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Fishery Bulletin 99(1) 
resulting posterior marginal distribution. Its standard de- 
viation roughly equaled threefold, and its posterior interval 
length, twofold, that from allozymes or microsatellites. 
Discussion 
Analysts, accustomed to using likelihood or least squares 
methods for stock-mixture problems, should not be deterred 
by the novelty of the proposed Bayes method. Instead, the 
Bayesian implementation should be seen as eminently prac- 
tical and sensible. The data augmentation algorithm rec- 
ognizes each mixture individual as an entity and labels 
it with a stock origin. Given the stock assignments, the 
observed stock proportions are obvious estimates of the 
stock composition. In concurrence, the means, variances, and 
covariances of the Bayes posterior distribution for stock pro- 
portions approximate closely the observed stock proportions, 
their estimated variances, and their estimated covariances, 
respectively, from frequentist methods. Given the current 
stock proportions and genetic parameters of an MCMC 
chain, the random labeling accurately reflects the uncer- 
tainty in stock origins. Each mixture individual is assigned 
to one of the baseline stocks, by using probabilities of stocks 
proportional to each stock’s contribution of its genotype to 
the mixture. Stock proportions and genetic parameters of 
the MCMC chain gravitate toward their true values because 
draws from the posterior, which integrates the baseline and 
stock-mixture information, are more probable nearby. 
One goal in developing a Bayesian method of stock-mix- 
ture analysis was to replace the conditional maximum like- 
lihood assumption of ignorable baseline sampling error by 
modeling that acknowledged the uncertainty in genetic com- 
position of the baseline stocks. Ignorable baseline sampling 
error is especially unrealistic in applications for which un- 
common genotypes are present. Stock-mixture individuals, 
particularly those with uncommon genotypes, may be con- 
tributed by stocks whose baseline samples imply their ab- 
sence. Current bootstrap resampling of the baseline samples 
does not accommodate reasonably this mismatch between 
stock-mixture presence and apparent absence of a genotype 
in a baseline stock. The individual is presumed to come 
from a stock different from that of the contributor. Such mis- 
matches become frequent when many rare and uncommon 
genotypes occur, such as in mtDNA data. The simulations 
for harbor porpoise showed that as greater numbers of rare 
haplotypes occurred in the populations, the bias of the CML 
estimator became severe. When none of the baseline sam- 
ples can explain presence of a stock-mixture genotype, the 
CML assumption leaves only an outside source. Data sets 
generated during bootstrap resampling can require an ap- 
parent outside source even when the original samples did 
not. Pooling of uncommon types to circumvent their effects 
on estimation should be preceded by careful study to assure 
information useful to stock-mixture composition is not lost. 
Some potential to improve stock-mixture assessment by 
Bayesian methods arises from the prior for 9=(p, Q), which 
has no counterpart in the likelihood approach. The Bayes 
proposal for stock-mixture analysis emulates the objectiv- 
ity of likelihood methods by letting stock-mixture sample 
information dominate that of the neutral low-information 
prior for stock proportions. If information about p is truly 
unavailable, or the researcher prefers to withhold it and 
let the “data do the talking,” the neutral low-information 
prior will be adequate. However, the resulting composition 
estimates may be so imprecise as to be of limited practi- 
cal value. If additional information is available, either cus- 
tomizing the prior to include it, or updating the posterior 
(it becomes the prior) with the additional information may 
improve precision. As an example of updating, the three 
independent data sets for Sashin Creek steelhead trout 
could be integrated sequentially into a single posterior for 
population proportions. 
In attempting to maintain objectivity for description of the 
uncertainty in genotypic composition of the separate stocks, 
the empirical Bayes method for specifying the baseline prior 
parameters was examined. The empirical Bayes method for 
choosing prior parameters for the haplotype, allele, or geno- 
type RFs provided excessive weight to the prior mean for 
harbor porpoise, with the prior “sample size” parameter, /3., 
for harbor porpoise of nearly 2000, many-fold the total of 
actual sample sizes. In addition, the empirical Bayes meth- 
od consistently weighted prior means heavily on several ap- 
plications examined and not reported. Information in these 
typical baseline samples is evidently inadequate for estima- 
tion of the prior parameters. A full Bayesian analysis, which 
views them as random variables (sec. 5.3 of Gelman et al., 
1995), would require an informative prior for them. Because 
such an informative prior was not evident, the pseudo-Bayes 
approach (Bishop et ah, 1975) was adopted. 
Under the pseudo-Bayes approach, the posterior mean 
for HAG RFs of stocks interpolates between observed val- 
ues for individual stocks and a baseline central value for all 
stocks, with the shrinkage, or weighting, determined by the 
values of the baseline prior parameters. The best choice for 
values of the prior parameters remains an open question. 
Possibly, the prior parameter values could be chosen for 
their performance in experiments of simulated stock-mix- 
ture analyses. However, the computations involved would 
be extensive and without guarantee beforehand of a clear 
solution. This proposal included an objective criterion — 
minimum squared-error risk of baseline allele RFs — by 
which to determine weighting between the prior and ob- 
served HAG RFs from the baseline samples alone. Re- 
searchers who find choice of weighting a deterrent to ap- 
plication of the Bayes method can set the baseline prior 
parameters to zero with the qualification that variation of 
stock proportions may be understated as with the CML 
method. In return, the Bayes algorithm easily includes the 
information in the stock-mixture and baseline samples in 
assessing stock proportions. The simulations for harbor 
porpoise showed that the weighting from the pseudo-Bayes 
method resulted in good frequency performance. 
In many practical applications, fisheries managers re- 
quire a point estimate of stock composition. The well- 
known bias of the conditional maximum likelihood esti- 
mate has been troublesome for this reason. Any corrections 
for its bias have referred estimated stock proportions to 
simple one-dimensional graphical relationships between 
simulation averages and known stock proportions. The 
