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test
1.
2.
0
1
3.
0
1 1. (n M)
4.
0
1 1. (n M) 2. (Fisher g α- (α) )
5.
0
1 1. (n M) 2. (Fisher g α- (α) ) 3. (M, g, (α) )
6.
0
1 1. (n M) 2. (Fisher g α- (α) ) 3. (M, g, (α) ) 4.
7.
1 n
M 2 p(x; ξ)
8.
1 n
M 2 p(x; ξ) M = p(x; ξ)|ξ = (ξ1 , · · · , ξn ) ∈ Ξ ⊂ Rn open n
9.
1 n
M 2 p(x; ξ) M = p(x; ξ)|ξ = (ξ1 , · · · , ξn ) ∈ Ξ ⊂ Rn open n (ξ1 , · · · , ξ n ) M n
10.
1 n
M 2 p(x; ξ) M = p(x; ξ)|ξ = (ξ1 , · · · , ξn ) ∈ Ξ ⊂ Rn open n (ξ1 , · · · , ξ n ) M n
11.
2
3
12.
2
3
13.
2.1 Fisher
4 M gi j
14.
2.1 Fisher
4 M gi j ∂lξ ∂lξ gi j (ξ) :=Eξ ∂ξi ∂ξ j Eξ [ f ] := f (x)p(x; ξ)dx( f ) l(x; ξ) := log p(x; ξ)
15.
2.1 Fisher
4 M gi j ∂lξ ∂lξ gi j (ξ) :=Eξ ∂ξi ∂ξ j ∂l(x; ξ) ∂l(x; ξ) = p(x; ξ)dx ∂ξ i ∂ξ j Eξ [ f ] := f (x)p(x; ξ)dx( f ) l(x; ξ) := log p(x; ξ)
16.
*1
5 *1 [3]
17.
Fisher
6
18.
Fisher
6 p(x; ξ)dx = 1 ξi
19.
7 (
)=0 ∂ ( )= i p(x; ξ)dx ∂ξ
20.
7 (
)=0 ∂ ( )= i p(x; ξ)dx ∂ξ ∂ = p(x; ξ)dx ∂ξ i
21.
7 (
)=0 ∂ ( )= i p(x; ξ)dx ∂ξ ∂ = p(x; ξ)dx ∂ξ i ∂l(x; ξ) = p(x; ξ)dx ∂ξ i
22.
,
8 ∂l(x; ξ) p(x; ξ)dx = 0 ∂ξ i
23.
,
8 ∂l(x; ξ) p(x; ξ)dx = 0 ∂ξ i ξj
24.
9 (
)=0 ∂ ∂l(x; ξ) ( )= p(x; ξ)dx ∂ξ j ∂ξi ∂ ∂l(x; ξ) = p(x; ξ) dx ∂ξ j ∂ξ i ∂2 l(x; ξ) = p(x; ξ)dx ∂ξ j ∂ξ i ∂l(x; ξ) ∂l(x; ξ) + p(x; ξ)dx ∂ξi ∂ξ j
25.
10 ∂l(x; ξ) ∂l(x;
ξ) ∂2 l(x; ξ) p(x; ξ)dx = − p(x; ξ)dx ∂ξ i ∂ξ j ∂ξ j ∂ξ i
26.
10 ∂l(x; ξ) ∂l(x;
ξ) ∂2 l(x; ξ) p(x; ξ)dx = − p(x; ξ)dx ∂ξ i ∂ξ j ∂ξ j ∂ξ i Fisher ∂2 l(x; ξ) gi j (ξ) = −E (1) ∂ξ j ∂ξi (1) Fisher
27.
2.2 α-
11 α∈R (α) ∂2 l(x; ξ) 1 − α ∂l(x; ξ) ∂l(x; ξ) ∂l(x; ξ) Γi j,k =E + ∂ξ j ∂ξ i 2 ∂ξi ∂ξ j ∂ξk
28.
2.2 α-
11 α∈R (α) ∂2 l(x; ξ) 1 − α ∂l(x; ξ) ∂l(x; ξ) ∂l(x; ξ) Γi j,k =E + ∂ξ j ∂ξ i 2 ∂ξi ∂ξ j ∂ξk α- (α) ∂ ∂ g (α) ∂ , k = Γ(α) j ∂ξ ∂ξi ∂ξ i j,k
29.
2.3 α-
12 1. (torsion tensorT )
30.
2.3 α-
12 1. (torsion tensorT ) 2. (−α) (α) (α) (−α) (Xg(Y, Z) = g( X Y, Z) + g(Y, X Z) )
31.
2.3 α-
12 1. (torsion tensorT ) 2. (−α) (α) (α) (−α) (Xg(Y, Z) = g( X Y, Z) + g(Y, X Z) ) 3. α = 0 α- Fisher g Levi-Civita
32.
2.4
13 R : X(M) × X(M) × X(M) (X, Y, Z) −→ R(X, Y)Z ∈ X(M) R(X, Y)Z := X YZ − Y XZ − [X,Y] Z (X, Y, Z ) (1,3)
33.
14
TpM 2 Πp {X, Y} g(R(X, Y)Y, X) K(Π p ) := g(X, X) · g(Y, Y) − (g(X, Y))2 p
34.
3
15
35.
3.1
16 M: g: M :g M ∗ (M, g, )
36.
3.2
17 p n n p(x; θ) = exp C(x) + θ F s (x) − ϕ(θ) s s=1 C(x) ∈ F (X), F s (x) 0 ∈ F (X), ϕ(θ) ∈ F (Θ) 3.3 1.
37.
2.
18 3. Poisson 4. Gamma 5. Beta 6. 7. etc. Cauchy
38.
19
39.
3.4 Fisher
α- 20 n p(x; θ) = exp C(x) + θ F s (x) − ϕ(θ) s s=1 n l(x; θ) = C(x) + θ s F s (x) − ϕ(θ) s=1
40.
θi , θ
j 21 ∂i ∂ j l = −∂i ∂ j ϕ Fisher g gi j (θ) = ∂i ∂ j ϕ
41.
θi , θ
j 21 ∂i ∂ j l = −∂i ∂ j ϕ Fisher g gi j (θ) = ∂i ∂ j ϕ Fisher
42.
α-
22
43.
α-
22 n p(x; θ) = exp C(x) + θ F s (x) − ϕ(θ) s s=1 x
44.
α-
22 n p(x; θ) = exp C(x) + θ F s (x) − ϕ(θ) s s=1 x 1 n 1= exp C(x) + θ F s (x) dx s exp ϕ(θ) s=1
45.
23
n exp ϕ(θ) = exp C(x) + θ F s (x) dx s s=1
46.
23
n exp ϕ(θ) = exp C(x) + θ F s (x) dx s s=1 θi *2 *2
47.
24
∂ϕ ( ) = exp ϕ(θ) · i (θ) ∂θ n ( )= C(x) + exp θ F s (x) Fi (x)e(ϕ(θ)−ϕ(θ)) dx s s=1 ϕ(θ) C(x)+ n θ s F s (x)−ϕ(θ) =e e s=1 Fi (x)dx = exp ϕ(θ) · p(x; θ)Fi (x)dx = exp ϕ(θ) · E [Fi ]
48.
25 ∂i ϕ =
E [Fi ] ( ∂i = ∂/∂θi )
49.
25
∂i ϕ = E [Fi ] ( ∂i = ∂/∂θi ) exp ϕ · ∂i ϕ = exp ϕ · E [Fi ] θj
50.
26 (
) = ∂ j (exp ϕ · ∂i ϕ) = ∂ j ∂i ϕ + ∂i ϕ · ∂ j ϕ ( ) = ∂ j exp ϕ · E [Fi ] = (∂ j exp ϕ) · E [Fi ] + exp ϕ · (∂ j E [Fi ]) A
51.
A = ∂j
p(x; θ) · Fi dx 27 = Fi ∂ j pdx = Fi p · (F j − ∂ j ϕ)dx = pFi F j dx − Fi p∂ j ϕdx = E[Fi F j ] − ∂ j ϕE[Fi ]
52.
28 = (∂ j
exp ϕ) · E [Fi ] + exp ϕ · (E[Fi F j ] − ∂ j ϕE[Fi ]) = exp ϕ · E[Fi F j ]
53.
28 = (∂ j
exp ϕ) · E [Fi ] + exp ϕ · (E[Fi F j ] − ∂ j ϕE[Fi ]) = exp ϕ · E[Fi F j ] ∂i ∂ j ϕ + ∂i ϕ · ∂ j ϕ = E[Fi F j ]
54.
28 =
(∂ j exp ϕ) · E [Fi ] + exp ϕ · (E[Fi F j ] − ∂ j ϕE[Fi ]) = exp ϕ · E[Fi F j ] ∂i ∂ j ϕ + ∂i ϕ · ∂ j ϕ = E[Fi F j ] E[Fi F j Fk ] = ∂i ∂ j ∂k ϕ + ∂i ∂ j ϕ · ∂k ϕ + ∂ j ∂k ϕ · ∂i ϕ + ∂k ∂i ϕ · ∂ j ϕ + ∂i ϕ · ∂ j ϕ · ∂k ϕ
55.
29
1−α Γ(α) i j,k = E ∂i ∂ j l + ∂i l · ∂ j l ∂k l 2
56.
29
1−α Γ(α) i j,k = E ∂i ∂ j l + ∂i l · ∂ j l ∂k l 2 1−α 1−α Γ(α) i j,k = ∂i g jk = ∂i ∂ j ∂k ϕ 2 2
57.
α-
30 (α) 1−α ∂i j ∂ = ∂ s gi j · g st ∂t 2
58.
4
31
59.
4
31
60.
4
31
61.
4.1
32 1 (x − µ)2 p(x; ξ) = √ exp − 2πσ 2σ2 (ξ = (µ, σ), µ ∈ R, σ ∈ R+ )
62.
33
x2 − 2µx + µ2 √ ( ) = exp − 2 − log 2πσ) 2σ 1 µ µ2 √ = exp −x 2 +x 2 − + log( 2πσ) 2σ 2 σ 2σ 2
63.
33
x2 − 2µx + µ2 √ ( ) = exp − 2 − log 2πσ) 2σ 1 µ µ2 √ = exp −x 2 +x 2 − + log( 2πσ) 2σ 2 σ 2σ 2 µ F1 (x) = −x2 ,F2 (x) = x,θ1 = 1 2σ2 θ2 = σ2
64.
33
x2 − 2µx + µ2 √ ( ) = exp − 2 − log 2πσ) 2σ 1 µ µ2 √ = exp −x 2 +x 2 − + log( 2πσ) 2σ 2 σ 2σ 2 µ F1 (x) = −x2 ,F 2 (x) = x ,θ 1 = 1 2σ2 θ2 = σ2 µ2 √ (θ2 )2 1 π ϕ(θ) = + log( 2πσ) = + log 1 2σ 2 4θ 1 2 θ p(x; θ) = exp F1 (x)θ1 + F2 (x)θ2 − ϕ(θ)
65.
θ ∈ Θ
= θ = [θ1 , θ2 ]|θ1 ∈ R+ , θ2 ∈ R 34
66.
4.2 Fisher
α- 35 Fisher dµ2 + 2dσ2 3 ds2 = * σ2 *3
67.
α-
∂µ = ∂/∂µ, ∂σ = ∂/∂σ 36 (α) 1−α ∂µ µ ∂ = ∂σ 2σ (α) (α) 1+α ∂µ σ ∂ = ∂ =− ∂σ µ ∂µ σ (α) 1 + 2α ∂σ σ ∂ =− ∂σ σ α- α- 1 − α2 R(α) (∂µ , ∂σ )∂σ = − ∂µ σ 2
68.
37
1 − α2 1 g(R(α) (∂µ , ∂σ )∂σ , ∂µ ) = − · 2 σ2 σ 1 2 g(X, X) · g(Y, Y) − (g(X, Y)) = 2 · 2 − 0 2 σ σ
69.
1−α2 2
− σ4 σ4 = − c(α) 38 2 α- 1−α2 − 2 *4 *4 k R(X, Y)Z = k{g(Y, Z)X − g(X, Z)Y}
70.
39 [1] Shun-Ichi Amari,Hiroshi
Nagaoka Methods of Information Geometry Oxford University Press [2] , [3] [4] ,
71.
40 Thank you very
much for your attention!!
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