2

Benjamin Bolker, Toshinori Okuyama, Karen Bjorndal, and Alan Bolten. 2003. Sea turtle stock estimation using genetic markers: accounting for sampling error of rare genotypes. Ecological Applications 13:763–775.

Appendix B. A description of Dirichlet distributions and shape parameters.

The probability distributions involved in the MCMC method described here are multinomial (for the number of turtles in a sample of known size of particular haplotypes or coming from a particular rookery) and Dirichlet (for the frequency of turtles with a particular haplotype or coming from a particular rookery). While the multinomial is probably familiar to most readers, the Dirichlet may not be. A Dirichlet distribution gives the probability density of a particular set of frequencies p1, ..., pn, where $0 \leq p_i \leq 1$ and $\sum_i p_i=1$. The Dirichlet distribution is described by shape parameters; if one starts out completely ignorant and samples N1, ..., Nn individuals, the posterior distribution is Dirichlet(N1 + 1, ..., Nn + 1). The relative magnitudes of the shape parameters control the expected mixture of haplotypes or contribution, while the magnitude of the sum controls the amount of variance around the expected frequencies. Thus the Dirichlet shape parameters for picking a new set of rookery haplotype frequencies, Frh + Irh + Fprh (Table 2, STEP B) can be thought of as a combination of the numbers of turtles of different haplotypes sampled in the rookeries (Frh), the numbers of turtles of different haplotypes in the mixed populations imputed to be from each rookery (Irh), and the prior (Fprh). Similarly, the shape parameters for picking a new set of rookery contributions are a combination of the number of turtles imputed to be from each rookery (Ir) and the prior (Cpr).



[Back to A013-011]