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# Sampling from Dirichlet populations: estimating the number of species

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## Sampling from Dirichlet partitions: estimating thenumber of species

1

2

n

n

(3)

n

1

n

n

m=1

m

n

S1,...,Sn

1

n

n

n m=1

θ−1m

Pnm=1sm−1

n

1

n

1

n

n

θ

n

m=1

mqm

n m=1

m

n m=1

m

n

d

n

n

m

d

n

n

1n

n1

n

m

n1

m

n2(nθ+1)n−1

n1

−(θ+1)/θ

1

θ−1

m

(n)

(1)

(n)

n

(1)

(n)

(1)

n1

1n

n

n

(n)

(∞)

n

n

d

n

n

n

n

1

m−1

1

m

m

n

(4)

m

m

m

m

n

1

k

n

n,k

n,k

n,k

l

n,k

k

l=1

l

θ

l

n

m

m

m

θ

n,k

n

m

k

n,k

n

m=1

m

n

1

n

n,k

θ

n,k

n

n

n m=1

m

n m=1

mkm

n

θ

n,k

n

θ

n,k

n

n

n

m=1

nkm

k

(5)

k

k

k

0

n

n,k

n

n

Sn

n

n,k

n

n

m=1

m

n m=1

(θ+km m)−1

(Pn

m=1sm−1)

n

n,k

n

d

n

n

n

1

n

θ

m

n,k

n

m

1

k

k

θ

1

1

θ

1

1

n

θ

m1

θ

2

2

1

1

2

θ

k

k

1

k−1

k−1

l=1

l

k

1

k

θ

1

1

k

k

k−1

l=1

l

j=1

j

l+1

n

m=1

km

m

k

l=1

l

1

k

n,k

a.s.

k→∞

n

n,k

θ

n m=1

Kmn,k(m)

n

n m=1

m

m

k

θ

n

m=1

Kmn,k(m)

m

m

n

m=1

m

m d

1

n

n

m=1

m

1

n

n

θ

n m=1

Kmn,k(m)/k

θ

n m=1

1/km

m

k

k→∞

θ

n m=1

m

k

n m=1

m

m

k

k→∞

θ

n m=1

Xmem

θ

n m=1

Smm

n,k

a.s.

k→∞

n

n

n,k

n

(6)

n

1

k

1

k

θ

n

m

θ

θ

n

θ

m

n,k

k

l=1

l

n,k

n

m=1

n,k

n,k

n,k

n,k

n,k

n,k

n,k

n,k

n,k

n m=1

n,k

k

i=0

n,k

k

i=1

n,k

n,k

k

i=1

n,k

n,k

n,k

n

1

p

q

p

q=1

q

θ

n,k

1

n,k

p

n,k

p q=1

q

k

p q=1

bq

1

k

k

i=1

i

k

i=1

i

θ

n,k

1

n,k

k

n,k

k

i=1

ai

i

k

k i=1

aii

(7)

n,k

θ

n,k

k

k,p

k,p

bq>1 Pp

q=1bq=k

p q=1

bq

q

ai>0 Pk

i=1ai=p Pk

i=1iai=k

k

i=1

ai

i

k i=1

aii

θ

n,k

n,k;m

((n−m)θ)(nθ) k

k

n,k

θ

n,k

p q=1

p−q

n,k;n−q

k+1

k+1

θ

k+1

n,k

1

n,k

p

n,k

1

p

(n−1)θnθ+1

nθ+1θ+1

1n

n1

k+1

r

r

θ

k+1

r

n,k

1

n,k

p

n,k

r

θ

n,k+1

n,k

θ

n,k+1

n,k

p

r=1

r

n,k

θ

n,k+1

θ

n,k

θ

n,k

k,p

k+1,p

k,p−1

k,p

(8)

k,0

0,p

0,0

k,p

1

1

2

2

k

k

k,1

k

θ

n,k

k

k

1

n,k

1

n,k

p

n,k

n p

n+k−1 k

1

p

k−1 p−1

q

q

k,p

k!p! k−1p−1

1

n,k

n p

k−1

p−1

n+k−1 k

1

n,k

1

n,k

k

n,k

np

n+k−1 k

k

i=1

i

n,k

1

n,k

p

n,k

p q=1

q

k

k,p

n,k

k,p

k

n,k

1

n,k

k

n,k

−k

k

i=1

ai

i

γ

k

1

k

p

k

p

k

p

q=1

q

k,p

γ

k

p

k,p

k

r

k

k

a.s.

k→∞

γ

k

1

k

k

k

p

k

k

i=1

ai

i

(9)

n,k

n,k

n,k

n,k

θ

n,k

1

n,k

p

n,k

k,p

p q=1

bq

q

θ

n,k

1

n,k

k

n,k

k,p

k i=1

aii

ai

i

n,k

θ

n,k

θ

n,kb

θ

bn−1,k

k

k

k,P−1

k,P

θ

k,P−1

k,P

k

k

n,k;1

k,p

enk

θ

2

(10)

k

k,p

k,P−1

k,P

k,p

k

k,p

γ

γ

k

k

k−1

l=0

k

P r

k→∞

k p=1

p

k,p

k

l

k−l

k

l

k−l,P−1

k,P

k,p

γ

k−l,P−1

k,P

k

k−l

p=1

p

k−l,p

l

1

sk−1,P−1s

k,P

1

γ(γ))k−1

k

γ+k−1γ

n,k

1

k

1

k

1

2

l1,l2

n m=1

l1

l2

(11)

l1

l2

n

n,k

k l16=l2=1

l1,l2

n,k

n,k

Pn,k

q=1

n,k

n,k

Pn,k

q=1

n,k

2

n,k

Pn,k

q=1

n,k

n,k

n,k

n,k

n,k

l1,l2

1

2

θ

n,k

θ

l1,l2

n,k

n,k

q

n,k

P q=1

q

2

1

1

1

1

1

1

k

1

1

k

1

1

1

k,P−1

1

k,P

1

1

1

1

1

1

1

1

1

n,k

q

n

n m=1

mα

n

n

m=1

m2

n

n

m=1

m

n

n

n m=1

m

2

n

θ

n m=1

m2

θ

12

(12)

n,k

m

n,k

n m=1

n,k

n

n,k

d

k→∞

n

θ

n,k

k→∞

θ

n

P

n,k

q=1

q

n,k

n m=1

m

m

m

n,k

P

n,k

q=1

q

q

q

q

SCq

n,k

Pn,k

q=1

q

q

q

q

k

n,k

n,k

d k→∞

n

n,k

n,k

q

q

q

n,k

n,k

n,k

P

n,k

q=1

B

n,k(q) k

Bn,kk(q)

k

2

n,k

P

n,k

q=1

B

n,k(q) k

2

Bn,kk(q)k

θ

n,k

k→∞

θ

n

n,k

P q=1

B

q

k

2

Bkq

k

q

n,k

q

2

2

2

2

2

2

k

2

2

k

(13)

2

2

2

k,P−1

2

k,P

2

q

(n)

n

(n)

(n)

(n)

n

(n)

k

n

θ

(n)

θ

(n)

n

θ

(n)

k

1

0

θ

(n)

k

1

0

θ

(n)

1k

(n)

θ

(n)

n

n,θ

n,θ

∗nθ,s

t=1

θ,s

θ−1

1

(n)

n−1+

+

1

n

1

0

1/kn−1 +

1

1−n1

k

1/kn−1

k−1

1 0

n−1

k−1

k−1k−1

j=0

−j

−1

n+

n,k

θ

n+

θ

n,k

θ

n,k

θ

n+

k

n−1

q=1

q−1

k

(14)

θ

n+

k

k,n

1

n+

k n

n+k−1 n

k−1 n−1

n+k−1 k

n+

k,n

k

1

1

ε

1

ε

1

1

1

1

1

1

2

1

2

1

1

ε

ε

ε

−1/2

ε

1/2

−1/2

ε

ε

1

k−1,P−1

k,P

i

θ

n,k

i

n,k−i;1

i

i

θ

n,k

i

k−i,P−1

k,P

i

(15)

k,P

i

i

i

n,k−i;1b

i

n,k

2

k i=1

i

i

2

i

2

k i=1

i

i

2

i

i

γ

k

2

k i=1

i

i

2

i

i

### =

(k−i)!k! i(bγ+i−1)bγ

12

32

2n3

3n2

n,3n 2

n,n

12

32

n,k

12

32

(16)

n,k

1

1

n,k

2

2

n,k

1

1

n,k

2

2

−5

n,k

1

1

n,k

2

2

n,k

1

1

n,k

2

2

n,k

1

1

n,k

2

2

(17)

1

2

n,k

n,k

(18)

### Wilks, S. (1962). Mathematical statistics. Wiley, New-York.

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