1. 2. C2

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1 (I) (II) ( 47, 1999) C1

2 1. 2. C2

3 1 ˆk AIC T C3

4 1.1 ( : 3 ) Y N ( µ(x a,x b,x c ),σ 2) µ(x a,x b,x c )=β 0 + β a x a + β b x b + β c x c x a,x b,x c α α {a, b, c} Θ α = {(σ, β) σ >0,β i =0,i α c }, α +2 C4

5 α M α M [ ] {a, b, c} M = {φ, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} 2 M = {{a, b}, {a, c}, {b, c}} α(k), k {1, 2,...,K} α k M α = f( Θ α ), α M M 1,M 2,...,M K C5

6 M α M β α β M α M β M α M β [ ] φ {a} {a, b} {a, b, c} α β β α M α M β [ ] {a, b} {b, c} ( ) C6

7 l n (α) l n (k) α k l(ˆθ α ) dim M α l(ˆθ k ) dim M k ˆα ˆk T M (T M) P {α T} P {k T} 1 P P C7

8 HALD : : 4 : n =13 : K =16 ( 2 4 =16 ) BOSTON : ( ) : 13 a= f= k= : m= : n = 506 : K = 286 ( 3 ) C8

9 HALD AIC p- α AIC LR 1 <abd> <abc> <ab> <acd> <abcd> <ad> <bcd> <cd> <bc> <d> <bd> <a> <ac> <c> <> C9

10 BOSTON 20 AIC p- p-value 100 α LK BA BP KH MC MS LR 1 <afm> <akm> <ahm> <agm> <adm> <alm> <ajm> <abm> <aim> <acm> <aem> <klm> <fjm> <jkm> <fkm> <flm> <hjm> <dkm> <bkm> <ekm> LK: log L BA: BP: KH: MC: ( ) MS: LR: C10

11 S S S S-2 <> <a> <ab> <c> <ac> <bc> <abc> <d> <ad> <bd> <abd> <cd> <acd> <bcd> <abcd> HALD S S S S-2 <m> <am> <bm> <abm> <cm> <acm> <bcm> <dm> <adm> <bdm> <cdm> <aem> <bem> <cem> <dem> <fm> <afm> <bfm> <cfm> <dfm> <efm> <gm> <agm> <bgm> <cgm> <dgm> <egm> <fgm> <hm> <ahm> <bhm> <chm> <dhm> <ehm> <fhm> <ghm> <im> <aim> <bim> <cim> <dim> <eim> <fim> <gim> <him> <jm> <ajm> <bjm> <cjm> <djm> <ejm> <fjm> <gjm> <hjm> <ijm> <km> <akm> <bkm> <ckm> <dkm> <ekm> <fkm> <gkm> <hkm> <ikm> <jkm> <lm> <alm> <blm> <clm> <dlm> <elm> <flm> <glm> <hlm> <ilm> <jlm> <klm> X BOSTON C11

12 DNA human seal cow rabbit mouse opossum h= ANLLLLIVPILIAMAFLMLTERKILGYMQLRKGPNVVGPYGLLQPFADAMKLFTKEPLKP INIISLIIPILLAVAFLTLVERKVLGYMQLRKGPNIVGPYGLLQPIADAVKLFTKEPLRP INILMLIIPILLAVAFLTLVERKVLGYMQLRKGPNVVGPYGLLQPIADAIKLFIKEPLRP INTLLLILPVLLAMAFLTLVERKILGYMQLRKGPNIVGPYGLLQPIADAIKLFTKEPLRP INILTLLVPILIAMAFLTLVERKILGYMQLRKGPNIVGPYGILQPFADAMKLFMKEPMRP INLLMYIIPILLAVAFLTLVERKVLGYMQFRKGPNVIGPYGILQPFADALKLFIKEPLRP X h : 6 human, (harbor seal, cow), rabbit, mouse, opossum : 10 split : n = 3416 : K =15 C12

13 2 <ai> a i 5 2 <bf> f b 3 4 <dg> d g <ah> a h 2 <ae> a e 3 <de> d e <cf> f c 2 <cg> g c 3 <ci> c i <dj> j d 2 <bj> j b 3 <bh> h b human seal cow harbor 2 e a rabbit mouse opossum 1 <ef> f e 3 2 <ij> j i 1 1 <gh> g h a = {123 45}, b = {134 25}, c = {234 15}, d = {124 35}, e = {12 345}, f = {34 125}, g = {24 135}, h = {13 245}, i = {23 145}, j = {14 235}. C13

14 Mammal Phylogeny: p-values p-values α log L LR BA BP KH MC MS MB tree 1 {a, e} (((12)3)45) 2 {a, i} ((1(23))45) 3 {a, h} (((13)2)45) 4 {e, f} ((12)(34)5) 5 {c, f} (1(2(34))5) 6 {c, i} (1((23)4)5) 7 {b, f} ((1(34))25) 8 {i, j} ((14)(23)5) 9 {d, e} (((12)4)35) 10 {b, j} (((14)3)25) 11 {b, h} (((13)4)25) 12 {d, j} (((14)2)35) 13 {c, g} (1((24)3)5) 14 {g, h} ((13)(24)5) 15 {d, g} ((1(24))35) LR: BA: BP: KH: MC: ( ) MS: MB:New Test C14

15 α log L ˆθ α {aefi} {ae} {bcef} {a} {ef} {e} {} full 95.2 (6.14,2.32,2.78,2.28,2.24,2.94,1.21,1.90,2.13,2.23) i f a 2 1 e <aefi> 3 (5.77) 1 e a 4 (2.28) <ae> c 2 3 f <bcef> 1 (6.18) a 4 <a> 5 (2.16) 1 e f 3 4 (1.76) <ef> (2.52) 5 2 e 3 4 <e> 61.3 (5.27,,,,1.70,3.07,,,2.43, ) 38.5 (5.86,,,,2.23,,,,, ) b e 34.6 (,2.17,3.05,,2.51,1.43,,,, ) 28.4 (6.18,,,,,,,,, ) 21.0 (,,,,1.88,2.31,,,, ) (,,,,2.52,,,,, ) < > 1 0 (,,,,,,,,, ) C15

16 4 2 S S <g> <gh> <dg> <h> <d> <> <dj> <j> <bh> <c> <cg> <bj> <de> <e> <f> <bf> <ef> <cf> <bcef> <ij> <ci> <ah> <a> <ai> <ae> <aefi> <a...j> S S <a...j> S-2 <e> <de> <ae> <ef> <d> <dg> <> <g> <h> <gh> <f> <bh> <bcef> <bf> <dj> <c> <cg> <j> <bj> <cf> <aefi> <a> <ah> <ai> <ij> <ci> S C16

17 1.2 ˆk x ˆk(X) ˆk = ˆk(x) ˆk(X) k P (ˆk(X) =k ) 1 (n ) AIC AIC C17

18 ( ) 4.6 AIC k (=ˆk) (n = 500) k A n = 2000 ˆk k A l 500 (k) k =10 l 2000 (k) k =12 C18

19 ( ) : (1000 ) n =10, 20, 50, 100, 200, 500, 1000 IC α = 2 l(ˆθ α )+c n dim M α c n : A =2 (AIC), B = log n (BIC), C =2n 0.1, D =2n 0.5, E =2n 0.6 correct selection count CB A D E CB A D E D B C A E D E C B A D E DE DE CB A CB A CB A correct selection count CB A D E CB A D E D D B B BE DB E DB E C C C C C A A A A D A E E correct selection count CB A D E E E D D D E D CB A B B B B C C A A C A CA D E E E D C B A sample size sample size sample size 3 C19

20 ( ) q^ p(). b q q^ q p* ^p a a p(). a p^b p* b p* 2 p^2 p 1 * p(). 2 ^ p1 p(). 1 1( ) 1 ( ) β =(β a,β b,β c ) 1: β =(1.0, 0.9, 0.1), M = {φ, {a}, {b, c}} C20

21 ( ) q q p* 2 p() 2 p* p() 3 2 p() 1 p 1 * 3 p(). 1 p(). 2 2: β =(1, 1, 0), M = {φ, {a}, {a, c}} 3: β =(1, 1, 0), M = {φ, {a}, {b, c}} C21

22 IC M α M β IC = 2 ( l(ˆθ α ) l(ˆθ β ) ) c n ( dim M α dim M β ) IC > 0 M α IC < 0 M β IC 2 ( l(θ α) l(θ β )) + ( (θ α, ˆθ α ) (θ β, ˆθ β ) ) c n ( dim M α dim M β ) ˆθ α (θ M α ) [ ] l l(θ) l(ˆθ α )+(θ ˆθ α ) θ ˆθ α + 1 [ 2 (θ ˆθ 2 ] l α ) θ (θ ˆθ α ) θ ˆθ α l(ˆθ α ) l(θ α )+ 1 2 (θ α, ˆθ α ) C22

23 IC 1 l(θ α) l(θ β )= n i=1 ( log f(xi θ α) log f(x i θ β )) n l(θ α) l(θ β ) N ( l (θ α) l (θ β ),nj α,β l (θ α) l (θ β )=n ( I(g; f(θ β )) I(g; f(θ α)) ) ) J α,β = V [ log f(x θ α) log f(x θ β )] I ( f(θ α),f(θ β )) + I ( f(θ β ),f(θ α) ) Ĵ α,β = 1 n n i=1 ( log f(xi ˆθ α ) log f(x i ˆθ β ) ( l(ˆθ α ) l(ˆθ β ) ) /n ) 2 C23

24 IC 1 n IC ( l(θ α) l(θ β )) N ( l (θ α) l (θ β ),nj ) α,β I(g; f(θ β )) I(g; f(θ α)) > 0 P ( IC > 0) 1 I(g; f(θ β )) I(g; f(θ α)) < 0 P ( IC < 0) 1 I(g; f(θ β )) I(g; f(θ α))=0 2 3 C24

25 IC 2 f( θ α)=f( θ β ) J α,β = V [ log f(x θ α) log f(x θ β )] =0 I(g; f(θ β )) I(g; f(θ α)) = 0 x IC 1 0 IC ( (θ α, ˆθ α ) (θ β, ˆθ β ) ) c n ( ) dim M α dim M β (θ α, ˆθ α ) χ 2 dim M α, (θ β, ˆθ β ) χ 2 dim M β d = dim M α dim M β > 0 M β c n M β P ( IC < 0) 1 C25

26 IC 2( ) (M β M α ) IC χ 2 d c nd, d = dim M α dim M β M β P ( IC < 0) = P (χ 2 d <c nd) c n =2,d=1 P ( IC < 0) 0.84 c n =2,d=10 P ( IC < 0) 0.97 d c n P ( IC < 0) 1 C26

27 IC 3 I(g; f(θ β )) I(g; f(θ α))=0 f( θ α) f( θ β ) IC 2 ( l(θ α) l(θ β )) c n ( ) dim M α dim M β 1 l (θ α) l (θ β )=0 l(θ α) l(θ β ) N ( 0,nJ α,β ) d = dim M α dim M β > 0 M β c n /n 1 2 M β P ( IC < 0) 1 C27

28 IC α = 2 l(ˆθ α )+c n dim M α 1 c n /n 0 2 c n 3 c n /n 1 2 AIC (c n =2) 1 BIC (c n = log n) 1 2 C28

29 1.3 ( 1) n P 1,P 2,...,P K ; p-value, p- C29

30 (p-value) M k, k =1,...,K [0, 1] 1 P k ( ) ( p-value) C30

31 x = {x 1,...,x n } ˆk(x) ( x i x ) x = { x 1,..., x n } ˆk( x) C31

32 ( B = 1000) x[1], x[2],..., x[b] ˆk( x[1]), ˆk( x[2]),...,ˆk( x[b]) M k P k = # {ˆk( x[b]) = k, b =1,...,B } K k=1 B P k =1, 0 P k 1, k =1,...,K M k P k C32

33 M α = f( Θ α ), α M M 1,M 2,...,M K ˆk M k M k C33

34 M k M 1,...,M K H k : k = k (l n (k) =l n (k ) k H k ) H 1,H 2,...,H K T = { k {1,...,K} H k P } C34

35 C35

36 AIC M α M β AIC = 2 ( l(ˆθ α ) l(ˆθ β ) ) 2 ( dim M α dim M β ) AIC > 0 M α AIC < 0 M β AIC (Linhart 1988) (Vuong 1989) (Kishino-Hasegawa 1989) ln(α) =ln(β) I(f(θ α); f(θ β )) > 0 n AIC ˆV { AIC} N(0, 1) ˆV { AIC} =4nĴ α,β + v α,β C36

37 ( ) M α AIC β AIC α > 0 AIC β AIC α ˆV {AIC β AIC α } <c M α M β c P Φ(c) =1 P C37

38 (K 3) K AIC M 1,M 2,...,M K AIC 1 AIC 2 AIC K AIC k AIC 1 ˆV {AIC k AIC 1 } <c M k M 1 AIC k AIC 1 AIC k AIC 1 = max k =1,...,K (AIC k AIC k ) C38

39 x1 x max(x1...x10) max(x1,...,x10) (x1,...,x10) N(0,I) samples C39

40 p- aic = (AIC 1, AIC 2,...,AIC K ) N(E{aic},V{aic}) E{AIC k } k δ k (aic) = max k =1,...,k 1,k+1,...,K AIC k AIC k ˆV {AIC k AIC k } (1) B AIC aic( x[b]), b =1,...,B (2) AIC Ê{aic} δ k δ k [b] =δ k ( aic( x[b]) Ê{aic} ),b=1,...,b (3) M k H k p- P k = # { δ k [b] >δ k (aic), b=1,...,b } B, k =1, 2,...,K C40

41 H k P P k <P H k P k P H k H k : δ k (E{aic}) 0 H k : E(AIC 1 ) E(AIC k ),...,E(AIC K ) E(AIC k ) H k H k P ( P k <P ) P E(AIC 1 )=E(AIC 2 )= = E(AIC K ) C41

42 T = {k {1, 2,...,K} P k P } k ( ) P ( k T ) 1 P ( ) k T P k P H k T ˆk T ˆk (1993) Shimodaira (1998) C42

43 f( θ) n ξ(θ) =(log f(x 1 θ),...,log f(x n θ)) M k dim M k {ξ(θ) θ Θ k } ξ(ˆθ k ) (PCA) C43

44 2 : X 1,X 2,...,X n g( ) : X 1,X 2,...,X n f( θ) : x =(x 1,x 2,...,x n ) : f( ˆθ(x)) f( ˆθ(x)) g( ) E { I(g( ); f( ˆθ(X))) } ( ) f( x) E {I(g( ); f( X))} C44

45 AIC, TIC, GIC, EIC, RIC ( ) EM PDIO C45

46 2.1 g( ) Θ T (g( )) T (f( θ )) = θ T (ĝ( )) GIC = l(t (ĝ)) + E { log f(x θ) θ θ T (1) (X g) } (EIC) C46

47 M- n i=1 ψ(x i ˆθ) =0 T (1) (x g) =M(ψ g) 1 ψ(x θ ); M(ψ g) = E { ψ(x θ) θ } θ GIC = l(ˆθ)+tr ( E { ψ(x θ ) log p(x θ) θ } θ M(ψ g) 1 ) l(θ) λk(θ) λ? ψ(x θ) = log f(x θ) θ λ k(θ) θ C47

48 2.2 : f(θ) : f(θ x) f(x θ)f(θ) f(θ x) = f(x θ)f(θ) f(x θ)f(θ) dθ f( x) f(z x) = f(z θ)f(θ x) dθ IC = n i=1 log f(x i x)+tr(gh 1 ) C48

49 [ ] IC IC IC n i=1 log f(x i ˆθ)+ 1 2 ( tr(gh 1 ) + dim θ ) q q^ IC = n i=1 log f(x i ˆθ)+tr(GH 1 ) p( θ* ) p^ p^b < 0 > 0 = 0 /2 =tr(gh 1 ) dim θ p() C49

50 2.3 x =(y, z) y z : f(x θ) : f(y θ) f(y θ) = f(y, z θ) dz ˆθ ( EM ) l Y (θ) = n i=1 log f(y i θ) AIC f(y ˆθ) g(y) AIC = l Y (ˆθ) + dim θ g(y) = g(y, z) dz C50

51 f(x ˆθ) g(x) IC = l Y (ˆθ)+tr ( I X IY 1 ) I X (θ) = f(x θ) 2 log f(x θ) θ θ dx, I Y (θ) = x y θ Fisher f(y θ) 2 log f(y θ) θ θ dy q^ ^ q Y observed manifold q q* model manifold p() I Z Y = I X I Y p** p* p^ tr(i X IY 1 ) = dim θ +tr(i Z Y I 1 Y ) C51

52 2.4 z : f(y z, θ) : y : z z g 0 (z), z g 1 (z) n l w (θ) = w(z i ) log f(y t z t, θ) i=1 ˆθ w f(y z, ˆθ w )g 1 (z) g(y z)g 1 (z) C52

53 : y = z + z 3 + ɛ, ɛ N(0, ) y OLS WLS WLS(opt) y z N(0.5, ) λ =0, 0.77, 1 z z N(0, ) λ =0 z C53

54 (y, z) g(y z)g 0 (z) f(y z, ˆθ w )g 1 (z) g(y z)g 1 (z) Ĵ w = 1 n IC w = Ĥ w = 1 n n i=1 n i=1 n i=1 g 1 (z i ) g 0 (z i ) log f(y i z i, ˆθ w )+tr(j w H 1 w ) g 1 (z i ) g 0 (z i ) w(z i) log f(y i z i, θ) log f(y i z i, θ) θ ˆθ w θ w(z i ) 2 log f(y i z i, θ) θ θ ˆθ w ˆθ w w(z) = ( ) λ g1 (z), λ [0, 1] g 0 (z) C54

55 2.5 x 1,x 2,...,x n,... f(x θ) f(θ) n (x 1,...,x n ) f(x 1,...,x n )= f(x 1 θ) f(x n θ)f(θ) dθ BIC = n i=1 log f(x i ˆθ)+ log n 2 dim θ log f(x 1,...,x n ) ( ) BIC C55

56 AIC BIC f(x 1,...,x n ) = f(x n x 1,...,x n 1 )f(x n 1 x 1,...,x n 2 ) f(x 3 x 1,x 2 )f(x 2 x 1 )f(x 1 ) log f(x 1,...,x n )= n i=1 IC log f(x i x 1,...,x i 1 ) 1 2i dim θ i =1,...,n n i=1 1 2i dim θ log n 2 dim θ BIC 2 AIC x 1,...,x n f(x n+1 ˆθ(x 1,...,x n )) C56

(I) (II) ˆk AIC T ( 47, 1999) C1 C ( : 3 ) Y N ( µ(x a,x b,x c ),σ 2) µ(x a,x b,x c )=β 0 + β a x a + β b x b + β c x c x a,x b,x c

(I) (II) ˆk AIC T ( 47, 1999) C1 C ( : 3 ) Y N ( µ(x a,x b,x c ),σ 2) µ(x a,x b,x c )=β 0 + β a x a + β b x b + β c x c x a,x b,x c 000 7 6 (I) (II) ˆk IC T ( 7, 999) C C.. ( : ) Y N ( µ(x a,x b,x c ),σ ) µ(x a,x b,x c )=β 0 + β a x a + β b x b + β c x c x a,x b,x c. α α {a, b, c} Θ α = {(σ, β) σ >0,β i =0,i α c }, α + C C α M α M