151 
Bayesian methods for analysis of stock mixtures 
from genetic characters 
Abstract— An implementation of Bay- 
esian methods to assess general stock 
mixtures is described. An informative 
prior for genetic characters of the sep- 
arate stocks in a mixture is derived 
from baseline samples. A neutral, low- 
information prior is used for the stock 
proportions in the mixture. A Gibbs 
sampler — the data augmentation algo- 
rithm — is used to alternately generate 
samples from the posterior distribution 
for the genetic parameters of the sepa- 
rate stocks and for the stock proportions 
in the mixture. The posterior distribu- 
tion incorporates the information about 
genetic characters in the baseline sam- 
ples, including relatedness of stocks, 
with that in the stock-mixture sample 
to better estimate genotypic composi- 
tion of the separate stocks. Advantages 
over usual likelihood methods include 
greater realism in model assumptions, 
better flexibility in applications, espe- 
cially those with missing data, and 
consequent improved estimation of 
stock-mixture proportions from the con- 
tributing stocks. Two challenging appli- 
cations illustrate the technique and its 
advantages. 
Manuscript accepted 13 September 2000. 
Fish. Bull. 99:151-167 (2001). 
Jerome Pella 
Auke Bay Laboratory 
Alaska Fisheries Science Center 
National Marine Fisheries Service, NOAA 
1 1305 Glacier Hwy. 
Juneau, Alaska 99801-8626 
E-mail address: Jerry.Pella@noaa.gov 
Michele Masuda 
Auke Bay Laboratory 
Alaska Fisheries Science Center 
National Marine Fisheries Service, NOAA 
11305 Glacier Hwy. 
Juneau, Alaska 99801-8626 
Fisheries that exploit mixed stocks are 
very common, and their management 
oftentimes requires assessment of com- 
position of the mixed catches (Begg et 
ah, 1999). Multilocus genotypes of fish 
are a natural tag by which to infer 
their origins. The unknown proportions 
from stocks comprising a stock mix- 
ture, or its stock composition, can be 
estimated from genotype counts in a 
random sample of the stock-mixture 
individuals if relative frequencies (RFs) 
of the genotypes vary among the con- 
tributing stocks. Larger differences in 
genotypic RFs among stocks result 
in more accurate and precise stock 
composition estimates. The conditional 
maximum likelihood (CML) method 
(Fournier et ah, 1984; Millar, 1987; Pella 
and Milner, 1987) has most commonly 
been used for stock-mixture analysis. 
Baseline samples drawn from the sepa- 
rate contributors are used in estimating 
the RFs of the observed stock-mixture 
genotypes in each stock. The CML stock 
composition estimate maximizes a like- 
lihood function of the stock-mixture 
genotypes as if their RFs in the base- 
line stocks were known without error. 
The baseline multilocus genotype RFs 
determine the outcome of a stock-mix- 
ture analysis. Larger errors in these 
estimated RFs result in larger stock 
composition errors. Usually the vari- 
ation in CML stock composition esti- 
mates from baseline and stock-mixture 
sampling is evaluated by the bootstrap 
method. 
The estimation of the baseline mul- 
tilocus genotype RFs depends on the 
mode of inheritance of the observed 
markers. Among molecular markers de- 
veloped for fish, allozymes, mitochondri- 
al DNA (mtDNA), minisatellite DNA, 
and microsatellite DNA are widely 
known for their utility in stock-mixture 
analysis. For mtDNA, the entire hap- 
lotype passes as a unit from female to 
offspring and the baseline multilocus 
haplotype RFs are estimated directly 
by their observed RFs in the baseline 
samples. For allozymes, minisatellite 
DNA, and microsatellite DNA, the mul- 
tilocus genotypes pass from parents to 
offspring under the usual rules of dip- 
loid inheritance. The expected multilo- 
cus genotype RFs for diploids equal the 
products of the genotypic RFs at the in- 
dividual loci (or subsets of them) that 
pass independently from parents to off- 
spring. In the special case when Hardy- 
Weinberg equilibrium holds at a locus, 
its expected genotypic RFs are deter- 
mined by its allele RFs; the homozy- 
gote RFs equal the squares of their 
allele RFs and the heterozygote RFs 
equal twice the product of their allele 
RFs. To compute the estimated base- 
line multilocus genotype RFs for dip- 
loids, observed RFs of alleles or geno- 
types in the baseline samples replace 
corresponding unknown RFs. The few- 
