Pella and Masuda: Bayesian methods for analysis of stock mixtures from genetic markers 
167 
where y/(w) = log HiM is the digamma function. 
dw 
The elements of the information matrix are 
d 2 LogL 
dPM 
-c !//■'( /h) + t i/'(n, + /).)- 
c 
PP-wW), 
(4) 
where the Kronecker delta is defined as 8 tu = 0 if t*u, 8 lu =l 
if t=u, and 
d u t— 1, 2, . . . , T, (6) 
/=1 
1 is the unit column vector of T “l”s, and 
c 
b = ct//'()3.) - y + /3.) is a scalar. (7) 
A quasi-Newton search for the maximum likelihood esti- 
mate of P=(/3 X , j3 2 ,...j3 7 4' is performed. At the Mh step 
P ( * + i> = P <,;) + [D 1 * 1 -6 ( * ) ir]‘ 1 S 1 *’, (8) 
y/'(w) = —5— log Hh;) 
<5m 
is the trigamma function. The observed information ma- 
trix, or negative Hessian of the total support function, can 
be written as 
' d 2 LogL ' 
D-611', 
(5) 
where p lk) = the approximation of the maximum likeli- 
hood estimate at the Mh step, 
D< k > - denotes the matrix D at Equation 6 when 
evaluated at P' kl , 
folk) _ denotes the minimum of either the scalar, b, 
at Equation 7 when evaluated at P' k) or the 
ratio, (l-e)/|l'(D (k) ) _1 1] with s being an arbi- 
trary constant in (0,1), and 
S lk) = the vector of scores at Equation 3 evaluated 
at P lk) . 
where D is a diagonal matrix with main diagonal ele- An arbitrary choice for P' 11 such as the unit column, 1, can 
ments be used to start the search. 
Appendix 2 — Minimum squared-error risk estimate 
of J3. with the prior mean fixed (an extension of sec. 
12.2.3 of Bishop et ai., 1975). 
Let the baseline risk criterion be the expected value of the 
squared distance of any matrix estimator of baseline RFs 
from the true values, 
c 1 
R( Q,Q) = y ^ n ,E(q„ ~q„f 
Q, Q ) — 
2 T 
1 -X</( 
V ^=i / 
^{n,+P- 
1 
Y. 
The value of /3. that minimizes the risk is found by set- 
ting the derivative of the risk function (with respect to /3.) 
equal to zero and solving. The minimizing value of /3. must 
satisfy the following equation: 
Denote the random version of the posterior mean of q | y 
by 
q, =q |(,S.,A) = 
n, + /L 
fi- 
n , + P- 
where y ; = (y iv y i2 , ■■■ , y lT Y = the array of sample allele 
counts from the zth stock, 
X = the baseline center, y=( y v 
y 2 . • • • y?Y, 
•li = (< 7,n 9,2- • •• . <7 iT Y and 
y, — — y, t / n t , t — L 2, . . . , T, = the arithmetic average of 
c (=1 the observed RFs of the 
tth allele among stocks. 
With p. and A viewed as fixed, the baseline risk is 
P- = 
The equation includes the unknown RFs, q t , and the 
observed RFs are substituted for their unknown values 
for estimation. The equation can be iterated to solve for 
the optimal p.. Beginning with an arbitrary value, /3.=1, on 
the right-hand side, the first revised value for optimal p. 
results on the left-hand side. This new approximation for 
optimal p. is used on the right-hand side to compute the 
next revision, and so on to convergence. If the resulting 
solution for optimal p. is less than 1, setting /3. equal to 1 
seems to be a practical remedy when numerical problems 
occur in sampling of HAG RFs from their Dirichlet poste- 
rior during MCMC computations. 
s- 
(/!, +p.) 
( n,+p.r 
