8172019 snowlepdf

httpslidepdfcomreaderfullsnowlepdf 12

1 snowl 211

111

50000

bullbull

bull

Bradley-Terry +MM

f( )

112

( )

K ( )Bradley-Terry K 1000

K ( )

12

2 Bradley-Terry

Bradley-Terry i

θi i θi i j

P (i j ) = θiθi+θj

Bradley-Terry 2 n

8172019 snowlepdf

httpslidepdfcomreaderfullsnowlepdf 22

P (i ) = θiθ1+θ2++θn

foralli isin 1n

( )3 (123421567)

Bradley-Terry 123

P (1 2 3 4 2 1 5 6 7 ) = θ1θ2θ3θ1θ2θ3+θ4θ2+θ1θ5θ6θ7

3 MM

Bradley-Terry

L(θ) =mprodi=1

mprodj=1

983080 θiθi+θj

983081wij

m θ1 θ2θm wij i j

θ(k+1)i = W i

983131sumj=i

N ij

θ(k)i +θ

(k)j

983133minus1

W i = sumj=iwij i N ij = wij + wjisumi

θ(k+1)i = 1

( )

θ(k+1)i = W i

983131sumjlti

N ij

θ(k)i +θ

(k+1)j

+ sumjgti

N ij

θ(k)i +θ

(k)j

983133minus1

[1] David R Hunter MM algorithms for generalized Bradley-Terry models The Annals of

Statistics32(1)384-406 (2004)

8172019 snowlepdf

httpslidepdfcomreaderfullsnowlepdf 22

P (i ) = θiθ1+θ2++θn

foralli isin 1n

( )3 (123421567)

Bradley-Terry 123

P (1 2 3 4 2 1 5 6 7 ) = θ1θ2θ3θ1θ2θ3+θ4θ2+θ1θ5θ6θ7

3 MM

Bradley-Terry

L(θ) =mprodi=1

mprodj=1

983080 θiθi+θj

983081wij

m θ1 θ2θm wij i j

θ(k+1)i = W i

983131sumj=i

N ij

θ(k)i +θ

(k)j

983133minus1

W i = sumj=iwij i N ij = wij + wjisumi

θ(k+1)i = 1

( )

θ(k+1)i = W i

983131sumjlti

N ij

θ(k)i +θ

(k+1)j

+ sumjgti

N ij

θ(k)i +θ

(k)j

983133minus1

[1] David R Hunter MM algorithms for generalized Bradley-Terry models The Annals of

Statistics32(1)384-406 (2004)