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!!