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Fishery Bulletin 99(1 ) 
er the number of unknown RFs that need to be estimated 
from the available baseline samples, the more reliable the 
estimated baseline multilocus genotype RFs become (Al- 
tham, 1984). Therefore, under Altham’s principle, allele 
RFs are to be estimated if Hardy- Weinberg equilibrium 
holds, and genotype RFs at the locus otherwise. If alleles 
among a linked subset of loci are not inherited indepen- 
dently, the multilocus genotype RFs for the subset have to 
be estimated directly from their observed RFs in the base- 
line samples. 
The search for genetic variation by which to distinguish 
among populations of fish and other marine organisms 
has provided an embarrassment of riches. The numbers 
of mtDNA haplotypes (Epifanio et al., 1995; Bowen et al., 
1996; Rosel et al., 1999) and alleles at minisatellite and 
microsatellite loci (O’Connell and Wright, 1997) can often 
be so large that reliable estimation of their RFs in stocks 
is a concern. Baseline sample sizes are usually limited, 
and the relative precision of estimated RFs for the numer- 
ous haplotypes, alleles, or genotypes (HAGs) declines with 
the magnitude of their RFs. Resulting stock composition 
estimates for the stock mixture may suffer. Grouping or 
binning of HAGs may help to control variation in estima- 
tion of their baseline RFs (O’Connell and Wright, 1997). 
However, for stock-mixture analysis, the practical details 
of grouping so as to balance the loss of information about 
the mixture against the gain in precision of the baseline 
RFs are unresolved. Bayesian methods developed for es- 
timating allele RFs at a locus (sec. 3.7 of Lange, 1997) 
and for estimating cell probabilities in contingency tables 
(Bishop et al., 1975; Sutherland et al., 1975; Leonard, 
1977) offer another attack on the problem. The Bayesian 
methods are applied later in estimating the RFs by using 
genetic similarities of stocks. 
Conditional maximum likelihood does not use the infor- 
mation in the stock-mixture sample to improve the esti- 
mates of baseline multilocus genotype RFs, and the omis- 
sion becomes ever more meaningful with the accumulation 
of mixture individuals from a series of analyses performed 
on stock mixtures of the same baseline populations. The 
unconditional maximum likelihood (UML) (Pella and Mil- 
ner, 1987) or unconditional least squares (ULS) (Xu et al., 
1994) methods have been suggested to remedy this short- 
coming for analysis of a single mixture. For either ap- 
proach, estimates are provided both for the stock propor- 
tions and baseline genotypic RFs by optimizing a criterion 
of fit to counts in both the baseline and stock-mixture sam- 
ples. However, the fitting criteria may have local optima 
(Smouse et al., 1990), and effective search for the global 
optimum and corresponding estimates from both methods 
is unresolved. A practical compromise for either method is 
to find a particular local optimum by starting the search 
from the CML estimate of stock proportions and stock ge- 
notypic RFs, the latter evaluated from the baseline sam- 
ples alone. 
None of the past approaches — CML, UML, or ULS — 
makes use of the genetic similarities among stocks to es- 
timate the relative frequencies of haplotypes, alleles, or 
genotypes more accurately in the separate stocks. Com- 
mon HAGs are shared universally among stocks; HAGs 
with moderate RFs are shared at least regionally, and rare 
HAGs occur only sporadically. Instead, the similarities in 
HAG RFs are viewed solely as limiting success in distin- 
guishing the origins of the stock-mixture individuals. Im- 
proved estimates of HAG RFs, to replace simple observed 
values, would generally benefit accuracy and precision of 
stock composition estimation, especially as the number of 
rare or uncommon HAGs increases (e.g. Xu et al., 1994). Es- 
timation of RFs for rare HAGs in separate stocks from base- 
line samples is especially problematic. Even when present 
in a population, they may well be absent from the baseline 
sample. The Bayesian proposal for stock-mixture analysis 
will shrink the observed baseline HAG RFs of individual 
stocks toward better-established grand, regional, or group 
means in order to control HAG RF estimation error. 
All past approaches — CML, UML, or ULS — produce es- 
timates of stock proportions that become increasingly bi- 
ased as the true stock-mixture proportions become more 
uneven (Pella and Milner, 1987; Xu et ah, 1994). Contribu- 
tions from abundant stocks are underestimated and those 
from less common or even absent stocks are overestimat- 
ed. No effective general solution for this bias has been pro- 
posed. The Bayesian proposal results in a probability dis- 
tribution for the stock composition estimates, the location 
of which can be characterized by various measures, such 
as the mean, median, and mode, which differ in their bias 
when viewed as potential point estimators. 
Finally, the previous estimation methods appear limit- 
ed in capacity to attack practical problems that fail to fit 
the standard mold of a sampled stock mixture and com- 
plete baseline. In particular, missing information for en- 
tire stocks from the baseline is very difficult to accommo- 
date (Smouse et al., 1990). Despite their availability for 
a decade or more, nothing has been accomplished using 
these methods to incorporate genetic similarities of base- 
line stocks to deal more realistically with missing data. 
The Bayesian proposal will initially fill in missing base- 
line HAG RFs with appropriate grand, regional, or group 
means, proxies that are revised later during analysis of 
the stock-mixture sample. 
Bayes methods have the potential to correct for these 
shortcomings better than the likelihood or least squares 
methods. In our study we describe the rationale for this 
new approach to stock-mixture analysis, develop the sta- 
tistical models, and outline the numerical algorithms by 
which to quantify uncertainty in stock proportions of 
the mixture as well as in the baseline HAG RFs in the 
separate contributing stocks. Software developed for per- 
forming the computations and summarizing results is 
available at our anonymous ftp site, with address ftp:/ / 
wwwabl.afsc.noaa.gov / sida / mixture-analysis / bayes. Two 
applications with special difficulties are used as illustra- 
tions. First, a winter stock mixture thought to be composed 
of four Northwest Atlantic harbor porpoise ( Phocoena pho- 
coena) populations is assessed. These porpoise populations 
are characterized by mtDNA haplotypes, the number of 
which exceeds baseline and stock-mixture sample sizes. 
Second, a Southeast Alaska steelhead trout (Oncorhynchus 
my kiss) stock mixture is resolved to two populations, only 
one of which could be sampled separately. Allozymes, mi- 
