Page 64 - Applied Probability

3. Newton’s Method and Scoring
is positive definite, it follows that A n − c n v n v is positive definite as well.
This necessary and sufficient condition suggests that c n be replaced by
−1
t
v n )} in updating A n , where is some constant in
min{c n , (1 − )/(v A
n
n
(0, 1).
3.6 The Dirichlet Distribution t n 47
In this section, we briefly discuss the Dirichlet distribution. In the next
section, we use it to construct an empirical Bayes procedure for estimating
allele frequencies when genotype data are available from several different
populations. As is often the case, the Bayes procedure provides an inter-
esting and useful alternative to maximum likelihood estimation.
The Dirichlet distribution is a natural generalization of the beta dis-
t
tribution [12]. To generate a Dirichlet random vector Y =(Y 1 ,...,Y k ) ,we
take k independent gamma random variables X 1 ,...,X k of unit scale and
form the ratios
X i
= .
Y i
k

j=1 X j
e
By “unit scale” we mean that X i has density x α i −1 −x i /Γ(α i )on(0, ∞)
i
k
for some α i > 0. Clearly, each Y i ≥ 0 and Y i =1.

i=1
We can find the joint density of (Y 1 ,... ,Y k−1 ) by considering the larger
t
random vector Z =(Y 1 ,...,Y k−1 ,S) , where S = k X i . The inverse
i=1
transformation
x = T(z)
 
y 1 s
=  . 
.
 . 
y k s
k−1
with y k =1 − y i has Jacobian
i=1
s 0 ... 0 y 1
 
 0 s . . . 0 y 2 
. . . . . 

det(dT) = det  . . . . . . . . . 
 .
0 0 ... s y k−1
 

−s −s ... −s y k
s 0 ... 0 y 1
 
 0 s . . . 0 y 2 
 . . . . . 
= det  . . . . . . . . . 
 .

00 ... s y k−1
 
00 ... 0 1
= s k−1 .