Pella and Masuda: Bayesian methods for analysis of stock mixtures from genetic markers 
165 
simulations for harbor porpoise showed that the Bayes- 
ian posterior mode had considerably less bias in situa- 
tions for which the CML estimate was severely biased. The 
mode has intuitive appeal as the most frequent estimate 
from the posterior. Its promise for situations requiring 
point estimates needs to be explored by simulation in fur- 
ther applications. In addition, computation of the multi- 
dimensional mode requires some smoothing of the poste- 
rior samples, such as the binning used here (see footnote, 
Table 1). Stock grouping, followed by summing of individ- 
ual stock proportions for group totals, may also be neces- 
sary because finding posterior modes becomes more prob- 
lematic with large numbers of stocks. 
The Dirichlet distribution was the basis for probability 
modeling because it is a natural choice. First, it is defined 
for random compositions (i.e. arrays with nonnegative 
elements that sum to one). Second, the posterior for 
multinomial data can be written in closed form and is 
also Dirichlet. Third, the prior parameters can be inter- 
preted as additional data. Fourth, and last, it is easy to 
sample by computer. Flowever, compositional data are al- 
so nicely modeled with the logistic normal density (Aitchi- 
son, 1986; Billheimer et al., 1998), whose flexibility and 
relation to normal theory may have advantages in stock- 
mixture analysis. Use of geographical structure for the 
stock proportions in complex stock mixtures comprising 
many stocks is an area for exploration with the logistic 
normal and Bayesian hierarchical methods. 
Acknowledgments 
We much appreciated the opportunity to examine the ge- 
netic samples of harbor porpoises and steelhead trout. The 
harbor porpoise samples were made available by Patricia 
E. Rosel of the National Ocean Service, NOAA, 219 Fort 
Johnson Road, Charleston, SC 29412, and the steelhead 
trout samples were made available by Frank Thrower, 
National Marine Fisheries Service, Auke Bay Laboratory, 
Juneau, AK 99801-8626. The manuscript was revised after 
many insightful comments received from colleagues. Our 
special thanks for their reviews and ideas go to Eric Ander- 
son of the University of Washington, Joel Reynolds of the 
Alaska Department of Fish and Game, Peter Smouse of 
Rutgers University, Jim Murphy and Richard Wilmot of 
the Auke Bay Laboratory, and three anonymous reviewers 
of the Fishery Bulleti n. 
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