Pella and Masuda: Bayesian methods for analysis of stock mixtures from genetic markers 
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crosatellites, and mtDNA were available for independent 
and confirming assessments of stock-mixture proportions 
from the two populations. 
Methods 
The premise of the Bayes method for estimating an un- 
known quantity, 6, is that some information about 6 is 
available before sampling begins. This information is in 
the form of a prior probability density, JiiO). After sam- 
pling, the new data obtained, Y , are used to revise the 
prior to the posterior probability density for the unknown, 
M6 \ Y). The posterior is obtained by application of Bayes’s 
theorem, which states that the posterior is proportional to 
the product of the prior and the likelihood of the sample, 
7riF| 6), viz. 
7r(0 1 Y ) - k ''/r(0)/r(Y | 0), (1) 
where k = f 7T(0)/r( Y | O)c/0. 
6 
Once the posterior for 6 is known, a variety of point esti- 
mates — mode, median, or mean — as well as the Bayesian 
posterior probability interval (the apparent counterpart 
of frequentist confidence intervals, but which actually is 
a direct probability statement about the unknown) for 9 
can be derived from it. In stock-mixture analysis, the un- 
knowns separate into two evident blocks [9 = (p,Q)]: 1) 
the stock proportions of the mixture, p, and 2) the param- 
eters, Q , needed to determine the genetic composition — 
stock-mixture haplotype or multilocus genotype RFs — of 
the baseline stocks. For haploids, Q represents the base- 
line haplotype RFs. For diploids, Q represents the array 
of baseline allele and genotype RFs that are needed under 
Altham’s principle to compute the stock-mixture genotype 
RFs in the baseline stocks. The new information comes 
from the baseline and stock-mixture samples for which 
the likelihood functions are unchanged from earlier likeli- 
hood methods. The stock-mixture sample provides counts 
of haplotypes or multilocus genotypes, and the baseline 
samples provide counts from the separate stocks of the 
haplotypes or alleles and genotypes at the loci comprising 
the mixture multilocus genotypes. 
The standard stock-mixture analysis for complete base- 
line and stock-mixture samples by Bayes methods will be 
outlined here, with details given in following sections. Ex- 
tension to nonstandard applications will be indicated by 
example. First, a prior for 9 = (p,Q ) is developed, which is 
a product of block priors for its components,/) and Q The 
prior proposed for p, which will be called “uninformative,” 
allows any substantive stock-mixture sample information 
regarding p to overwhelm that from the prior. The prior 
for Q , used to analyze the stock-mixture sample, is infor- 
mative and will be derived from the baseline samples to 
quantify uncertainty in the genotypic composition of the 
contributing stocks. Second, the standard likelihood func- 
tion for the haplotype or multi locus genotype counts seen 
in the stock-mixture sample is described. Third, and last, 
the data augmentation algorithm, a Gibbs sampler, is used 
to alternately generate a sequence of samples from the 
posterior distributions for p and Q. The stock identities 
of the mixture-sample individuals are reassigned at each 
sampling cycle by using a chance mechanism that reflects 
their uncertainty. The stock identities simplify greatly the 
revision of the prior distributions for p and Q to account 
for the stock-mixture sample information; just as with the 
baseline samples, counts of mixture individuals and their 
HAGs by stock are available at each cycle. Assignment of 
individuals to stock origin contrasts with their fractional 
allocation by the CML method (Pella and Milner, 1987). 
These samples from the posterior distributions are used to 
quantify the final uncertainty in p and Q after observing 
the stock-mixture sample. 
Prior for stock-mixture proportions, nip) 
The prior for p can incorporate information about the 
stock-mixture composition other than that in the stock- 
mixture sample if such is available. More commonly, how- 
ever, such information is either unavailable or else the 
researcher prefers to let that of the stock-mixture sample 
dominate, just as it does with the earlier likelihood or 
least squares methods. Therefore, the prior proposed will 
be restricted, providing no useful information about the 
stock-mixture composition. Such an uninformative prior 
for the stock proportions of a c-stock mixture must be 
defined over the stock composition simplex, 
S(p) = Ip: 0 < p, < 1, ^ p, = 1 j, 
and have negligible effect on the posterior distribution. 
The Dirichlet probability density can accommodate these 
requirements, and it is natural to use it as a prior with 
compositional count data both for computational conve- 
nience and for its interpretation as additional data. Prior 
draws of p from the Dirichlet probability density, 
a , > 0,i - l,...,c, 
have means, variances, and covariances given by 
E(p ; ) = a , / a 0 , var( p, ) = «,(«„ - or, )/ («,;(«„ + D), 
cov(p,,p ; ,) = -a i a l . /(a'g(a 0 + - 1,2,... ,c, and (3) 
«n = 
/= 1 
