x3.4 - x3.5

@liang4673

2013/3/13()

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1 RiemannRiemann

2 Fisher-3

Legendre

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abstract

1 - - - -

2

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review

-

n S = fpg

()ij;k := E

@i@jl +1 2

@il@jl

(@kl)

< r()@i @j ; @k >= ()ij;k

r(0) : Fisher Riemannr(1) : (exponential family)r(e)r(1) : (mixture family)r(m)

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review

(S; g;r;r) X;Y; Z 2 T (S)

Z < X;Y >=< rZX;Y > + < X;rZY >

r r g (dual)@kgij = ki;j + kj;ir()r() Fishier

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alpha-ane manifold

-:

S = fp(xj )j 2 g Rp(xj )dx = constant

L()(u) :=

(2

1u(1)=2 ( 6= 1)

log u ( = 1)

l()(xj ) := L()(p(xj ))

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alpha-ane manifold

-:

Fisher

gij() =Z@il

()(xj ) @jl()(xj )dx

()ij;k =Z@i @j l

()(xj ) @k l()(xj )dx

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alpha-ane manifold

-:

S = fp(xj )j 2 g -[i]

@i @j l()(xj ) = 0

x fC;Fig

l()(xj ) = C(x) + iFi(x)

S

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alpha-ane manifold

-:

!?(S; g;r();r())

i :=ZFi(x)l()(xj )dx

[i] i

() =

(2

1+

Rp(xj )dx ( 6= 1)R

p(xj )(log p(xj ) 1)dx ( = 1)

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alpha-divergence

-:

- D()(pjjq)8p;q 2 P;8 2 R

D()(pjjq) := 41 2

Z 1 2

p+1 + 2

q

p(1)=2q(1+)=2dx ( 6= 1)

D(1)(pjjq) = D(1)(qjjp) :=Z

q p+ p log pq

dx

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alpha-divergence

-:,

D(1)(jj) : Kullback-Leibler divergence (ex.EM-algorithm : mixture model)

D(0)(jj) : Hellinger

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alpha-family

-:

: S = fp(xj)j 2 g (3.35)

S := fp(xj )j 2 ; > 0g

- : S - S = fp(xj)j 2 g

p(xj ) =(

nXi=0

ci()Fi(x)

)2=(1)

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alpha-family

-:

(1)- : (1)-(1)-1- : 1-(1)- S () M

M S e- () M - S M ( 3.11)

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foliation

: ( 3.4)

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