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Appendix 1 — The empirical Bayes method for 
maximum likelihood estimation of the Dirichlet 
prior parameters from the prior predictive 
distribution (adapted from Lange, 1997). 
If the allele RFs at a locus for the ith stock, q | =(q il , . . . , 
q iT )', are distributed as the Dirichlet density, D(q i \ /3 ; ,/3 2 , 
. . . , (i T ), and the allele counts, y i =(y il ,...,yi T )', in a random 
sample of nj alleles have the multinomial distribution, 
Multfn^q,), then the prior predictive distribution (Gelman 
et al.. 1995) for the allele counts, obtained by integrating 
the product of the probability distributions, Multfn^q) 
and D(q t \ /3 ; ,/3.„ . . . ,/3 r ), over the simplex, S(q t ), is 
Estimation of the /3’s begins with the logarithm of Equa- 
tion 1, which can be viewed as the ith component of the 
loglikelihood or support function for the unknown /3’s, 
LogL,(/3„...,P T ) = 
log 
V Yii- ■ ■ ■ Yit- ) 
+ log f(/3.) - log r(?;, + /3.) + 
(2) 
i 
X(log r(y„ + /3, ) - log R/3,)). 
/=1 
The total support function for all baseline samples is the 
sum of the individual support functions at Equation 2, 
FT - n/3.) 
1 1 Q„ <?„ 
n r( A } 
/-I 
n,\ n/3.) T“f r(y j( + /3, ) 
y (1 !...y, T ! r(n,+/3.)ll F(/3,) 
n 
where S(q, ) = \ q,:0 < q„ < 1, ^/1„ 
= 1 
( 1 ) 
LogUl 3i,...,/3 r ) = I LogL i (/3j,...,/3 r ). 
1=1 
The score function for /3,, s t , is the first derivative of the 
total support function, 
dLogL _ Y’ dLogL j 
cy/(/3.)-^y/(?2 ; + /3.) +^(i//(y„ + /3, )-</d/3, )), 
(Lange, 1997). 
(3) 
