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11 (p A ) (p B ) (p C ) 1 2, 3, 4 2, 3, 4 {2, 3, 4} 1 2 (p A ) (p B ) (p C ) d d {1, 2} (p A,p B )=0<d {1,2,3,4} (p A,p B ) (p 1,...,p n ) (q 1,...,q n ) d(p, q) = n j=1 (p j q j ) 2, d(p, q) = 1 2 n j=1 p j q j

12 2 2, 3, 4 2, 3, 4 {1, 2, 3, 4} 0.13 {1, {2, 3, 4}} 0.14 d(p, q) = 1 i pi q i {1, 2} {3, 4} (p A ) (p B ) (p C ) (p A ) (p B ) (p C ) {1, 2, 3, 4} {1, 2} d(p, q) = ( ) i pi q i

13 8/9 8/10 9/10 p A p B p C d(p B,p A ) d(p B,p C ) 0 0 {1, 2} i

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17 Θ={θ 1,...,θ n } (Θ) := {p R n + j p j =1,p j 0, j} Θ (Θ) = {p (Θ) p j > 0, j} p q (Θ) p i p(i) p i D Θ : (Θ) (Θ) R + {+ } Θ D Θ {+ } D Θ (Θ) (Θ) D Θ Θ p, q (Θ) D Θ (p, q) =0 p = q.

18 p, q D(p, q) =0 p = q Θ 1, Θ 2 γ :Θ 1 Θ 2 p, q (Θ 1 ) D Θ2 (p γ 1,q γ 1 )=D Θ1 (p, q), p γ 1 Θ 2 p γ 1 (θ 2 ):=p(γ 1 (θ 2 )) Θ n = Θ D n n n := {x R n i x i =1,x j 0} Θ Θ γ :Θ Θ D Θ R n p 1,p 2,q 1,q 2 n D n (λp 1 +(1 λ)p 2,λq 1 +(1 λ)q 2 ) {D n (p 1,q 1 ),D n (p 2,q 2 )}, λ [0, 1]

19 p 1,q 1,p 2,q 2 p 1 = p p 2 = q 1 = q 2 D n (λp +(1 λ)q, q) D n (p, q) p = p 1 = p 2 D n (p, λq 1 +(1 λ)q 2 ) {D n (p, q 1 ),D n (p, q 2 )} D n p, q n D n 1 ((p 1 + p 2,...,p n ), (q 1 + q 2,...,q n )) D n ((p 1,p 2,...,p n ), (q 1,q 2,...,q n )). n, m N p n q m p q nm nm p q p q =(p 1 q 1,...,p 1 q m,...,p j q 1,...,p j q m,...,p n q 1,...,p n q m ). n, m N p (1),p (2) n q (1),q (2) m D nm (p (1) q (1),p (2) q (2) )=D n (p (1),p (2) )+D m (q (1),q (2) ).

20 i =1,...n 1 p, q n i i +1 D n (p, (q 1,...,q i ϵ, q i+1 + ϵ,..., q n )) D n (p, q) i D n (p, q) :=. ϵ 0 ϵ i D n (p, q) q i +1 i i j id n(p,q) j D n(p,q) i, j {1,...,n 1} j D n (p, q) 0 i D n (p, q) j D n (p, q) = g(p i,p i+1,p j,p j+1 ; q i,q i+1,q j,q j+1 ), i, i +1,j,j+1

21 D Θ Θ p, q (Θ) D Θ (p, q) =a ( ) p(θ) p(θ) + b q(θ) θ Θ θ ( ) q(θ) q(θ), p(θ) a, b 0 ( D Θ (p, q) =a p(θ) z R \ { 0.5, 0.5} θ ( ) ) z 0.5 p(θ) q(θ) a>0 z > 0.5 a<0 z < 0.5 D(p, q) p, q n n p j = q j =0 0 α /0 β =0

22 0 (0/0) = 0 p j > 0=q j α, β > 0 p α j /0 = + 0/p β j =0 p j (p j /0) = + 0 (0/p j )=0 Supp(p) p Supp(p) ={θ Θ p θ > 0} p, q n j p j /q j D n (p, q) Θ n D(p, q) p q D(p, q) =D(q, p) p, q D(p, q) = j (p j q j ) ( pj q j ) = j p j ( pj q j ) + j ( ) qj q j ; p j ( ) D(p, q) = pj q j. j D(p, q) = j f j(p j,q j )

23 D(p, q) < + Supp(p) Supp(q) D KL (p q)+d KL (q p) D KL (p q) = j p j ( pj q j ) p q q p p q q Θ q p p LR q p(θ)q(θ ) p(θ )q(θ), θ θ. Θ f j :[0, 1] [0, 1] R

24 p LR q LR r Θ p i q j q i p j q i r j r i q j, i j. p, q, r n p< LR q< LR r, D(p, q) D(p, r). Θ p< FOSD q< FOSD r D(p, q) D(p, r) D αd α>0 q p j j q i p i, i=1 i=1 j, p FOSD q

25 ( D z j p j (p, q) := ( j p j ( pj q j ) z 0.5 ) ( pj q j ) z 0.5 ) z > 0.5, z < 0.5. z z D z D z p, q, r n D z (p, q) <D z (p, r) D z (p, q) >D z (p, r). D z j ( ) z 0.5 pj p j = ( ) qj p j φ z, φ z (x) =x 0.5 z. q j p j j p q (p 1 /q 1,...,p n /q n ) φ z p q z p j φ z (q j /p j )= j j ( ) z 0.5 pj p j = ( ) z 0.5 qj q j = q j φ z+1 (p j /q j ). q j p j j j φ z q j /p j p, q j p j qj p j =1 qj p j q j p j

26 z 0.5, 0.5 z > 0.5 φ z ( ) z < 0.5 φ z ( ) D z z q1 p 1! P q j p j j z p j z z q2 p 2 q3 p 3 q 1 /p 1 q 2 /p 2 q 3 /p 3 z>0.5 z(x) z(x) z(x) z < 0.5 φ z x z< 0.5 φ z x z>0.5 φ z x φ z z R \{0.5, 0.5} (D z ) z R\{ 0.5,0.5} p, q n z 0.5, 0.5 D z (p, q) =D z (q, p) z R z z 0.5 Dz (p, q) =( p(θ)/q(θ) p z 0.5),

27 w f : Θ R + f(θ) p w := ( i f(θ i) w p(θ i )) 1/w D z z =0.5 z = 0.5 D z (p, q) z 0.5 z 0.5 = j p j ( pj q j ) =: D 0.5 (p, q) z 0.5 D z (p, q) z 0.5 = j ( ) qj q j =: D 0.5 (p, q). p j z D z D z p q D z z f z p f z p z<z f D z R \ { 0.5, 0.5} D 0.5 D 0.5 (p, q) =D KL (p q) D 0.5 (p, q) =D KL (q p) p, q n D z n z < 0.5 D z (p, q) =+ Supp(p) Supp(q) =. z 0.5 z 0.5 D z (p, q) =+ Supp(p) \ Supp(q). z 0.5 D z (p, q) =+ Supp(q) \ Supp(p). n n \ n

28 D z n n n n n p D z (p, ) : n R q D z (p, q) z 0.5 D z (p, ) : n R p n D z θ Supp(p) ( D z j Supp(p) p j (p, q) = j Supp(p) p j ( pj ( pj q j ) z 0.5 ) q j ) z>0.5. z =0.5 D z (,q): n R q n z 0.5 D z (,q): n R q n D z θ Supp(q) ( D z j Supp(q) p j (p, q) = j Supp(q) p j ( pj ( pj q j ) z 0.5 ) q j ) z>0.5. z =0.5 D z (p, ) : n R p n z < 0.5 D z : n n R

29 D z (p, q) = j Supp(p) Supp(q) ( ) z 0.5 pj p j, q j j =0 n D z p, q n z 0.5 D z n z < 0.5 n x := j x j n z 0.5 ϵ, δ > 0 p, q,, n p q > x,y n x y δ, <ϵ, D z (, ) >D z ( p, q) z < 0.5 D z (p, q) ϵ δ p q δ D z (p, q) <ϵ D z z > 0.5 R n

30 z < 0.5 D z D z z > 0.5 p ϵ =(1 ϵ, ϵ) q ϵ =(1 ϵ 3,ϵ 3 ) ϵ 0 p ϵ,q ϵ (1, 0) p ϵ q ϵ 0 ( ) z 1 ϵ ( ϵ ) z D z (p ϵ,q ϵ )=(1 ϵ) + ϵ +, ϵ 0, 1 ϵ 3 ϵ 3 z>0.5 (1, 0) p ϵ q ϵ 1 θ 2 θ 2 D z z < 0.5 R n z > 0.5 D z (p, q) z>0.5 p

31 ( 1 α 1 j p j R α (p, q) = ( j p pj j q j ) ( pj q j ) α 1 ) α 1 α =1 C β (p, q) = j pβ j q1 β j,β (0, 1) D KL (p q) = ( ) j p pj j q j D S (p, q) = ( ) j (p pj j q j ) q j ( ) D B (p, q) = j pj q j D z 1 α 1 Dα 0.5 (p, q) α 0, 1 R α (p, q) = D 0.5 (p, q) α =1 C β (p, q) =( D β 0.5 (p, q)) D KL (p q) =D 0.5 (p, q) D S (p, q) =D 0.5 (p, q)+d 0.5 (p, q) D B (p, q) =D 0 (p, q)

32 p q α<0 D 0.5 (p, q) = ( j q pj j q j ) D 0.5 (p, q) D 0.5 (p, q)

33 g D(p, q) =g 1 ( j p j g(d(p i,q i )) ), D(p i,q i ) i Θ p, q (Θ) D Θ (p, q) =0 p = q. D D(p, q) > 0 p q

34 Θ 1, Θ 2 γ :Θ 1 Θ 2 p, q (Θ 1 ) D Θ2 (p γ 1,q γ 1 )=D Θ1 (p, q), p γ 1 Θ 2 p γ 1 (θ 2 ):=p(γ 1 (θ 2 )) D Θ n = Θ D Θ Θ

35 ( D n ) n 2,n N D n : (Θ) (Θ) R + {+ }, Dn (p, q) :=D Θ (p, q), Θ Θ = n (D n ) n 2 n D Θ Θ A Θ p(a) =q(a) p(a) (0, 1) p( A) A p( A c ) D n (p, q) D A (p( A),q( A)) D A c (p( A c ),q( A c )) D n (p, q) {D A (p( A),q( A)),D A c (p( A c ),q( A c ))}, A Θ p(a) =q(a) (0, 1) A A

36 p 1,p 2,q 1,q 2 p 1,p 2,q 1,q 2 n D n (λp 1 +(1 λ)p 2,λq 1 +(1 λ)q 2 ) {D n (p 1,q 1 ),D n (p 2,q 2 )}, λ [0, 1] D n : n n R {+ } D n p, q n λ [0, 1] D n (p, λp +(1 λ)q) D n (p, q). p, q, r n λ [0, 1] D n (λp +(1 λ)r, λq +(1 λ)r) D n (p, q). p, q 1,q 2 λ [0, 1] D n (p, λq 1 +(1 λ)q 2 ) {D n (p, q 1 ),D n (p, q 2 )}. A s p 1 = p( s), p 2 = p( s c ) q 1 = q( s), q 2 = q( s c ) λ = P p (s) = P p (s) D(p(s),q(s)) <D(p, q) D(p, q) D(p(s c ),q(s c )) P p (s) =P p (s)

37 B(p, ρ] :={q n D n (p, q) ρ} D n C n C C n D n (p, q) =D n ( p, q), p,q C p, q C p, q n D n (p, q) D n (e 1,e 2 ) (= D n (e j,e i ) i, j), e 1 =(1, 0, 0,...,0) e 2 =(0, 1, 0,...,0) n p, q n θ 1,θ 2 {θ 1,θ 2 } p,q n 1 p p =(p 1 + p 2,p 3,...,p n ) q (q 1 + q 2,q 3,...,q n ).

38 θ 1,θ 2 p, q n D n 1 ((p 1 + p 2,...,p n ), (q 1 + q 2,...,q n )) D n ((p 1,p 2,...,p n ), (q 1,q 2,...,q n )). A =(A j ) j Θ D A (p A,q A ) D n (p, q), p A (A) p A (A j ):= θ A j p(θ j ), A j A. n D n n Θ 1 Θ 2 n = Θ 1 m = Θ 2 n, m p (1) n q (1) m r (1) := p (1) q (1) (A j ) j Θ A j A i =, A j =Θ. j

39 (Θ 1 Θ 2 ) r (1) (θ 1,θ 2 ):=p (1) (θ 1 )q (1) (θ 2 ). r (1) p (1) q (1) Θ 1 Θ 2 n, m N p (1),p (2) n q (1),q (2) m D nm (p (1) q (1),p (2) q (2) )=D n (p (1),p (2) )+D m (q (1),q (2) ). p 1,q 1,p 2,q 2 n D nm (p 2 r, q 2 s) D nm (p 1 r, q 1 s) =D n (p 2,q 2 ) D n (p 1,q 1 ) r, s m. 1 2 p 1 p 2 q 1 q 2 P Q D(p 2,q 2 ) D(p 1,q 1 ) P Q r s p 1 r q 1 s p 2 r q 2 s r s 1

40 2 D, n p, q m D n (, ) D m ( p, q) D n n ( r, s) D m n ( p r, q s), r, s n p, q n r m D n (p, q) =D n m (p r, q r). D D

41 D D p, q n U q q U p p p U p q U q D(p,q) D(p, q) D(p, q ). D D φ : R + R + D := φ(d) D n p, q n D n (p, q) =D n+1 ((p 1,...,p n, 0), (q 1,...,q n, 0)).

42 p, q n Supp(p) Supp(q) = p j q j =0 j D D(p, q) =+. θ j p, q n p 1 q 1 = p 2 q 2. D n (p, q) =D n 1 ((p 1 + p 2,p 3,...,p n ), (q 1 + q 2,q 3,...,q n )). p n q 1 = (q 1 + q 2, 0,q 3,...,q n ) q 2 =(0,q 1 + q 2,q 3,...,q n ) [q 1,q 2 ]={r n r = λq 1 +(1 λ)q 2,λ [0, 1]}. p [q 1,q 2 ] q [q 1,q 2 ]

43 q1 q2 = p 1 p 2, r, r [q 1,q 2 ] r [r,q ] D n (p, r) D n (p, r ). p, q q [q 1,q 2 ] p r r 1 r 2 = p 1 p 2 q p D(p, q ) p [q 1,q 2 ] 3 p q 1 q 2 q q 1 2 n =3 D(p, q ) p q [q 1,q 2 ]

44 p, q n i D n (p, q) := ϵ 0 D n (p, (q 1,...,q i ϵ, q i+1 + ϵ,..., q n )) D n (p, q) ϵ i =1,...,n 1. i D n (p, q) D n q i i +1 (p, q) ( i D n (p, q)) n 1 i=1 D n (p, ) : n R + q D n (p, q) i, j D n (p, (q 1,...,q i ϵ,..., q j + ϵ,..., q n )) D n (p, q) ϵ 0 ϵ ( i D(p, q)) n 1 i=1 i, i +1,j,j+1 i, j {1,...,n 1} j D n (p, q) 0 i D n (p, q) j D n (p, q) = g(p i,p i+1,p j,p j+1 ; q i,q i+1,q j,q j+1 ),

45 i i +1 j j +1 n 1 j n 1 p, q D((p 1,...,p j, p j+1,..., p n ), (q 1,...,q j, q j+1,..., q n )) D((p 1,...,p j, p j+1,..., p n ), (q 1,...,q j, q j+1,..., q n )), D((p 1,...,p j, p j+1,..., p n ), (q 1,...,q j, q j+1,..., q n )) D((p 1,...,p j, p j+1,..., p n ), (q 1,...,q j, q j+1,..., q n )), p, q p, q n h D n (p, q) =D n (p, (...,q i + ϵ, q i ϵ,..., q j + h(ϵ),q j+1 h(ϵ))), h i, i+1,j,j+1

46 jd(p,q) i D(p,q) = h (0) i, i+1,j,j+1 1/z j (Π i ) i

47 D zj+0.5 (, ) p j D zj+0.5 (p j, Π) + D 0.5 z j (Π,p j ) D zj 0.5 (Π,p j ) π =(S, f(s θ)) S f( θ) (S) θ Θ f(s θ) s θ p(s) =(p 1 (s),...,p n (s)) p(θ j s) =p j (s) = f(s θ j)p j i f(s θ i)p i, s P p (s) := i f(s θ i)p i. p, q n p q D

48 π =(S, f(s θ)) s S D(p(s ),q(s )) <D(p, q). D p, q n π =(S, f(s θ)) s S D(p(s),q(s)) >D(p, q). p π E π p(d(p(s),q(s))) := s P p (s)d(p(s),q(s)). E π p(d(p(s),q(s))) D(p, q)

49 E π p[d(p(s),q(s))] E π p[d(p( s),q( s))], π π Θ π =(S, f(s θ)) π =( S,g( s θ)) π π π π g( s θ) = s λ s, s f(s θ), (λ s, s ) s, s s λ s, s =1 π π π π p θ q D p q n

50 π π E π p[d(p(s),q(s))] E π p[d(p( s),q( s))]. p p, q n E π p[d(p(s),q(s))] E π p[d(p( s),q( s))], D D p, q p q

51 D(p, q) =+ Supp(p) \ Supp(q). p, q D(p, q) = Supp(p) Supp(q) D(p, q) < + Supp(p) \ Supp(q) p, q 2 p =(p 1, 1 p 1 ) q =(0, 1) D(p, q) < q n = ( 1 n, 1 1 n) n D(p, q n ) <D(p, q) < q n [q, p] D(p, q n ) n s 0 p(s 0 ) (1, 0) q n (s 0 ) (0, 1) n ϵ q n D(p(s 0 ),q n (s 0 )) >D((1 ϵ, ϵ), (ϵ, 1 ϵ)). D((1 ϵ, ϵ), (ϵ, 1 ϵ)) ϵ 0 E π p(d(p(s),q n (s))) + D(p, q n ) D(p, q n ) < E π p(d(p(s),q n (s))). n ϵ

52 Supp(p) \ Supp(q) p q q(θ s) =0 θ Supp(p)\Supp(q) θ D(p, q) = D(p, q) < E π p[d(p(s),q(s))]. D D(p, q) =ad z (p, q) z 0.5 a>0 D D D(p, q) =ad z (p, q) z 0.5 a>0 π π p, q n E π p[d(p(s),q(s))] E π p[d(p( s),q( s))].

53 Θ Θ = n< + J J N x R + j J p j (Θ) u j : R + R j u j x 0 u (x) = x u (x) =0 j U j ( ) :=E p j(u j ):= i p j i u j(x j i ), x j i i = (x j 1,...,x j n) θ 1,...,θ n i I Θ 1 x i 0 i I I 2 I I =Θ (Π i ) i I i I Π i =1 (p j ) j p j (Θ)

54 Π i (Π i ) i I (x j i )j J i I j J (x j i ) i I R n p j i u j(x j i ) i i I Π i x i 1, i x j i = J. j J

55 j V j ( ) := i p j i (xj i 1), j G j ( ) :=U j ( ) U j ( ), =(1,...,1) x j i i p j i /Π i (p j 1/Π i,...,p j n/π n ) G j ( ) p Π G j ( ) D(p, Π) i j 1 i x j i i xj i 1

56 (u j ) j J 1 z j u j (x) = x1 1 1 z j > 0, x > 0. z j u j u j (x) u j (x) = z jx, z j (0, 1) z j (0, 1) z j > 1 p 1,...,p J n (u j ) j z j (0, 1) (Π i ) i 1 D zj+0.5 (q, p j ( I)), q (I) 1 z j j 1 1 xu (x) u (x) = 1 z j [0.07, 0.62]

57 V j ((x j i ) i)=(d z j+0.5 (p j ( I), Π) + D z j+0.5 (Π,p j ( I))) 1 j J; G j ((x j i ) i)= 1 ( ( )) 1 1 Dzj 0.5 (Π,p j ( I)), j J. z j 1 z j z j (0, 1) z j > 1 G j ((x j i ) i)= 1 ( 1 1 z j 1 ( D z j 0.5 (Π,p j ( I)) z j )), D z j 0.5 V j ((x j i ) i)=(d z j+0.5 (p j ( I), Π) D z j+0.5 (Π,p j ( I))) 1. z j

58 0 p j m N p j,(m) n m + j 1,j 2 J m 1,m 2 >m p j 1,(m 1 ) p j 2,(m 2 ) =0. ϵ m p j,(m)

59 ϵ j J m> m j (p j,(m) ) m n m + j n n (p j,(m) ) j J m N ε>0 pj,(m) i >ε>0 i =1,...,n j J m N m j u j 1 x j i 1 u j 1 u j(1) = 1 j z j = 1 u (1) 1 z j ũ j (x) := x1 1 1 z j u (1) = ũ j(1) u (1) = ũ j (1) I =Θ n

60 I Θ (u j ) j J 1 u j(1) = 1 1/u j (1) =: z j (0, 1) (p j,(m) ) j J m N m ((x j,(m) i ) j J i=1,...,n, (Πm i ) i ) m Π m q n j 1 1 z j D zj+0.5 (q, p j,(m) ). j m D z j +0.5 (p j,(m),π m ) D z j +0.5 (p j,(m), Π m ) =1 pj Π m Π m m i pj,(m) i (x j,(m) i 1) D z j+0.5 (p j,(m), Π m )+D z j+0.5 ( Π m,p j,(m) ) =1. m i pj,(m) i u i (x j,(m) i ) 1 1 z j D z j 0.5 ( Π m,p j,(m) ) =1.

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67 Θ={θ 1,...,θ n } π =(S, (f(s θ)) s S,θ Θ ) S f( θ) (S) θ s θ f(s θ) > 0 f( θ) S(f( θ)) := {s S f(s θ) > 0}, S(f( θ)) < θ S Θ E Θ,S π =(S, (f(s θ)) s S,θ Θ ) S S(f( θ)) f( θ) θ π E Θ,S Θ A (A) A σ σ

68 π = ( f(s1 ) θ 1 )... f(s 1 θ n) f(s 2 θ 1 )... f(s 2 θ n) f(s m θ 1 )... f(s m θ n) f( θ) f(s θ) 0 s, θ s f(s θ) =1 θ Θ π π E Θ,S π s S f(s θ) =f(s θ ), θ, θ Θ. s f(s θ) =0 θ S(f( θ)) = S(f( θ )) θ, θ Θ, θ s θ S f(s θ θ) > 0

69 µ (Θ) µ(θ) > 0 θ Θ f(s θ) > 0 θ µ(θ s) = µ(θ)f(s θ) θ µ(θ )f(s θ ). µ(s) =µ s µ(θ) > 0 µ(θ s) > 0 s µ(θ s) Θ S c Θ,S : E Θ,S [0, + ] c Θ,S µ =(1, 0,...,0) (0, 1, 0,...,0) c Θ,S ((1 ϵ)π+ϵπ ) c Θ,S (π) π, π E Θ,S ϵ 0 + ϵ E Θ,S (1 ϵ)π + ϵπ E Θ,S

70 c Θ,S (π) =0 π π c Θ,S (π) < π 0 π c Θ,S (π) = + c Θ,S (π) c Θ,S Θ S f(s θ) c Θ,S

71 Θ, Θ S, S π =(S, (f(s θ)) s S,θ Θ ) π =(S, (g(s θ )) s S,θ Θ ) γ :Θ Θ λ : S S f(s θ) =g(λ(s) γ(θ)), s S, θ Θ. π π π π π =(S, (f(s θ)) s S,θ Θ ) π =(S, (g(s θ )) s S,θ Θ ) c Θ,S (π) =c Θ,S (π ) c Θ,S S c Θ Θ c Θ c Θ Θ = Θ Θ Θ c n π =(S, (f(s θ)) s S,θ Θ ) Θ n N E n n E n λπ +(1 λ)π E n π, π E n λ [0, 1]

72 π π n π, π E n c n (λπ +(1 λ)π ) {c n (π),c n (π )} λ [0, 1]. π π λπ +(1 λ)π c n (π) c c π = (S, (f(s θ)) s S,θ Θ ) A = (A i ) i S s S A i A s π =(A,f(A θ) A A,θ Θ ) π θ π π A j S A i A j = i j i A i = S

73 S S γ : S S π =(S, (f(s θ)) s S,θ Θ ) π := γ π =(S,g(s θ) s S,θ Θ) g(s θ) = s γ 1 (s ) f(s θ). γ π π γ π π γ π s 1 s 2 s 1 π = f(s 1 θ 1 )... f(s 1 θ n) f(s 2 θ 1 )... f(s 2 θ n) f(s 3 θ 1 )... f(s 3 θ n) f(s m θ 1 )... f(s m θ n) γ π ( = ) f(s 1 θ 1 )+f(s 2 θ 1 )... f(s 1 θ n)+f(s 2 θ n) f(s 3 θ 1 )... f(s 3 θ n) f(s m θ 1 )... f(s m θ n) π E n γ π E n γ : S S π = γ π π π S s γ π π γ : S S c n (γ π) c n (π). s 1,s 2 S f(s θ)/f(s θ ) θ, θ π s 1 s 2 {s 1,s 2 } γ 1 (s) s s S γ(s) =s

74 s 1,s 2 f(s 1 θ)f(s 2 θ )=f(s 1 θ )f(s 2 θ) θ, θ γ : S S γ(s 1 )=γ(s 2 ) γ(s) γ(s ) s s s, s s 1,s 2 c n (γ π) =c(π). π π = f(s 1 θ 1 )... f(s 1 θ n) f(s 2 θ 1 )... f(s 2 θ n) f(s m θ 1 )... f(s m θ n) kf(s m θ 1 )... kf(s m θ n) , k>0, π = ( f(s1 ) θ 1 )... f(s 1 θ n) f(s 2 θ 1 )... f(s 2 θ n) (1+k)f(s m θ 1 )... (1+k)f(s m θ n) c n ( π) c n (π) c n ( π) =c n (π) f(s θ) =0 θ

75 π π c(π) c( π) π = (S, (f(s θ)) s S,θ Θ ) π = (S, (g(s θ)) s S,θ Θ) g(s θ) = s λ s,s f(s θ), λ s,s 0 s S λ s,s =1 π π π π π π π π π π c n c n (π) c n ( π)

76 π =(S, (f(s θ)) s S,θ Θ ) S 1,S 2 S S 1 S 2 = S 1 S 2 = S π 1 =(S 1, (f(s θ)) s S1,θ Θ) π 2 =(S 2, (f(s θ)) s2 S,θ Θ) π = π 1 π 2 π 1 π 2 π 1 c n π c n π 1 π π 1, c n π c n π 2 π π 2, π 2 π 1 π 2 s S 1 f(s θ) 1 π ( ) π1 π = π θ Θ π 2 f(s 1 θ)+ f(s 2 θ) =1, s 1 S 1 s 2 S 2

77 ( ) c n (π) =h g(f(s θ 1 ),...,f(s θ n )), s S g :[0, 1] n [0, ] h :[0, + ] [0, + ] Θ Θ π E n f(s 1 θ 1 )... f(s 1 θ n ) π = f(s 2 θ 1 )... f(s 2 θ n ), π(θ 1,...,θ n 1 ) E n 1 f(s 1 θ 1 )... f(s 1 θ n 1 ) π(θ 1,...,θ n 1 )= f(s 2 θ 1 )... f(s 2 θ n 1 )

78 Θ = {θ 1,...,θ n 2, {θ n 1,θ n }} {θ n 1,θ n } θ n θ n 1 θ n 1 π π(θ 1,...,θ n 1 ) θ n θ 1,...,θ n 1 π π(θ 1,...,θ n 1 ) θ 1,θ n θ 2,θ n (θ n 1,θ n ) f(s 1 θ i ) f(s 1 θ j ) π(θ i,θ j ):= f(s 2 θ i ) f(s 2 θ j ), {θ i,θ j } π n>2 π E n c n (π) =c n 1 (π(θ 1,...,θ n 1 )) + h n 1 (c 2 (π(θ 1,θ n )),...,c 2 (π(θ n 1,θ n ))), h n :[0, ] n 1 [0, ] n h n h n

79 f( θ) c n n n 1 n th π 1 = (S 1, (f(s θ)) s S1,θ Θ) π 2 = (S 2, (f(s θ)) s S2,θ Θ) π := π 1 π 2 π =(S 1 S 2, (f(s 1 θ)f(s 2 θ)) (s1,s 2 ) S 1 S 2,θ Θ). π 1 π 2 π 1 π 2 π 1 π 2 π 1 π 2 π 1 π 2 S 1 S 2 S 1 S 2

80 c n (π π) =c n (π)+c n ( π). (c n ) n (c n ) n φ : E n R {+ } φ π φ(π) =0 φ(π) = (Θ) µ µ(θ) > 0 θ Θ δ j θ j δ j =(0,..., 1 j th,...,0) φ(π 1 π 2 )=φ(π 1 )+φ(π 2 ), π 1,π 2 E n.

81 H : (Θ) R H(µ) >H(δ j ) µ (Θ) j =1,..., Θ µ (Θ) φ µ : E n R φ µ (π) :=H(µ) P µ (s)h(µ(s)). s S φ µ (π) H H(µ) := θ µ(θ) (µ(θ)), φ µ H H(µ) > 0=H(δ j ) µ (Θ) φ µ H Θ Θ > 3 S c Θ,S ( ) f(s θi ) (f(s θ i ) f(s θ j )) = f(s θ s i,j j ) i,j D(f( θ i ),f( θ j )), P µ (s) θ µ(θ)f(s θ) µ s

82 D D(p, q) = ( ) i p i pi q i + ( i q i qi p i ) θ, θ Θ D(f( θ),f( θ )) s θ θ D(f( θ),f( θ )) D(p, q) = i p i ( pi q i ) + i q i ( qi p i ), p q q p ( ) c Θ,S

83 f(s θ) f(s θ ) θ θ c Θ,S (π) = π s S θ, θ Θ f(s θ) > 0=f(s θ ) Θ =2 Θ > 2 n = Θ Θ =2 Θ =2

84 c n c n 1 c 2 h n n =2 Θ =2 c 2 c KL 2 (π) = s [f(s θ 1) f(s θ 2 )][(f(s θ 1 )) (f(s θ 2 ))]; ( ) c B 2 (π) = s f(s θ1 )f(s θ 2 ). c B 2 (π) =+ π c KL 2 (π) =+ π 2 Θ =2 KL B π θ 1 s f(s θ 1 )=0<f(s θ 2 )

85 Ω D : (Ω) (Ω) [0, + ] Ω S (Ω) (S) f( θ) θ θ 1,θ 2 D(p, q) =D(q, p) p, q (Ω) θ 1 θ 2 θ 1 θ 2 2 <n< h n c n c 2 (π(θ 1,θ n )),...

86 c n f n f n (x 1,...,x n )= n x i, (x i ) i [0, ] n. i=1 c Θ,S = θ,θ c 2 (π(θ, θ )). Θ c Θ,S (π) = θ,θ c KL 2 (π(θ, θ )), c Θ,S (π) = θ,θ c B 2 (π(θ, θ )). c Θ,S Θ Θ > 2 c B 2 (π) π(θ, θ ) θ θ

87 n = Θ > 2 c Θ,S (π) = θ,θ c B 2 (π(θ, θ )). c(π) = s ( ) f(s θi ) (f(s θ i ) f(s θ j )) = f(s θ i,j j ) i,j D(f( θ i ),f( θ j )), c( ) c n ( ) c Θ,S ( ) c( ) π n E Θ,S f n (s θ) π E Θ,S f(s θ) π n c Θ =2 Θ > 2

88 π π n π f n (s θ) f(s θ) 0. s S,θ Θ π (π n ) n π n π c(π n ) (π n ) n c(π n ) 0 π n π c( ) f(s θ)/f(s θ ) 0 c 2 B (f(s θ)) s,θ µ

89 µ (Θ) P µ (s) := θ µ(θ)f(s θ) s µ(θ s) θ s µ (Θ) g(µ) := Θ µ(θ1 ) µ(θ n ) µ(θ 1 ),...,µ(θ n ) µ(θ) > 0 θ Θ µ (Θ) µ(θ) > 0 θ Θ π c(π) = s P µ (s) θ µ(θ s)n µ(θ) ((µ(θ s) [a(µ(s))])) =: s P µ (s)φ I (µ(s); µ), φ I (µ(s); µ) := θ µ(θ s)n µ(θ) ((µ(θ s) [a(µ(s))])) u : A Θ R V (µ) := a θ u(a, θ)µ(θ) µ π P µ (s)v (µ(s)) c(π), s c

90 P µ (s)(v (µ(s)) φ I (µ(s); µ)), (µ(s)) s s s P µ (s)µ(s) =µ, π c H (π; µ) := s P µ (s)(h(µ) H(µ(s))) =: s P µ (s)φ H (µ(s); µ), H(µ) = θ µ(θ) (µ(θ)) µ(s) φ I φ H µ (Θ) φ H (µ; µ) =φ I (µ; µ) =0 φ H ( ; µ) φ I ( ; µ) x (Θ) φ H (x; µ) < x (Θ) φ I (x; µ) = (Θ) (Θ) (Θ) = {µ (Θ) µ(θ), θ}

91 µ(s) =µ (Θ)

92

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97

98 M = W =+ M = {m 0,m 1,m 2,...} W = {w 0,w 1,w 2,...} m j w n > mj W {m j } M {w n } W {m j } M {w n } w 1 > mj w 2 > mj m j > mj w 3 m j w 1 w 2 w 3 m j w n A W {m j } A M {w n } x A {m j } A {w n } x> mj y, y A \{x} > wn ).

99 m j A W {m j } W {m j } w 0 w j w j j>j j 0 w 0 {w 1,w 2,...,w n,...} A A A P (m j ):w j1,w j2,...,w jl,m j,w jl+1,...,w jm,...; m j w j1 w jk w jk k < k m j m j P (m j ):w j1,w j2,...,w jl,

100 P (m j ) m (M,W, ) M = {m 0,m 1,...} W = {w 0,w 1,...} =(P (x)) x M W x W x M P (x) W {x} M {x} P (x) µ µ : M W M W,

101 x M W µ(µ(x)) = x x µ(x) µ(x) M x W µ x µ (M,W, ) x x> x µ(x); (m, w) m> w µ(w) w> m µ(m). µ x y y x

102 (M,W, ) P (x) x (M,W, )

103 P (m 1 ):w 1 P (w 1 ):m 1,m 2 P (m 2 ):w 1,w 2 P (w 2 ):m 2,m 3 P (m 3 ):w 2,w 3 P (w 3 ):m 3,m P (m i ):w i 1,w i P (w i ):m i,m i w 1 m 1

104 m 2 m 2 m 2 w 2 m 3 m 2 m 3 i m i+1 w i w i+1 m i+2 µ M µ (j) ( ) j µ M j µ (j) (m) (µ (j) (m)) + j=1 µ M (m) := m µ M (m) =w m w µ M (w) =m µ M (w) =w j µ (j) (m) (µ (j) (m)) + j=1 W

105 µ M µ M µ W µ µ > M µ µ µ(m) m µ (m) m M; µ> M µ µ(m) > m µ (m) m M;

106 > W µ> M µ µ µ µ µ > M µ M µ M > M µ µ M M µ µ W W µ, µ. > M > W µ µ (M,W, ) µ> M µ µ > W µ. µ µ µ> M µ µ > M µ µ µ

107 µ µ M M µ M µ W, µ W (A, ) A A (A, )

108 a, b A c := a b c a c b c a c b c c a b a b a b c := a b c a c b c a c b c c x {µ(x),ν(x)} µ ν(x) := x {µ(x),ν(x)} x M x W x {µ(x),ν(x)} µ ν(x) := x {µ(x),ν(x)} x M x W. m µ ν(m) m µ ν(w) m (µ ν)(m) m µ(m),ν(m) m (µ ν)(m). µ ν µ ν µ ν (M,W, )

109 λ = µ ν λ := i I ν i µ ν µ ν λ := i I ν i λ i I ν i

110 µ M(µ) :={m M µ(m) m}, W (µ) :={w W µ(w) w}.

111 µ µ M(µ) =M(µ ), W (µ) =W (µ ), (M,W, ) µ µ M(µ) M(µ ) P (m 1 ):w 1 P (w 1 ):m 2,m 1 P (m 2 ):w 2,w 1 P (w 2 ):m 3,m 2 P (m 3 ):w 3,w 2 P (w 3 ):m 4,m P (m i ):w i,w i 1 P (w i ):m i+1,m i µ M µ W µ M (w i )=m i i N

112 µ W (w i )=m i+1 i N µ M (m 1 )=m 1 M(µ M ) M(µ W ), m 1 M(µ M ) \ M(µ W ). µ µ µ M µ M(µ) M(µ ). µ M(µ W ) M(µ) M(µ M );

113 W (µ M ) W (µ) W (µ W ). M(µ W ) M(µ) M(µ M ) ; W (µ M ) W (µ) W (µ W ). µ W (µ) = M(µ) M(µ) =M(µ ) µ µ A B A = B A = B A A B A B A = B

114 M = µ W µ M (M). M = µ W µ M (M) W = µ M µ W (W ). M µ W µ M (M) m M \ µ W µ M (M) µ M (m) W µ M (m) W \ µ M µ W (W ) µ M µ W m µ(m) W µ (m) =m µ, µ µ M (m) W f g x f(g(x)) A f(a) ={y y = f(x), x A}

115 µ W (m) =m µ M µ W µ M µ W µ M µ W µ M µ W µ M µ W M(µ) < + µ M = j M j W = j W j j M j, W j < + m M j m W j m> m w, m M j,w W \ W j. M W Z

116 P (m i ):w i,w i 1 P (w i ):m i+1,m i i Z µ M m i w i µ W m i w i 1 M(µ W )=M W (µ M )=W µ µ M(µ) W (µ) M(µ ) W (µ ). µ M µ W W M = {...,m i,...,m 1,m 0,m 1,...,m i,...}

117 P (m 1 ):w 3 P (w 1 ):m 3 P (m 2 ):w 4 P (w 2 ):m 4 P (m 3 ):w 5,w 1 P (w 3 ):m 5,m P (m i ):w i+2,w i 2 P (w i ):m i+2,m i µ M µ W {m 3,...,m i,...} = M(µ W ) M(µ M )={m 1,...,m i,...}, µ µ µ M,µ W P (m 1 ):w 3,w 1 P (w 1 ):m 3,m 1 P (m 2 ):w 4,w 2 P (w 2 ):m 4,m 2 P (m 3 ):w 5,w 3,w 1 P (w 3 ):m 5,m 3,m 1

118 P (m i ):w i+2,w i,w i 2 P (w i ):m i+2,m i,m i {m 3,...,m i,...} = M(µ W ) M(µ M )={m 1,...,m i,...}, µ µ(m i )=w i M W µ M(µ) M(µ ), µ

119 M t W t t M t W t µ ( t N M t, t N W t, )

120 t M t t W t t t t t t t t m M t t t m> m w, w W s,s<t,

121 m> m w w W s,s>t, (M,W, 1 ) (M,W, 2 )

122 (M,W, ) (M,W, ) µ M µ W

123

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128 β

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132

133 D(p, q) p q 0 i D(p, q) h i i +1 α p, q i i D(p, q) = h(p i,p i+1,q i,q i+1 ). α(p, q)

134 h n p(p i,p i+1,q i,q i+1 ) h(p 1,p 2,q 1,q 2 )=i(p 1,q 1 ) i(p 2,q 2 ), i :[0, 1] 2 R i(x, y) i(x, y) =G(x/y), G :(0, + ) R G G (x) =cx α α, c R. G (x) = 1 x ( a + b x) G (x) α(p, q)

135 p q j =1,...,n 1 j D(p, q) 0 n =2 p q 1 D(p, q) 0 1 D(p, q) =0 p, q q [p, q] γ(q, q ): 2 2 R D(p, q) =D(p, q )+D(p, p + ϵ(q, q )z), z =(1, 1) 1 D(p, q) =0 1 D(p, p + ϵ(q, q )z) = 1 D(p, p + ϵ(q, q )z) ϵ(q, q ) q =0, q [p, q] q q ϵ(q, q ) 0 y (0,δ) R 1 D(p, p + yz) 0 ϵ(q,q ) q =0 q q q q ϵ(q, q )=h(q ) q = q ϵ(q, q )=0 h(q )=0 q [p, q] q D(p, q) =D(p, q ) D(p, q) (p, q] 0=D(p, p) =D(p, q) D(p, q) (p, q] q (p, q) D(p, q ) < D(p, q) = D(p, q) q [p, q) 1 D(p, q) =0 D(p, ) [ q, q] p q 1 D(p, p + yz) =0 y (0,δ) D(p, p + yz) =D(p, p) =0

136 1 D(p, q) =0 q q [p, q] q D(p, q ) = D(p, q) q n =2 n j D(p, q) =0 j z i,j =(0,...,1, 0,..., 1, 0,...,0) D(p, q) q q [p, q] q [p, q] D(p, q) =D(p, q) D(p, q ) <D(p, q) q [p, q] D [p, q] D(p, ) q [ q, q] q p D(p, q) =0 n =2 [p, q] D(p, ) q D n (p, q) h :[0, 1] 4 R α(p, q) : n n R i D(p, q) = h(p i,p i+1,q i,q i+1 ), α(p, q) p, q n q p D(p, q) q p D(p, q + ϵ(q p)) D(p, q) q p D(p, q) =. ϵ 0 ϵ h x =(x 1,x 2,x 3,x 4 ) x 1 + x 2 1 x 3,x 4 1 h

137 (α, h) ( α, h) α(p, q) =k α(p, q) h(p i,p i+1,q i,q i+1 )=k h(p i,p i+1,q i,q i+1 ), k 0 p q n j j D(p, q) 0 h(p i,p i+1,q i,q i+1 ):=g(p i,p i+1,p j,p j+1,q i,q i+1,q j,q j+1 ), g h p j,p j+1,q j,q j+1 h(p i,p i+1,q i,q i+1 )=0 i D n (p, q) =0 i i D(p, q) i D(p, q) h(p i,p i+1,q i,q i+1 ) i, i D n (p, q) h(p i,p i+1,q i,q i+1 ) = jd n (p, p) = k D n (p, q) h(p k,p k+1,q k,q k+1 ) α(p, q) α(p, q) := h(p i,p i+1,q i,q i+1 ), i D(p, q) (α, h n p) ( α, h n p)

138 p, q h(i) := h(p i,p i+1,q i,q i+1 ) i D := i D(p, q) α := α(p, q) (α, h) ( α, h) h(j) j D = α α = h(i) i D, jd = h(j) i D h(i), α α = h(i) h(i). p i,q i,p i+1,q i+1 p, q α α = k = h(i) h(i), k p, q i :[0, 1] 2 R h(p 1,p 2,q 1,q 2 )=i(p 1,q 1 ) i(p 2,q 2 ), p 1,p 2,q 1,q 2 [0, 1] 4 i,j D(p, q) z ij = e i e j

139 i,i+1 D(p, q)+ i+1,i+2 D(p, q) = i,i+2 D(p, q), h h(p i,p i+1,q i,q i+1 ) α(p, q) + h(p i+1,p i+2,q i+1,q i+2 ) α(p, q) = h(p i,p i+2,q i,q i+2 ). α(p, q) h(p i,p i+1,q i,q i+1 )+h(p i+1,p i+2,q i+1,q i+2 )=h(p i,p i+2,q i,q i+2 ). h D p i h j j h 1 (p i,p i+1,q i,q i+1 )=h 1 (p i,p i+2,q i,q i+2 ), p i+1,q i+1,p i+2,q i+2 h 1 p i q i h 3 p i q i h i (1) i (2) h(p 1,p i+1,q i,q i+1 )=i (1) (p i,q i ) i (2) (p i+1,q i+1 ). h 1 (x, y, w, z) := h x h 2,h 3,h 4

140 i (1) i (2) x := p i = p i+1 = p i+2 y := q i = q i+1 = q i+2, 2h(x, x, y, y) =h(x, x, y, y) h(x, x, y, y) =0, x, y [0, 1] 2. i (1) i (2) i (1) (x, y) i (2) (x, y) =0, x, y [0, 1] 2, i (1) (x, y) =i (2) (x, y) i i(x, y) x/y i(x, y) =G(x/y), G : R + R p, q n q 1 =(q 1 + q 2, 0,q 3,...,q n ) q 2 =(0,q 1 + q 2,q 3,...,q n ) [q 1,q 2 ]={r n r = λq 1 +(1 λ)q 2,λ [0, 1]}.

141 r [q 1,q 2 ] D(p, r) r r 1 r 2 = p 1 p 2 D 1, D(p, r) = h(p 1,p 2,r 1,r 2 ) α(p, r) =0, i(p 1,r 1 ) i(p 2,r 2 ) α(p, q) =0, p 1 p 2 = r 1 r 2, i(p 1,r 1 )=i(p 2,r 2 ) p 1 r 1 = p 2 r 2, i i(p 1,r 1 )=G ( p1 r 1 ). G p =(p 1,...,p n ) n q =(q 1,...q n ) n λ := (λ, 1 λ)

142 2 γ =(γ,1 γ) 2 p λ =(λp 1, (1 λ)p 1,...,λp n, (1 λ)p n ), q γ =(γq 1, (1 γ)q 1,...,γq n, (1 γ)q n ), D(p λ, q γ) =D(p, q)+d(λ, γ), p ϵz i ( ( pi pi+1 )) G( ) G q i q i+1 α(p,q) RHS := ( ( ) ( )) p G i q i G pi+1 q i+1. α(p, q) LHS := ( ( ) ( )) ( ( ) ( )) λp λ G i γq i G λpi+1 γq i+1 +(1 λ) G (1 λ)pi (1 γ)q i G (1 λ)pi+1 (1 γ)q i+1 α(p λ, q γ) LHS = RHS α(p λ, q γ) α(p, q) = ( λ G ( ) ( λp i γq i G λpi+1 )) ( ( ) γq i+1 +(1 λ) G (1 λ)pi (1 γ)q i G ( ( ) ( )) p G i q i G pi+1 q i+1 ( )) (1 λ)pi+1 (1 γ)q i+1

143 r i = p i q i R 1 = λ R γ 2 = 1 λ 1 γ α(p λ, q γ) α(p, q) = λ (G (R 1r i ) G (R 1 r i+1 )) + (1 λ)(g (R 2 r i ) G (R 2 r i+1 )) (G (r i ) G (r i+1 )) α(p λ,q γ) α(p,q) r i r i+1 r i,r i+1 R + λ (G (R 1 r i ) G (R 1 r i+1 )) + (1 λ)(g (R 2 r i ) G (R 2 r i+1 )) (G (r i ) G (r i+1 )) = λ ( G (R 1 r i) G ( )) ( R 1 r i+1 +(1 λ) G (R2 r i) G ( R 2 r i+1)) ( G (r i ) G ( r i+1)) K(λ, γ) λ (G (R 1 x) G (R 1 y)) + (1 λ)(g (R 2 x) G (R 2 y)) (G (x) G (y)) = K(λ, γ), x, y [0, + ) R 1 := λ γ R 2 := 1 λ 1 γ G(x) G x 0 G(x) = x + G(x) = x 0 G(x) x + G(x) G G x, x G(x) = G(x ) G(y) =G(x) y [x, x ] [q 1,q ] [q,q 2 ]

144 G(x) 0 R >0 f(x) = f(rx), x + x + G G G(x) =+. x (0,+ ) G c 1 <G(x) <c 2 c 1,c 2 R x + I := x G(x) G λ (I G (R 1 y)) + (1 λ)(i G (R 2 y)) (I G (y)) = K(λ, γ). y 0 J := y 0 G(y) K(λ, γ) = λ (I J)+(1 λ)(i J) (I J) =1. K(λ, γ) =1 λ γ λ (G (R 1 x) G (R 1 y)) + (1 λ)(g (R 2 x) G (R 2 y)) = (G (x) G (y)).

145 λ 0 λ (G (R 1 x) G (R 1 y)) 0, (G (R 1 x) G (R 1 y)) R 2 1 G 1 γ (0, + ) ( ) ( ) x y G G = G(x) G(y). 1 γ 1 γ y + ( ) x>0 γ (0, 1) G = G(x), G x 1 γ G G (x) =cx α, α R c R G (x) = a x 2 + b x, a, b R G G p =0 G G (0, + ) x + x 0 x 0 G(x) = x 0 G(x) (, ) x + G(x) =±

146 λ (G (R 1 x) G (R 1 y)) + (1 λ)(g (R 2 x) G (R 2 y)) (G (x) G (y)) = K(λ, γ), x 0 λ (G (R 1 x) G (R 1 y)) + (1 λ)(g (R 2 x) G (R 2 y)) K(λ, γ) =, x 0 (G (x) G (y)) G(x) x 0 y R 1 R 2 λ (G (R 1 x) G (R 1 y)) + (1 λ)(g (R 2 x) G (R 2 y)) x 0 (G (x) G (y)) λg (R 1 x)+(1 λ)g (R 2 x) =, x 0 G (x) λ γ G(Rx) x 0 G(x) R. λg (R 1 x)+(1 λ)g (R 2 x) x 0 G (x) λg (R 1 x)+(1 λ)g (R 2 x) = G(Rx) x 0 G (Rx) G(x), x 0 λg(r 1 x)+(1 λ)g(r 2 x) G(Rx) R x 0 G(Rx) G(x)

147 L(R) := x 0 G(Rx) G(x), L(R) =R α α R G(Rx) L(R) = x 0 G(x) = G(Rx) G(R x) x + G(R x) G(x) ( ) R = L L(R ). R λ (G (R 1 x) G (R 1 y)) + (1 λ)(g (R 2 x) G (R 2 y)) x + (G (x) G (y)) = λ ( ) α λ +(1 λ) γ ( ) α 1 λ. 1 γ λ (G (R 1 x) G (R 1 y)) + (1 λ)(g (R 2 x) G (R 2 y)) (G (x) G (y)) x x (0, + ) λ (G (R 1 x) G (R 1 y)) + (1 λ)(g (R 2 x) G (R 2 y)) (G (x) G (y)) = λ ( ) α λ +(1 λ) γ ( ) α 1 λ, 1 γ x, y γ,λ

148 D x ( G (x) λ ( ) α λ +(1 λ) γ x [0, + ) λ, γ (0, 1) ( ) α ) 1 λ =(λr 1 G (R 1 x)+(1 λ)r 2 G (R 2 x)), 1 γ α =0, 1 α 0, 1 α =0 α = 1 λ (G (R 1 x) G (R 1 y)) + (1 λ)(g (R 2 x) G (R 2 y)) = (G (x) G (y)), x G (x) =(λr 1 G (R 1 x)+(1 λ)r 2 G (R 2 x)), x φ(x) :=xg (x) φ(x) =λφ(r 1 x)+(1 λ)φ(r 2 x), λ = 1 2 ( ) ( x 2φ(x) =φ + φ 2γ x 2(1 γ) ).

149 γ 0= 2x ( ) x (2γ) 2 φ + (2γ) ( 2x (2(1 γ)) 2 φ x (2(1 γ)) ). z = x 2γ ) φ (z γ = 1 γ ( ) 2 γ φ (z), 1 γ φ (z) = c z 2 c R γ (0, 1) γ/(1 γ) R + z φ(z) = a +b z φ (z) φ( ) G (x) = a x 2 + b x, α 0, 1 φ(x) :=xg (x) ( φ(x) λ ( ) α λ +(1 λ) γ ( ) α ) 1 λ = λφ(r 1 x)+(1 λ)φ(r 2 x). 1 γ λ [0, 1] γ G D D

150 λ 0 c(r) := λ 0 λφ (λr). c(r) = d r d R r r c(r )= λφ (λr )= r ) (λ λ 0 r λr 0 λr r φ r r r = r r c(r). α>0 λ 0 ( ) ( ) 1 x 1 φ(x) (1 γ) = c + φ α γ (1 γ) x, φ (0, + ) c c(r) =0 c(r) = d r d = xφ(x) γ 1 (1 γ) x α γ φ ( ) 1 (1 γ) x, x 0 xφ(x) d x γ φ ( d(1 γ), γ xφ(x) 1 d = x 0 γ (1 γ) x α γ φ ( ) 1 (1 γ) x = d γ(1 γ) d(1 γ), α γ ) 1 x (1 γ) γ α 1, 0

151 d =0 c(r) 0 c(x/r) =0 ( ) 1 φ (1 γ) x = 1 (1 γ) α φ(x). x γ φ(x) α φ(x) =cx α, G (x) =cx α 1, α R G (x) 1, 2 α 0, 1 a = 0 b = 0 G G(x) =ax α + b α 0, 1 G(x) =a (x)+ b x + c, a, b, c, z R

152 G (x) =ax α α R G (x) = a x + b x 2. G (x) α(p, q) G(x) =a (x)+ b + c α(p, q) =K x G(x) =ax z + b p, q n α(p, q) =K ( j ( ) ) z pj p j. q j λ, γ m m m =2 α(p λ, q γ) α(p, q) = m j=1 ( ) α λj λ j, γ j α =0, 1 α(p λ,q γ) α(p,q) =1 α(, ) p, q, λ, γ α 0, 1 α(p λ, q γ) =α(p, q) m j=1 ( ) α λj λ j. γ j D(p λ, q γ) =D(λ p, γ q) p, q λ, γ α(p λ, q γ) =α(λ, γ) j ( ) z pj p j, q j

153 α(p, q) j p j ( pj q j ) z = α(λ, γ) m j=1 λ j ( λj γ j ) z, α(p,q) ( ) z j p pj j q j p, q D Θ Θ p, q (Θ) D Θ (p, q) =a ( ) p(θ) p(θ) + b q(θ) θ Θ θ ( ) q(θ) q(θ), p(θ) a, b 0 ( D Θ (p, q) =a p(θ) z R \ { 0.5, 0.5} θ ( ) ) z 0.5 p(θ) q(θ) a>0 z > 0.5 a<0 z < 0.5 z 0

154 i D z (p, q) = h(p i,p i+1,q i,q i+1 ) α(p, q) = a ( p i ) z ( q i pi+1 j p j q i+1 ) z ( pj q j ) z, a, z R q + ϵz i,i+1 p D(p, q) =a ( j ( ) ) z pj p j + K, q j K D(q, q) =0 K =0 D(p, q) 0 p, q n z ( 1, 0) a<0 z ( 1, 0) x z j ( ) ( ) z z pj p j q j =1, q j j a ( j p j a<0 ( pj q j ) z ) 0 a 0 z (, 1) (0, + ) a>0 z (0, ) x z j ( ) ( ) z z pj p j q j =1, q j j D(p, q) 0 a 0 z (, 1)

155 a ( j ( ) ) ( z pj ( ) ) z 1 qj p j = a q j, q j p j j z 1 (0, + ) z +1 (, 0) j ( ) ( ) z 1 z+1 qj q j p j =1 p j j G(z) =a (x)+ b x α(p, q) ( i D(p, q) =a ( pi q i ) ( pi+1 q i+1 )) ( qi + b q ) i+1, p i p i+1 a, b R p a j ( ) qj q j + b p j j p j ( pj q j ) + K, a b D(p, p) =0 K =0 a b D(p, q) 0 a b a p =(ϵ, 1 ϵ, 0,...,0)

156 q =(0.5, 0.5, 0,...,0) ϵ 0 b j p j ( pj q j ) = ϵ 0 b(ϵ (2ϵ)+(1 ϵ) (2(1 ϵ))) = b (2), ϵ 0 a j ( ) qj q j =, a<0 p j ϵ D(p, q) = a, b p = q D(p, p) =0 x 1,x 2,y 1,y 2,λ [0, 1] ( ) (λx1 +(1 λ)x 2 ) (λx 1 +(1 λ)x 2 ) (λy 1 +(1 λ)y 2 ) λx 1 ( x1 y 1 ) +(1 λ)x 2 ( x2 y 2 ), x x (x) =0

157 ( λx 1 λx 1 +(1 λ)x 2 ( ) y1 (1 λ)x 2 + x 1 λx 1 +(1 λ)x 2 ( ) λx 1 y1 λx 1 +(1 λ)x 2 x 1 ( y2 x 2 )) (1 λ)x 2 λx 1 +(1 λ)x 2 ( y2 x 2 ), D Θ (λp 1 +(1 λ)p 2,λq 1 +(1 λ)q 2 ) λd(p 1,q 1 )+(1 λ)d(p 2,q 2 ), x z 0.5 z > 0.5 z < 0.5 p 1,p 2,q 1,q 2 z > 0.5 ( ) z 0.5 ( ) z 0.5 ( ) z 0.5 p1 + p 2 p1 p2 (p 1 + p 2 ) p 1 + p 2. q 1 + q 2 q 1 q 2 ( p1 q 1 + p ) 0.5 z 2 q 2 p 1 p 1 + p 2 p 1 p 1 + p 2 p 2 p 1 + p 2 ( q1 p 1 ) 0.5 z + p 2 p 1 + p 2 ( q2 p 2 ) 0.5 z,

158 z > 0.5 x 0.5 z z < 0.5 ( p1 p 1 + p 2 ( q1 p 1 ) + p 2 p 1 + p 2 ( )) 0.5 z q2 p 2 p 1 p 1 + p 2 ( q1 p 1 ) 0.5 z + p 2 p 1 + p 2 ( q2 p 2 ) 0.5 z, z < z (0, 1) x 0.5 z (ab) =(a) +(b) i,j (p ip j ) z = i pz i j pz j D(p, q) =D(q, p) p, q D(p, q) = j (p j q j ) ( pj q j ) = j p j ( pj q j ) + j ( ) qj q j ; p j

159 ( ) D(p, q) = pj q j. j D(p, q) < + Supp(p) Supp(q) p =(1, 0) q =(q 1, 1 q 1 ) q D(p, q) D(p, q) D(q, p) z =0 p =(0.5, 0.5) q =(q 1, 1 q 1 ) D(p, q) D(q, p) a b p, q, r n p< LR q< LR r, D(p, q) D(p, r). D D(p, q) = j f j(p j,q j ) f j :[0, 1] [0, 1] R

160 D(p, q) = ( i p i p, q, r p< LR q< LR r, ( p i q i ) z 0.5 ) z>0.5 i ( ) z 0.5 pi p i ( ) z 0.5 pi p i. q i r i i ϵ (0, 1) (D(p,ϵr+(1 ϵ)q)) ϵ 0 D(p, r) D(p, q) ( i p i ( ) ) z 0.5 p i ϵr i +(1 ϵ)q i ϵ ( 0 ( ) z+0.5 q i, i p i (1 ϵ)q i + ϵr i ) z+0.5 r i i p i (1 ϵ)q i + ϵr i ( p i (1 ϵ)q i +ϵr i ) z+0.5 p< LR ϵr +(1 ϵ)q q< LR r z < 0.5 z< 0.5 D(p, q) = i p i ( pi q i ) D(p, q) = i q i ( qi p i ) z 0.5, 0.5 z z D z D z

161 p, q, r n D z (p, q) <D z (p, r) D z (p, q) >D z (p, r). z 0.5, 0.5 i D z (p, q) j D z (p, q) = ( p i ) z+1 ( q i pi+1 ( ) z+1 ( ) z+1, pj pj+1 q j q j+1 q i+1 ) z+1 D(p, q) =K = D(p, (q 1,...,q i + ϵ, q i+1 ϵ,..., q j + g z (ϵ),q j+1 g z (ϵ),...,q n )), g z(0) = id z (p, q) j D z (p, q) = ( p i ) z+1 ( q i pi+1 ( ) z+1 ( ) z+1. pj pj+1 q j q j+1 q i+1 ) z+1 z z p, q n g z(0) g z(0). q D(p, q) p z i z j

162 3 p D z D z q D z (p, q) D z (p, q) z z p q r q D z (p, q) <D z (p, r) D z (p, r) <D z (p, q). p, q n z 0.5, 0.5 D z (p, q) =D z (q, p) z R z z 0.5 Dz (p, q) =( p(θ)/q(θ) p z 0.5), w f : Θ R + f(θ) p w := ( i f(θ i) w p(θ i )) 1/w

163 D z z =0.5 z = 0.5 D z (p, q) z 0.5 z 0.5 = j p j ( pj q j ) =: D 0.5 (p, q) z 0.5 D z (p, q) z 0.5 = j ( ) qj q j =: D 0.5 (p, q). p j D z f ( w 0 f p w = p j j ( pj q j ) ), p(θ)/q(θ) z 0.5 D z (p, q) z 0.5 = D z (q, p) z 0.5 z 0.5 w= z D w (q, p) = w 0.5 w 0.5, z < 0.5 D z (p, q) =+ Supp(p) Supp(q) =. z 0.5 z 0.5 D z (p, q) =+ Supp(p) \ Supp(q). z 0.5 D z (p, q) =+ Supp(q) \ Supp(p).

164 D z (p, q) z < 0.5 D z (p, q) = ( j p z+0.5 j q 0.5 z j ), z < z>0 < 0.5 z p, q n D z (p, q) =+ j pz+0.5 j q 0.5 z j =0 0 0 z>0.5 q j θ j Supp(p) \ Supp(q) D z j Supp(p) Supp(q) q j =0 p j D(p, q) < + z 0.5 D z (p, ) : n R p n D z θ Supp(p) ( D z j Supp(p) p j (p, q) = j Supp(p) p j ( pj ( pj q j ) z 0.5 ) q j ) z>0.5. z =0.5 D z (,q): n R q n z 0.5 D z (,q): n R q n D z

165 θ Supp(q) ( D z j Supp(q) p j (p, q) = j Supp(q) p j ( pj ( pj q j ) z 0.5 ) q j ) z>0.5. z =0.5 D z (p, ) : n R p n z < 0.5 D z : n n R D z (p, q) = j Supp(p) Supp(q) ( ) z 0.5 pj p j, q j j =0 z 0.5 z>0.5 D z (p, ) n D z n (q (n) ) n q (n) j 0 p j =0 D z (p, q (n) ) j p j > 0 ( D z (p, q (n) ) p z+0.5 j (q (n) j ) 0.5 z ) (0.5 z) (q (n) j ) +, q (n) j 0 (0.5 z) < 0 D z (p, q (n) ) + D(p, r) =+ p j > 0 r j =0 D z (p, )

166 n D z (p, q) = j Supp(p) ( ) z 0.5 pj p j. q j D z (,q): n R q n q n D z (,q) n q / n D z (,q) q q 1 =0 q j > 0 j =2,...,n z>0.5 D z (p, q) p p (m) p (m) j := 1 j =1 m (m 1) > 1 m(n 1) D w (p (m),q)=+ m Supp(p) \ Supp(q), p m p = ( ) 1 0, n 1,..., 1, n 1 D z (p,q) < + Supp(q) \ Supp(p) =. z =0.5 p q z < 0.5 z +0.5 z 0.5

167 D z (p, q) = j Supp(p) Supp(q) p z+0.5 j q z 0.5 j. z 0.5 ϵ, δ > 0 p, q,, n p q > x,y n x y δ, <ϵ, D z (, ) >D z ( p, q) z < 0.5 D z (p, q) ϵ δ p q δ D z (p, q) <ϵ z > 0.5 n =2 n (p, 1 p, 0,...,0) x,y 2 x y =1 ϵ δ p, q n p q > 1 δ p, q n D( p, q) < + (p m ) m, (q m ) n p m q m ϵ D(p m,q m ) + m + q m := (1/m, 1 1/m) p m =(1/m+ϵ, 1 1/m ϵ) p m q m ϵ z>0.5 D z (p, q) ( p m 1 ( p m 1 q 1 m ) z 0.5 ) = [ ( 1 m + ϵ )( 1 + ϵ ) z 0.5 ] m, 1 m

168 m + z =0.5 ( p m D 0.5 (p, q) p m 1 1 q1 m ) = ( ) 1 m + ϵ ( 1 + ϵ ) m, 1 m m + ϵ>0 z< 0.5 p q D z z < 0.5 n =2 n 2 p =(x, 1 x) q =(x + ϵ, 1 x ϵ) x [0, 1 ϵ] α := z +0.5 α (0, 1) z < 0.5 D z (p, q) = ( x α (x + ϵ) 1 α +(1 x) α (1 x ϵ) 1 α), x α (x+ϵ) 1 α (1 x) α (1 x ϵ) 1 α x α (x + ϵ) 1 α +(1 x) α (1 x ϵ) 1 α x =0 x =1 ϵ D z (p, q) { (ϵ α +(1 ϵ) 1 α ); ((1 ϵ) α + ϵ 1 α )}. ϵ 0 (ϵ α +(1 ϵ) 1 α ) ((1 ϵ) α +ϵ 1 α ) 0 δ := { (ϵ α +(1 ϵ) 1 α ); ((1 ϵ) α + ϵ 1 α )},

169 n =2 n p, q p j = 0 j p 1 = 0 D z (p, q) D z (p, (0,q 1 + q 2,...,q n )), n 1 p n q p q ϵ p j q j ϵ j p 1 <q 1 p =(0,p 1 + p 2,p 3,...,p n ), q =(q 1 p 1,q 2 + p 1,q 3,...,q n ). n =2 D z (p, q) D z (p,q ), p / n n 1 D Θ Θ ( D n ) n 2,n N D n : (Θ) (Θ) R + {+ }, Dn (p, q) :=D Θ (p, q), q 2 + p 1 < 1 j q j + p 1 < 1 j 2

170 Θ Θ = n D n p, q n λ [0, 1] D n (p, λp +(1 λ)q) D n (p, q). p, q, r n λ [0, 1] D n (λp +(1 λ)r, λq +(1 λ)r) D n (p, q). p, q 1,q 2 λ [0, 1] D n (p, λq 1 +(1 λ)q 2 ) {D n (p, q 1 ),D n (p, q 2 )}. B(p, ρ] :={q n D n (p, q) ρ} D n p 1 = p 2 = p q 1 = p p 1 = p p 2 = r = q 2

171 q 1 = q p 1 = p 2 = p D n C n D n (p, q) =D n ( p, q), p,q C p, q C p, q n D n (p, q) D n (e 1,e 2 ) (= D n (e j,e i ) i, j), e 1 =(1, 0, 0,...,0) e 2 =(0, 1, 0,...,0) s := p,q C D n (p, q) s R + {+ } (p m,q m ) C D n (p m,q m ) s ( p m, q m ) p m, q m C [p m,q m ] [ p m, q m ] D n ( p m, q m ) D n (p m,q m ) m D n ( p m, q m ) s, m D n ( p m, q m )=s s C C ( p m, q m ) C (p,q ) D n (p,q )=s D D D a, b n [a, b] [a, b] :={q n q = λa +(1 λ)b, λ [0, 1]}.

172 φ : R + R + D := φ(d) D n (e 1,e 2 ) D n (p, q) p, q n e 1 =(1, 0,...,0) e 2 =(0, 1, 0,...,0) p, q n D n (p, q) =D n+1 ((p 1,...,p n, 0), (q 1,...,q n, 0)), n =2 D 2 (e 1,e 2 ) >D 2 (e 1, (x, 1 x)) x (0, 1) D 2 (e 1,e 2 )=D 2 (e 1, (x, 1 x)) x (0, 1) p, q n x (0, 1) D n (p, q) =D 2 (e 1, (x, 1 x)), φ(x) =D 2 (e 1, (x, 1 x)). φ(1) = 0 φ(x) x 0 p, q n x D n (p, q) =D 2 (e 1, (x, 1 x)) D 2 x D n ψ ψ(d n (p q, r s)) =

173 ψ(d n (p, r)) + ψ(d n (q, s)). D n x, y, z D 4 (e 1 e 1, (x, 1 x) (y, 1 y)) = D 2 (e 1, (z,1 z)). ψ ψ(d 2 (e 1, (x, 1 x))) + ψ(d 2 (e 1, (y, 1 y))) = ψ(d 2 (e 1, (z,1 z))) x y D 4 (e 1 e 1, (x, 1 x) (y, 1 y)) = D 4 (e 1 e 1, (x, 1 x ) (y, 1 y )), ψ(φ(x))+ψ(φ(y)) = ψ(φ(x ))+ψ(φ(y )) φ(x ) >φ(x) w D 4 (e 1 e 1, (w, 1 w) (x, 1 x)) = D 2 (e 1, (x, 1 x )), D 8 (e 1 e 1 e 1, (w, 1 w) (x, 1 x) (y, 1 y)) = D 4 (e 1 e 1, (x, 1 x ) (y, 1 y)),

174 D 8 (e 1 e 1 e 1, (w, 1 w) (x, 1 x) (y, 1 y)) = D 8 (e 1 e 1 e 1, (w, 1 w) (x, 1 x ) (y, 1 y )), x D 4 (e 1 e 1, (w, 1 w) (y, 1 y )) = D 2 (e 1, (y, 1 y)). φ(x ) >φ(x) φ(y ) <φ(y) w D 4 (e 1 e 1, (w, 1 w ) (y, 1 y )) = D 2 (e 1, (y, 1 y)), w D 2 (e 1, (w, 1 w)) = D 2 (e 1, (w, 1 w )), ψ(φ(y )) ψ(φ(y)) = (ψ(φ(x )) ψ(φ(x))) ψ(φ(x)) + ψ(φ(y)) = ψ(φ(x )) + ψ(φ(y )) p, q, r, s ψ(d nm (p q, r s)) = ψ(d n (p, r)) + ψ(d m (q, s)).

175 x p,r (0, 1) x q,s D n (p, r) =D 2 (e 1, (x p,r, 1 x p,r )) D m (q, s) =D 2 (e 1, (x q,s, 1 x q,s )). x p,r x q,s D nm (p q, r s) =D nm (e 1 e 1, (x p,r, 1 x p,r ) (x q,s, 1 x q,s )). D n (p, r) D 2 (e 1, (x p,r, 1 x p,r )) D n (p, r) D 2 (e 1, (x p,r, 1 x p,r )) D nm (p q, r s) =D nm (e 1 q, (x p,r, 1 x p,r ) s), q, s (e 1 e 1, (x p,r, 1 x p,r ) (x q,s, 1 x q,s )) ψ(d 4 (e 1 e 1, (x p,r, 1 x p,r ) (x q,s, 1 x q,s ))) = ψ(d 2 (e 1, (x p,r, 1 x p,r ))) + ψ(d 2 (e 1, (x p,r, 1 x p,r ))). D 4 (e 1 e 1, (x p,r, 1 x p,r ) (x q,s, 1 x q,s )) = D nm (p q, r s),

176 D nm (p q, r s) =D n (p, r)+d m (q, s). D 2 (e 1,e 2 ) >D 2 (e 1, (x, 1 x)), x (0, 1). D 2 (e 1,e 2 )=D 2 (e 1, (x, 1 x)) x (0, 1) φ(z) :=D 2 ((z,1 z), (x, 1 x)) z (0, 1) ψ ψ(d(, )) p, q, r, s 2 n e 1 x D n p, q n D n (p, q) =D n+1 ((p 1,...,p n, 0), (q 1,...,q n, 0)). m n D m+1 ( p m+1, q m+1 ) D m ( p m, q m ), D m ( p m, q m ) m

177 p (1, 0) p (1, 0) := (p 1,...,p n, 0,...,0) = p 2n q (1, 0) = q 2n D 2n ( p 2n, q 2n )=D 2n (p (1, 0),q (1, 0)) = D n (p, q)+d 2 ((1, 0), (1, 0)), D 2 ((1, 0), (1, 0)) = 0 D 2n ( p 2n, q 2n )=D n (p, q), D mn ( p mn, q mn )=D n (p, q), m N D m ( p m, q m ) m p, q n Supp(p) Supp(q) = p j q j =0 j D D(p, q) =+. p, q n D(p, q) D((1, 0), (0, 1)), Θ A 1 := Supp(p) A 2 =Θ\ Supp(p) D((1, 0), (0, 1)) = +

178 D((1, 0), (0, 1)) < (1, 0) (1, 0) = (1, 0, 0, 0) (0, 1) (0, 1) = (0, 0, 0, 1). D((1, 0, 0, 0), (0, 0, 0, 1)) = D((1, 0), (0, 1)), D((1, 0) (1, 0), (0, 1) (0, 1)) = 2D((1, 0), (0, 1)) D((1, 0), (0, 1)) = 2D((1, 0), (0, 1)), D((1, 0), (0, 1)) < D((1, 0), (0, 1)) = 0 D(p, q) > 0 p q D((1, 0), (0, 1)) = p, q n p 1 q 1 = p 2 q 2. D n (p, q) =D n 1 ((p 1 + p 2,p 3,...,p n ), (q 1 + q 2,q 3,...,q n )). D n (p, q) D n 1 ((p 1 + p 2,p 3,...,p n ), (q 1 + q 2,q 3,...,q n )) D n (p, q) D n 1 ((p 1 + p 2,p 3,...,p n ), (q 1 + q 2,q 3,...,q n )) D(p, q) D((p 1 + p 2,p 3,...,p n ), (q 1 + q 2,q 3,...,q n )).

179 D((p 1 + p 2,p 3,...,p n ), (q 1 + q 2,q 3,...,q n )) = D((0,p 1 + p 2,p 3,...,p n ), (0,q 1 + q 2,q 3,...,q n )) D((p 1 + p 2,p 3,...,p n ), (q 1 + q 2,q 3,...,q n )) = D((p 1 + p 2, 0,p 3,...,p n ), (q 1 + q 2, 0,q 3,...,q n )). p = λ(p 1 + p 2, 0,p 3,...,p n )+(1 λ)(0,p 1 + p 2,p 3,...,p n ), λ = p 1 p 1 +p 2 q = λ (q 1 + q 2, 0,q 3,...,q n )+(1 λ )(0,q 1 + q 2,q 3,...,q n ), λ = q 1 q 1 +q 2. p 1 p 2 = q1 q 2 λ = λ D(p, q) {D((0,p 1 + p 2,p 3,...,p n ), (0,q 1 + q 2,q 3,...,q n )), D((p 1 + p 2, 0,p 3,...,p n ), (q 1 + q 2, 0,q 3,...,q n ))} = D((p 1 + p 2,p 3,...,p n ), (q 1 + q 2,q 3,...,q n )),

180 p n q 1 = (q 1 + q 2, 0,q 3,...,q n ) q 2 =(0,q 1 + q 2,q 3,...,q n ) [q 1,q 2 ]={r n r = λq 1 +(1 λ)q 2,λ [0, 1]}. p [q 1,q 2 ] q [q 1,q 2 ] q1 q2 = p 1 p 2, r, r [q 1,q 2 ] r [r,q ] D n (p, r) D n (p, r ). p 1 q 1 = p 2 q 2 D(p, q )=D((p 1 + p 2,...,p n ), (q 1 + q 2,...,q n )), r [q 1,q 2 ] D(p, r) D((p 1 + p 2,...,p n ), (q 1 + q 2,...,q n )),

181 r [q 1,q 2 ] D(p, r) D(p, q ), r [q 1,q 2 ]. f(q) :=D(p, q) r r r [r,q ], D(p, r) {D(p, r ),D(p, q )}, D(p, q ) D(p, r ) D(p, r) D(p, r ) p, q n p q D π =(S, f(s θ)) s S D(p(s ),q(s )) <D(p, q). D p, q n π =(S, f(s θ)) s S D(p(s),q(s)) >D(p, q). s θ 1 f(s θ 1 )=1 f(s θ j )=0 j =2,...,n p, q

182 p(s )=q(s )=(1, 0,...,0) D(p(s ),q(s )) = 0 <D(p, q), p q p q s D(p (s ),q(s )) <D(p,q ) s f(s θ) = k f(s θ), θ, k f(s θ) < 1 p := p (s ) q := q (s ) p(s) =p q(s) =q D(p, q) =D(p (s ),q(s )) <D(p,q )=D(p(s),q(s)), p q n π π E π p[d(p(s),q(s))] E π p[d(p( s),q( s))]. z 0.5

183 z 0.5 p =(1, 0,...,0) S π f(s θ 1 ) > 0 S π s S f(s θ j )=0 j 2 E π p[d(p(s),q(s))] = 0, s f(s θ j ) > 0 θ j 2 s S g( s θ j ) > 0 g( s θ j )= s λ s, s f(s θ j ), s λ s, s =1 s p q j ( s) > 0=p j ( s) D z (p( s),q( s)) = E π p[d(p( s),q( s))] =, z < 0.5 s S p 1 (s) = 1 D z (p(s),q(s)) = (q 1 (s) z+0.5 ) s (0.5 z) s f(s θ 1 ) (q 1 (s)) (0.5 z) s g( s θ 1 ) (q 1 ( s)),

184 (0.5 z) < 0 z < 0.5 g( s θ 1 ) (q 1 ( s)) s s f(s θ 1 ) (q 1 (s)) s s λ s, s =1 f(s θ 1 ) (q 1 (s)) = s s λ s, s f(s θ 1 ) (q 1 (s)) = s s λ s, s f(s θ 1 ) (q 1 (s)), s g( s θ 1 ) f(s θ 1 ) (q 1 (s)) = s s λ s, s f(s θ 1 ) (q 1 (s)) s = s g( s θ 1 ) s λ s, s f(s θ 1 ) g( s θ 1 ) (q 1 (s)), λ s, sf(s θ 1 ) g( s θ 1 ) s λ s, s f(s θ 1 ) g( s θ 1 ) =1 ( ) s λ s, s f(s θ 1 ) ( (q 1 (s) 1 )) g( s θ 1 ) ( s λ s, s f(s θ 1 ) g( s θ 1 ) ) f(s; q) q 1 f(s θ 1 ) ( = s λ s, sf(s; q) q 1 g( s θ 1 ) ), s λ s, sf(s; q) =f( s; q) ( s λ ) s, sf(s; q) = (q 1 ( s)), q 1 g( s θ 1 )

185 f(s θ 1 ) (q 1 (s)) s s g( s θ 1 ) (q 1 ( s)), D D(p, q) =ad z (p, q) z 0.5 a>0 z 0.5 D z z<0.5 p, q q j =0<p j D(p, q) < + q =(1, 0) p =(α, 1 α) D D D(p, q) =ad z (p, q) z 0.5 a>0 π π p, q n E π p[d(p(s),q(s))] E π p[d(p( s),q( s))].

186 p, q n q / n p / n Θ p p, q n π π φ : n R p n P p (s)φ(p(s)) s S s S P p ( s)φ(p( s)). D z (p(s),q(s)) p(s) z 0.5 q(s) p(s) p q q j (s) = q j i p j p j (s) q i p i p i (s) = r jp j (s) p(s) r, r j := q j p j

187 R n ( ( ) ) z 0.5 pj (s) D z (p(s),q(s)) = p j (s) q j j (s) ( ( ) ) z 0.5 p(s) r = p j (s) j r j =(z 0.5) (p(s) r)+(p(s) r z ), r z j := 1 r z 0.5 j z>0.5 D z (p(s),q(s)) p(s) r z (p(s) r z ) (z 0.5) (p(s) r) r 0 z 0.5 > 0 z =0.5 D 0.5 (p(s),q(s)) = j ( ) pj (s) p j (s) q j (s) = j p j (s)((p(s) r) (r j )) = p(s) (r)+(p(s) r), (r) :=((r 1 ),...,(r n )). D 0.5 (p(s),q(s)) p(s) p(s) z<0.5 p, q π E π p(d z (p(s),q(s))) >D(p, q);

188 p, q π E π p(ad 0.5 (p(s),q(s)) + bd 0.5 (p(s),q(s))) > ad 0.5 (p, q)+bd 0.5 (p, q)) z<0.5 p 2 q n =(1/n, 1 1/n) n D z (p, q) < D z (p, (0, 1)) < + π(ϵ) ϵ 0 E π(ϵ) p (D(p(s),q n (s))) + n π π(ϵ) f(s 0 θ 1 )=1 ϵ, f(s 1 θ 1 )=ϵ, f(s 1 θ 2 )=ϵ f(s 0 θ 2 )=1 ϵ. s 1 p 2 (s 0 )= ϵp 2 ϵp 2 +(1 ϵ)p 1 ; q 2 (s 0 )= ( 1 1 n) ϵ ( ) 1 1 n ϵ + 1 (1 ϵ). n ϵ = n 0.5 n + p 2 (s 0 ) 0 q 2 (s 0 ) 1 n E π(n 0.5 ) p (D(p(s),q(s))) P p (s 0 )D z ((1 δ, δ), (δ, 1 δ)),

189 δ (0, 1) n P p (s 0 ) > 0 D z (p, q n ) <D z (p, (0, 1)) < n E π(n 0.5 ) p (D z (p(s),q(s))) >D z (p, q n ), z D z ((1 δ, δ), (δ, 1 δ)) δ 0 D(p, q) =ad 0.5 (p, q) +bd 0.5 (p, q) p =(0.5 ϵ, ϵ, 0.5) q =(0.5 ϵ, δ, 0.5+ϵ δ) ( D(p, q) =a δ ( ) δ +(0.5+ϵ δ) ϵ ( + b ϵ ( )) 0.5+ϵ δ 0.5 ( ϵ ) ( +0.5 δ ϵ δ )). π =({s 1,s 2 },f(s θ)) f(s 1 θ 1 )=f(s 1 θ 2 )=1 f(s 2 θ 3 )=1 ϵ, δ π θ 3 p(s 2 )=q(s 2 )= (0, 0, 1) ϵ, δ p(s 1 )= ( ) 0.5 ϵ 0.5, ϵ 0.5, 0 q(s 1 )= ( ) 0.5 ϵ 0.5 ϵ + δ, δ 0.5 ϵ + δ, 0.

190 P p (s 1 )=0.5 ϵ, δ 0.5D(p(s 1 ),q(s 1 )) = 0.5aD 0.5 (p(s 1 ),q(s 1 )) + 0.5bD 0.5 (p(s 1 ),q(s 1 )). δ 0 ϵ 0.5 D 0.5 (p, q) 0.5D 0.5 (p(s 1 )q(s 1 )) D 0.5 (p, q) 0.5D 0.5 (p(s 1 )q(s 1 )) D 0.5 (p, q) 0.5D 0.5 (p(s 1 )q(s 1 )) ( ϵ ) ( ) 0.5+ϵ δ = ϵ +(0.5+ϵ δ) δ 0.5 ( ( ) 0.5 ϵ ϵ ( ( ϵ ) ( ))) 0.5 ϵ + δ ϵ + δ 0.5 δ δ ( ) ( ) 0.5+ϵ δ 0.5 ϵ + δ = δ ϵ. 0.5 δ δ = 0.5 ϵ ϵ 0.5 D 0.5 (p, q) 0.5D 0.5 (p(s 1 )q(s 1 ))

191 D 0.5 (p, q) 0.5D 0.5 (p(s 1 )q(s 1 )) D 0.5 (p, q) 0.5D 0.5 (p(s 1 )q(s 1 )) ( ) ( ) δ 0.5+ϵ δ = δ +(0.5+ϵ δ) ϵ 0.5 ( ( ) 0.5 ϵ 0.5 ϵ + δ ϵ + δ δ ( ( ) ( )) ) δ δ δ ϵ + δ ϵ 0.5 ϵ + δ ϵ 0.5 δ = 0.5 ϵ D 0.5 (p, q) =δ ( ) δ ϵ + (0.5+ϵ δ) ( ) 0.5+ϵ δ 0.5 δ = 0.5 ϵ 0.5D(p(s 1 ),q(s 1 )) p 1,...,p J n (u j ) j z j (0, 1) (Π i ) i 1 D zj+0.5 (q, p j ( I)), q (I) 1 z j j

192 V j ((x j i ) i)=(d z j+0.5 (p j ( I), Π) + D z j+0.5 (Π,p j ( I))) 1 j J; G j ((x j i ) i)= 1 ( ( )) 1 1 Dzj 0.5 (Π,p j ( I)), j J. z j 1 z j j (x j i ) i i p j i (x j i )1 1 z j 1 1 z j Π i x i 0, i x j i p j i (xj i ) 1/z j λπ i =0, λ x j i x j i = ( p j i λπ i ) zj, i Π ix j i =1 λ z j = i ( ) p j zj i Π i, Π i

193 x j i = ( p j i k Π k Π i ) zj ( Π k p j k ) zj, j 1= j x i,j = j ( p j i k Π k Π i ) zj ( Π k p j k ) zj, i i i +1 j ( p j i Π i ) zj ( k Π k ( ) zj p j i+1 Π i+1 Π k p j k ) zj =0, j 1 i ( ) 1 D zj+0.5 (Π,p j ( I)), 1 z j 1 i D D Π z i,i+1 D

194 V j ((x j i ) i)= i ( p j i xj i 1= i pj i ( k Π k ) zj Π i p j i Π k p j k ) zj 1 = (D z j+0.5 (p j ( I), Π) + D z j+0.5 (Π,p j ( I))) 1. u j (1) = z j u j (x j i )= z j i ( k Π k ) zj Π i p j i ( ) zj Π k p j k p j i u(xj i )= z j 1 1 z j ( i p j i ( Πi p j i ) ) ( ( 1 zj Π k k Π k p j k ) zj ) 1 z j 1, G j ((x j i ) i)= 1 ( ( )) 1 1 Dzj 0.5 (Π,p j ( I)), j J. z j 1 z j (u j ) j J 1 u j(1) = 1 1/u j (1) =: z j (0, 1) (p j,(m) ) j J m N m ((x j,(m) i ) j J i=1,...,n, (Πm i ) i ) m Π m q n j 1 1 z j D zj+0.5 (q, p j,(m) ).

195 j m D z j +0.5 (p j,(m),π m ) D z j +0.5 (p j,(m), Π m ) =1 pj Π m Π m m i pj,(m) i (x j,(m) i 1) D z j+0.5 (p j,(m), Π m )+D z j+0.5 ( Π m,p j,(m) ) =1. m i pj,(m) i u i (x j,(m) i ) 1 1 z j D z j 0.5 ( Π m,p j,(m) ) =1. m p j,(m) x j,(m) i 1 u j ũ j u j 1 1 z j ũ j (x) := x1 1 1, z j z j = 1 u (1)

196 J := ((p 1 i ) i,...,(p J i ) i ) ((x j i ( )) i,j, (Π i ( )) i, (λ j ( )) j ) u j x j i Π i λ j (( x j i ( )) i,j, ( Π i ( )) i, ( λ j ( )) j ) ũ j x j i ( ) j, i Π i ( ) i λ j ( ) j =(p,...,p ) x j i ( )= x j i ( ), v =(v 1,...,v J ) j J i vj i =0 pj + ϵv j n ϵ ((x j i ( )) i,j, (Π i ( )) i, (λ j ( )) j ) (( x j i ( )) i,j, ( Π i ( )) i, ( λ j ( )) j ) u j ũ j p j,(m) i u j(x j,(m) i ) λ m j Π m =0, p j,(m) i ũ j( x j,(m) i ) λ m j Π m =0,

197 Π i ( )= Πi ( ) λ j ( )= λj ( ). i, j x j i ( ) 1 =1, ( ) 1 x j i n n. x j i =(u ) 1 ( λj ( )Π i ( ) p i,j ) x j i ( ) = ( λj ( ) Π i ( ) p i,j ) zj ũj φ( ) := λ j ( )Π i ( ) p i,j φ( ) φ( )= φ( ),. 0/0 (x j i ( ) 1) ( x j =1, i ( ) 1) 1 u ((u ) 1 (φ( ))) φ( ), z j φ( ) φ( )

198 φ( ) 1 φ( ) 1 1 u ((u ) 1 (φ( ))) = z j φ( ), φ( ) φ( ) 0 p j i i xj i ( )pj i i xj i ( )pj i =1, D zj+0.5 (p j, Π( )) + D zj+0.5 ( Π( ),p j ) u j ũ j Π Π p j i Π j i ( ) = λ j( )u (x j i ( )) p j i Π j i ( ) = λ j ( )ũ ( x j i ( )), i, j p j i Π j i ( ) =1, p j i Π j i ( ) φ

199 (x i ) i x i 0, x i 1. i n G n G n n I(x y) x y n N x, y G n I(x, y) 0 x i y i i I(x y) 0 x i y i i I(1, 1/2) = 1 n, m N x 1,y 1 G n x 2,y 2 G m I(x 1 x 2 y 1 y 2 )= I(x 1 y 1 )+I(x 2 y 2 ) x G n y G m x y := (x 1,...,x n,y 1,...,y m ) G n+m g : R R x 1,x 2,y 1,y 2 x 1 x 2

200 y 1 y 2 I(x 1 x 2 y 1 y 2 )=g 1 ( i x1 i ) g(i(x 1 x 2 )) + ( ) j x2 j i x1 i + j x2 j g(i(y 1 y 2 )) (P 6) (P 7) x, y x i y i i x = y (P 7) I(p p) =0 P (8) (P 9) x 1,y 1,x 2,y 2 G 1 =[0, 1] I(x 1 x 2 y 1 y 2 )=I(x 1 y 1 )+I(x 2 y 2 ). I x, y [0, 1] I(x y) = β (x/y) β R (P 9) I 1 x, y [0, 1] (P 9) (P 10) I(x y) x G n y G m x y n m x i + y j 1. i=1 j=1

201 z,z [0, 1] I(z z )=β (z/z ) (P 10) x, y G n I(x y) =g 1 ( j x j i x g(α (x j /y j )) i ), g I (P 9) g(x) :=γe αx (P 8) I(x y) = 1 α 1 ( ) α 1 j x xj j y j j x j, R α (p, q) ( R α (p, q) = 1 ( ) ) α 1 α 1 pj p j. q j j (P 10) (p i /q i )

202 (P 10) x y (x, y) I(x y) (x, y) I(y x) (x, y) D(x, y) (x, y) D(y, x) D(p, q) = j q j ( qj p j ) n x 1,...,x n R ( M p φ (x 1,...,x n )=φ 1 j p ) jφ(x j ), n φ( ) p q p q

203 s 1,s 2 f(s 1 θ)f(s 2 θ )=f(s 1 θ )f(s 2 θ) θ, θ γ : S S γ(s 1 )=γ(s 2 ) γ(s) γ(s ) s s s, s s 1,s 2 c n (γ π) =c(π). c n (γ π) c n (π) c n (γ π) c n (π) k>0 kf(s 1 θ) =f(s 2 θ) θ Θ,

204 γ π = (( (1+k)f(s1 )) θ 1 )... (1+k)f(s 1 θ n) f(s 3 θ 1 )... f(s 3 θ n) f(s m θ 1 )... f(s m θ n) = ( 1+k k f(s 1+k 2 θ 1 )... k f(s 2 θ n) f(s 3 θ 1 )... f(s 3 θ n) f(s m θ 1 )... f(s m θ n) ), (1+k)f(s 1 θ 1 )... (1+k)f(s 1 θ n) c n (γ π) =c n f(s 3 θ 1 )... f(s 3 θ n) = c n f(s m θ 1 )... f(s m θ n) k k f(s 1+k 2 θ 1 )... k f(s 2 θ n) f(s 3 θ 1 )... f(s 3 θ n) f(s m θ 1 )... f(s m θ n) , π π = 1 k +1 (1+k)f(s 1 θ 1 )... (1+k)f(s 1 θ n) f(s 3 θ 1 )... f(s 3 θ n) + k f(s m θ 1 )... f(s m θ n) k k k f(s 1+k 2 θ 1 )... k f(s 2 θ n) f(s 3 θ 1 )... f(s 3 θ n) f(s m θ 1 )... f(s m θ n) c n (π) { c n (1+k)f(s 1 θ 1 )... (1+k)f(s 1 θ n) f(s 3 θ 1 )... f(s 3 θ n) f(s m θ 1 )... f(s m θ n) , c n k k f(s 1+k 2 θ 1 )... k f(s 2 θ n) f(s 3 θ 1 )... f(s 3 θ n) f(s m θ 1 )... f(s m θ n) }, c n (π) c n (γ π),

205 π π c n c n (π) c n (π ) π = (S, (f(s θ)) s S,θ Θ ) π = (S, (g(s θ)) s S,θ Θ) λ s,s 0 s S λ s,s =1 g(s θ) = s λ s,s f(s θ). s s f(s θ) > 0 θ s s g(s θ) > 0 π c n (π) =c n λ s1,s 1 f(s 1 θ 1 )... λ s1,s 1 f(s 1 θ n) λ s1,s 2 f(s 1 θ 1 )... λ s1,s 2 f(s 1 θ n) λ s1, sf(s 1 θ 1 )... λ s1, sf(s 1 θ n) f(s 2 θ 1 )... f(s 2 θ n) f(s m θ 1 )... f(s m θ n) , s s =1 λ s 1,s = 1 π c n (π) =c n λ s1,s 1 f(s 1 θ 1 )... λ s1,s 1 f(s 1 θ n) λ s1,s 2 f(s 1 θ 1 )... λ s1,s 2 f(s 1 θ n) λ s1, sf(s 1 θ 1 )... λ s1, sf(s 1 θ n) λ s2,s 1 f(s 2 θ 1 )... λ s2,s 1 f(s 2 θ n) λ s2, sf(s 2 θ 1 )... λ s2, sf(s 2 θ n) λ sm,s 1 f(s m θ 1 )... λ sm,s 1 f(s m θ n) λ sm, sf(s m θ 1 )... λ sm, sf(s m θ n) , s

206 c n λ s1,s 1 f(s 1 θ 1 )... λ s1,s 1 f(s 1 θ n) λ s1,s 2 f(s 1 θ 1 )... λ s1,s 2 f(s 1 θ n) λ s1, sf(s 1 θ 1 )... λ s1, sf(s 1 θ n) λ s2,s 1 f(s 2 θ 1 )... λ s2,s 1 f(s 2 θ n) λ s2, sf(s 2 θ 1 )... λ s2, sf(s 2 θ n) λ sm,s 1 f(s m θ 1 )... λ sm,s 1 f(s m θ n) λ sm, sf(s m θ 1 )... λ sm, sf(s m θ n) (( c n s λ s,s f(s θ 1)... s λ 1 s,s f(s θn) 1 s λ s,s f(s θ 1)... s λ 2 s,s f(s θn) s λ s, s f(s θ 1)... s λs, sf(s θn) )), π c n (π) c n (π ) φ : E n R {+ } φ π φ(π) =0 φ(π) = π π := π π c n (π) =c n ( π) =c n (π π) =c n (π)+c n (π), c n (π) =0 c n (π) = H : (Θ) R H(µ) >H(δ j ) µ (Θ) j =1,..., Θ µ (Θ) φ µ : E n R φ(π 1 π 2 )=φ(π 1 )+φ(π 2 ), π 1,π 2 E n.

207 φ µ (π) := H(µ) s S P µ (s)h(µ(s)). φ µ (π) π s s P µ (s) > 0 µ(s) =δ j j H(µ) > s S P µ(s)h(µ(s)) φ µ (π) > 0 π φ µ (π) = H Θ Θ > 3 S c Θ,S ( ) f(s θi ) (f(s θ i ) f(s θ j )) = f(s θ s i,j j ) i,j D(f( θ i ),f( θ j )), D D(p, q) = ( ) i p i pi q i + ( i q i qi p i ) c Θ,S = θ,θ c 2 (π(θ, θ )), c Θ,S (π) = θ,θ c KL 2 (π(θ, θ )), P µ (s) θ µ(θ)f(s θ) µ s

208 c Θ,S (π) = θ,θ c B 2 (π(θ, θ )), c Θ,S (π) = θ,θ c KL 2 (π(θ, θ )), c Θ,S (π) = π s S θ, θ Θ f(s θ) > 0=f(s θ ) c Θ,S (π) =0 D(f( θ),f( θ )) = θ, θ Θ D(f( θ),f( θ )) = s f(s θ) > 0=f(s θ ) f(s θ) =0<f(s θ ) c 2 c KL 2 (π) = s [f(s θ 1) f(s θ 2 )][(f(s θ 1 )) (f(s θ 2 ))]; ( ) c B 2 (π) = s f(s θ1 )f(s θ 2 ). 2 Θ =2 KL B

209 p, q f( θ),f( θ ) D(p, q) < p, q D D(p, q) = D(q, p) n =2 θ s c n f n f n (x 1,...,x n )= n x i, (x i ) i [0, ] n. i=1 Θ =3 c 3 (π) =c 2 (π(θ 1,θ 2 )) + f 2 (c 2 (π(θ 3,θ 2 )),c 2 (π(θ 1,θ 3 ))).

210 c 3 (π) =c 2 (π(θ 2,θ 3 )) + f 2 (c 2 (π(θ 1,θ 2 )),c 2 (π(θ 1,θ 3 ))), c 2 (π(θ 1,θ 2 )) + f 2 (c 2 (π(θ 3,θ 2 )),c 2 (π(θ 1,θ 3 ))) = c 2 (π(θ 2,θ 3 )) + f 2 (c 2 (π(θ 1,θ 2 )),c 2 (π(θ 1,θ 3 ))). c 2 f( θ 3 ) c 2 (π(θ 1,θ 3 )) (S) c 2 c 2 (π(θ 1,θ 3 )) 0 x f 2 (c 2 (π(θ 3,θ 2 )),c 2 (π(θ 1,θ 3 ))) c 2 (π(θ 1,θ 3 )) = c 2 (π(θ 1,θ 3 )). c 2 (π(θ 1,θ 3 )) 0 x f 2 (c 2 (π(θ 3,θ 2 )),c 2 (π(θ 1,θ 3 ))) = 1. c 2 (π(θ 1,θ 3 )) c 2 (π(θ 1,θ 3 )) x f 2 f 2