(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 [ ] {a, b, c} M = {φ, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} M = {{a, b}, {a, c}, {b, c}} ln (α) l n (k) α k α(k), k {,,...,K} αk l(ˆθ α ) dim M α l(ˆθ k ) dim M k ˆα ˆk T M(T M) M α = f( Θ α ), α M P {α T} P {k T} P M,M,...,M K P C C7 M α M β HL α β M α M β M α M β [ ] φ {a} {a, b} {a, b, c} : : : n = : K =6 ( =6 ) α β β α M α M β [ ] {a, b} {b, c} ( ) BOSTON : ( ) : a= f=k= :m= : n = 06 : K = 86 ( ) C6 C8

HL IC p- α IC LR <abd> 0 0.97 0.80 0.86 <abc> 0.0 0.9 0.980.6 0.796 <ab> 0. 0.80.9 0. 0.90 <acd> 0.7 0.8 0.9 0.8 0.76 <abcd>.97 0.088 0. 0.00 6 <ad>.77 0.69 0.0 0.09 0.0 7 <bcd>.60 0.6 0. 0.0 0.08 8 <cd>.88 0.09 0.07 0.00 0.000 9 <bc> 6.06 0.00 0.000 0.000 0.000 0 <d>.88 0.000 0.000 0.000 0.000 <b>.0 0.000 0.000 0.000 0.000 <bd>.66 0.000 0.000 0.000 0.000 <a> 8. 0.000 0.000 0.000 0.000 <ac> 0. 0.000 0.000 0.000 0.000 <c>.09 0.000 0.000 0.000 0.000 6 <> 6.7 0.000 0.000 0.000 0.000 C9 - <c> <a> <> <ac> <bc> <b> <bd> <d> <bcd> <cd> - <ab> <acd> 0 0. S- S- 0 <ad> <abd> <abc> <abcd> -0. - 0 S- HL <klm> <jlm> <hlm> <glm> <clm> <lm> <elm> <blm> <dlm> <ilm> <jkm> <hjm> <ijm> <gjm> <ejm> <jm> <bjm> <cjm> <djm> <ckm> <chm> <cgm> <cm> <bcm> <cem> <cdm> <cim> <km> <him> <gim> <im> <eim> <bim> <dim> <bkm> <ekm> <ikm> <gkm> <hkm> <dkm> <m> <em> <bm> <dm> <bem> <dem> <gm> <ghm> <bgm> <ehm> <hm> <bhm> <egm> <flm> <dgm> <dhm> <bdm> <fjm> <fkm> <cfm> <fim> <efm> <fm> <fhm> <fgm> <bfm> <dfm> 0 X S- <afm> 0 S- 0 BOSTON <alm>. <ajm> <akm> <acm> <aem> <am> <abm> <adm> 0 <agm> <aim> S- <ahm> -. - C BOSTON 0 IC p- p-value 00 α LK B BP KH MC MS LR <afm> 0 70. 8.8 99. 99. 0.0 <akm> 0.9 9.7 9. 6. 98.8 98.7 0.0 <ahm> 9. 0.0 7. 8.0 86. 77. 0.0 <agm>. 0.0 0. 9.0 79.8.9 0.0 <adm>.7 0.0.. 80.7. 0.0 6 <alm> 6. 0.0 0.0.0 7.0. 0.0 7 <ajm> 7.7 0.0 0.0. 7..6 0.0 8 <abm> 9. 0.0 0.0.9 66.6 0.0 0.0 9 <aim> 9.8 0.0 0.0. 6.8 8. 0.0 0 <acm> 0. 0.0 0.0.0 6..0 0.0 <aem> 0. 0.0 0.0. 6.8. 0.0 <klm>. 0.0. 8..9. 0.0 <fjm>. 0.0 0..9.7 6.0 0.0 <jkm> 7.0 0.0 0.0. 7.9. 0.0 <fkm> 7. 0.0 0.. 7.9. 0.0 6 <flm> 9.9 0.0 0..7 0.6 7.7 0.0 7 <hjm> 0. 0.0 0..6 0. 0. 0.0 8 <dkm>.0 0.0 0.0.8 8..7 0.0 9 <bkm>. 0.0 0.0. 7..0 0.0 0 <ekm>. 0.0 0.0.7.7 6.6 0.0 LK: log L B: BP: KH: MC: () MS: LR: C0 N human seal cow rabbit mouse opossum h=0 0 NLLLLIVPILIMFLMLTRKILGYMQLRKGPNVVGPYGLLQPFMKLFTKPLKP INIISLIIPILLVFLTLVRKVLGYMQLRKGPNIVGPYGLLQPIVKLFTKPLRP INILMLIIPILLVFLTLVRKVLGYMQLRKGPNVVGPYGLLQPIIKLFIKPLRP INTLLLILPVLLMFLTLVRKILGYMQLRKGPNIVGPYGLLQPIIKLFTKPLRP INILTLLVPILIMFLTLVRKILGYMQLRKGPNIVGPYGILQPFMKLFMKPMRP INLLMYIIPILLVFLTLVRKVLGYMQFRKGPNVIGPYGILQPFLKLFIKPLRP X h : 6 human, (harbor seal, cow), rabbit, mouse, opossum : 0 split : n = 6 : K = C

<ai> a i <ah> a h <ae> a e <de> d e <ef> f e <bf> f b <cf> f c <cg> g c <ci> c i <ij> j i <dg> d g <dj> j d <bj> j b <bh> h b <gh> g h human seal harbor e cow a rabbit mouse opossum a = { }, b = { }, c = { }, d = { }, e = { }, f = { }, g = { }, h = { }, i = { }, j = { }. α logl ˆθ α {aefi} {ae} {bcef} {a} {ef} {e} {} full 9. (6.,.,.78,.8,.,.9,.,.90,.,.)..07 i f a e.7.70 <aefi> (.77) e.86. a (.8) <ae>.0c f.7.. <bcef> (6.8) a <a> (.6) e..88 f (.76) <ef> (.) e <e> 6. (.7,,,,.70,.07,,,., ) 8. (.86,,,,.,,,,, ) b e.6 (,.7,.0,,.,.,,,, ) 8. (6.8,,,,,,,,, ).0 (,,,,.88,.,,,, ).7 (,,,,.,,,,, ) < > 0 (,,,,,,,,, ) C C Mammal Phylogeny: p-values p-values α log L LR B BP KH MC MS MB tree {a, e} 0.0.000.9.8.9.9.8 ((())) {a, i}.7.000.06.7.60.8.80.7 ((())) {a, h} 7..000.00.08..77..7 ((())) {e, f} 7.6.000.000.0.00.69.0. (()()) {c, f} 8.9.000.000.00.066.9.96.8 ((())) 6 {c, i} 0..000.000.006.00.09.00.09 ((())) 7 {b, f} 0.6.000.000.0.08.07.8.9 ((())) 8 {i, j}..000.000.00.0.070.08.08 (()()) 9 {d, e}..000.000.000.00.09.0.0 ((())) 0 {b, j} 6..000.000.00.08.0..09 ((())) {b, h} 8.9.000.000.000.008.07.069.08 ((())) {d, j}.6.000.000.000.00.006.0.0 ((())) {c, g}.7.000.000.000.00.006.0.0 ((())) {g, h}.7.000.000.000.00.00.0.0 (()()) {d, g} 6..000.000.000.000.00.007.0 ((())) S- 0 - - S- -. 0. 7. <dg> <g> <d> <> <gh> <h> <dj> <j> <ij> <i> <bh> <b><c> <cg> <bj> <ci> <de> <e> <f> <bf> <ef> <cf> <bcef> <ah> <a> <ai> <ae> -. 0 S-. <aefi> <a...j> S- 0 - -. <a...j> S- <e> <de> 0 -. <ae> <ef> <d> <dg> <> <g> <h> <gh> <f> <b> <bh> <bcef> <bf> <dj> <c> <cg> <j> <bj> <cf> <aefi> <a> <ah> <ai> <i> <ij> <ci> -. 0. S- 7. LR:B: BP:KH: MC: () MS:MB:New Test C C6

. ˆk x ˆk = ˆk(x) ˆk(X) ˆk(X) k P (ˆk(X) =k ) (n ) IC () : (000 ) n =0, 0, 0, 00, 00, 00, 000 IC α = l(ˆθ α )+c n dim M α c n : = (IC), B =logn (BIC), C =n 0., =n 0., =n 0.6 correct selection count 0 00 00 600 800 000 B C CB CB B C CB CB C B correct selection count 0 00 00 600 800 000 B B B B B C C C C C CB CB correct selection count 0 00 00 600 800 000 CB B B B B C C C C CB C B IC C7 0 0 0 00 00 00 000 sample size 0 0 0 00 00 00 000 sample size 0 0 0 00 00 00 000 sample size C9 ().6 ICk (=ˆk) (n = 00) k 6 80 6 80 0 0 0 6 7 9 7 7 0 000.7 n = 000 ˆk k 6 80 6 80 0 0 0 6 0 9 6 688 000 l00 (k) k =0 l000 (k) k = () q q^ p* ^p a a p(). a p^b p* b ( ) p(). b β =(β a,β b,β c ) p* p^ q p * p(). p ^ q^ ( ) p(). : β =(.0, 0.9, 0.), M = {φ, {a}, {b, c}} C8 C0

() IC q q p* p(). l(θ α) l(θ n β )= ( log f(xi θ α) log f(x i θ β )) i= p() p* p() p() p * p(). 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)) ) : β =(,, 0), M = {φ, {a}, {a, c}} 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) ) : β =(,, 0), M = {φ, {a}, {b, c}} ( log f(xi ˆθ α ) log f(x i ˆθ β ) ( l(ˆθ α ) l(ˆθ β ) ) /n ) C Ĵ α,β = n i= C IC IC M α M β IC = ( l(ˆθ α ) l(ˆθ β ) ) c n ( dim M α dim M β ) IC > 0 M α IC < 0 M β n IC ( l(θ α) l(θ β )) N ( l (θ α) l (θ β ),nj ) α,β IC ( l(θ α) l(θ β )) + ( (θ α, ˆθ α ) (θ β, ˆθ β ) ) c n ( dim M α dim M β ) I(g; f(θ β )) I(g; f(θ α)) > 0 P ( IC > 0) ˆθ α (θ M α ) [ ] l l(θ) l(ˆθ α )+(θ ˆθ α ) θ ˆθ α + [ (θ ˆθ ] l α ) θ (θ ˆθ α ) θ ˆθ α I(g; f(θ β )) I(g; f(θ α)) < 0 P ( IC < 0) I(g; f(θ β )) I(g; f(θ α)) = 0 l(ˆθ α ) l(θ α )+ (θ α, ˆθ α ) C C

IC IC 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 0 IC ( (θ α, ˆθ α ) (θ β, ˆθ β ) ) c n ( ) dim M α dim M β (θ α, ˆθ α ) χ dim M α, (θ β, ˆθ β ) χ dim M β d =dimm α dim M β > 0 M β c n M β P ( IC < 0) C I(g; f(θ β )) I(g; f(θ α)) = 0 f( θ α) f( θ β ) IC ( l(θ α) l(θ β )) c n ( ) dim M α dim M β l (θ α) l (θ β )=0 l(θ α) l(θ β ) N ( ) 0,nJ α,β d =dimm α dim M β > 0 M β c n /n M β P ( IC < 0) C7 IC ( ) (M β M α ) IC χ d c nd, d =dimm α dim M β M β P ( IC < 0) = P (χ d <c nd) c n =,d= P ( IC < 0) 0.8 c n =,d=0 P ( IC < 0) 0.97 d c n P ( IC < 0) IC α = l(ˆθ α )+c n dim M α c n /n 0 c n c n /n IC (c n =) C6 BIC (c n =logn) C8

. ( ) n P,P,...,P K ; p-value, p- x = {x,...,x n } ˆk(x) ( x i x ) x = { x,..., x n } ˆk( x) C9 C (p-value) M k, k =,...,K [0, ] P k ( B = 000) x[], x[],..., x[b] ˆk( x[]), ˆk( x[]),...,ˆk( x[b]) ( ) ( p-value) M k P k = # {ˆk( x[b]) = k, b =,...,B }, k =,...,K B K P k =, 0 P k k= M k P k C0 C

M α = f( Θ α ), α M M,M,...,M K ˆk M k M k C C IC M k M,...,M K H k : k = k (ln (k) =l n (k ) k H k ) M α M β IC = ( l(ˆθ α ) l(ˆθ β ) ) ( ) dim M α dim M β IC > 0 M α IC < 0 M β IC (Linhart 988) (Vuong 989) (Kishino-Hasegawa 989) H,H,...,H K ln(α) =ln(β) I(f(θ α); f(θ β )) > 0 n IC ˆV { IC} N(0, ) T = { k {,...,K} H k P } C ˆV { IC} =nĵ α,β + v α,β C6

() M α IC β IC α > 0 IC β IC α ˆV {IC β IC α } <c M α M β cp Φ(c) = P 0 00 00 600 800 000 00 00 0 00 00 600 800 000 00 00 - - 0 x x - - 0 max(x...x0) max(x,...,x 0 ) (x,...,x 0 ) N(0,I) 0000 samples C7 C9 (K ) p- K IC M,M,...,M K IC IC IC K IC k IC ˆV {IC k IC } <c M k M IC k IC aic = (IC, IC,...,IC K ) N({aic},V{aic}) {IC k }k δ k (aic) = IC max k IC k k =,...,k,k+,...,k ˆV {IC k IC k } () B IC aic( x[b]), b =,...,B () IC Ê{aic} δ k δ k [b] =δ k ( aic( x[b]) Ê{aic} ),b=,...,b IC k IC = max k =,...,K (IC k IC k ) C8 () M k H k p- P k = # { δ k [b] >δ k (aic), b=,...,b }, k =,,...,K B C0

H k P P k <P H k P k P H k f( θ) n ξ(θ) =(log f(x θ),...,log f(x n θ)) M k dim M k H k : δ k ({aic}) 0 H k : (IC ) (IC k ),...,(IC K ) (IC k ) H k H k P ( P k <P ) P {ξ(θ) θ Θ k } ξ(ˆθ k ) (IC )=(IC )= = (IC K ) (PC) C C T = {k {,,...,K} P k P } : X,X,...,X n g( ) : X,X,...,X n f( θ) : x =(x,x,...,x n ) : f( ˆθ(x)) k ( ) P ( k T ) P f( ˆθ(x)) g( ) { I(g( ); f( ˆθ(X))) } ( ) k T P k P H k ( ) T ˆk T ˆk f( x) {I(g( ); f( X))} (99) Shimodaira (998) C C

IC, TIC, GIC, IC, RIC ( ) MPIO M- ψ(x i ˆθ) =0 i= { } T () (x g) =M(ψ g) ψ(x θ ψ(x θ) ); M(ψ g) = θ θ GIC = l(ˆθ)+tr ( { ψ(x θ ) l(θ) λk(θ) λ? ψ(x θ) = log f(x θ) θ log p(x θ) θ λ k(θ) θ } ) M(ψ g) θ C C7.. g( ) Θ T (g( )) : f(θ) : f(θ x) f(x θ)f(θ) T (f( θ )) = θ T (ĝ( )) f( x) f(θ x) = f(x θ)f(θ) f(x θ)f(θ) dθ f(z x) = f(z θ)f(θ x) dθ { } log f(x θ) GIC = l(t (ĝ)) + θ θ T () (X g) IC = log f(x i x) + tr(gh ) i= (IC) C6 C8

[ ] IC IC q p( θ* ) p() q^ p^ IC log f(x i ˆθ)+ ( tr(gh )+dimθ ) i= p^b IC = log f(x i ˆθ) + tr(gh ) i= < 0 > 0 = 0 / =tr(gh ) dim θ C9 f(x ˆθ)g(x) IC = l Y (ˆθ)+tr ( I X IY ) I X (θ) = f(x θ) log f(x θ) θ θ dx, I Y (θ) = x y θ Fisher q q* q^ p** p* p^ ^ q Y observed manifold model manifold p() f(y θ) log f(y θ) θ θ dy I Z Y = I X I Y tr(i X IY )=dimθ +tr(i Z Y I Y ) C. x =(y, z) y z : f(x θ) : f(y θ) f(y θ) = f(y, z θ) dz ˆθ ( M) l Y (θ) = log f(y i θ) i= IC f(y ˆθ) g(y) IC = l Y (ˆθ)+dimθ g(y) = g(y, z) dz. z : f(y z, θ) : y : z z g 0 (z), z g (z) l w (θ) = w(z i )logf(y t z t, θ) i= ˆθ w f(y z, ˆθ w )g (z) g(y z)g (z) C0 C

: y -.0-0. 0.0 0..0. OLS WLS WLS(opt) -0. 0.0 0..0. z N(0., 0. ) λ =0, 0.77, z y = z + z + ɛ, ɛ N(0, 0. ) y -.0-0. 0.0 0..0. -0. 0.0 0..0. z N(0, 0. ) λ =0 z. x,x,...,x n,... f(x θ) f(θ) (x,...,x n ) f(x,...,x n )= f(x θ) f(x n θ)f(θ) dθ n BIC = log f(x i ˆθ)+ log n i= dim θ log f(x,...,x n ) ( ) BIC C C IC BIC (y, z) g(y z)g 0 (z) f(y z, ˆθ w )g (z) g(y z)g (z) g IC w = (z i ) i= g 0 (z i ) log f(y i z i, ˆθ w ) + tr(j w Hw ) Ĵ w = g (z i ) n 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 ˆθ w Ĥ w = w(z i ) log f(y i z i, θ) n i= θ θ ˆθ w w(z) = ( ) λ g (z), λ [0, ] g 0 (z) C f(x,...,x n ) = f(x n x,...,x n )f(x n x,...,x n ) f(x x,x )f(x x )f(x ) log f(x,...,x n )= log f(x i x,...,x i ) i= IC i dim θ i =,...,n i= i dim θ log n dim θ BIC IC x,...,x n f(x n+ ˆθ(x,...,x n )) C6