Le critère BIC : fondements théoriques et interprétation
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Le critère BIC : fondements théoriques et interprétation Emilie Lebarbier, Tristan Mary-Huard. N° 5315 Septembre 2004. ISSN 0249-6399. ISRN INRIA/RR--5315--FR. THÈME 4. apport de recherche.
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(42) gnBgi|jhjac[ [Irvh nYfcdyr}nygm|vhpacw)dYfqacX)|p[)Z\[Ilxhw)kmlyrpgmw)|j I[ ®1f8° aql>hj[)|pnY|p Qh ghjackml t [ fsg1nY|pkmygmYacfqaqhj G =% %# "% t [ M ª@e° giny|jXCr@fcg1¥¦km|pZe3 dYfq[ t [¢ gCum[Cr) P (M |X) r)° Iw)|jaqh ¨8m© P (X|M )P (M ) . P (M |X) = P (X) 6 kxdyr rjdYnYnTk>rvkxlyr t gmlyr,hjkxdohj[&fsg\rvdyaÏhp[&±>dY[&fcg!fckma :%# % t [Ir Z!k t X)fc[Ir M [Crh lYkmlaclo¥¦km|pZ§ghpaqm9 [ 0 i. i. i. i. i. i. i. i. i. i. i. i. BIC. i. Mi. i. i. i. i. i. i. P (M1 ) = P (M2 ) = . . . = P (Mm ).. « aclyrja7ygmdywQdylZ!k t X)fc[1l° [Irvh ny|jacbaqfc )xaq xª@_bkxdyr w)[Qhvhp[&W>ubnTkihpWYXIrj[1[Qh t ° ginY|pXIr&¨ ©,[Qhe¨ ©QYfsg!|p[IwpWy[)| wpWY[ t dZ\[Iaqfcfq[IdY| Z\k t XIfq[lY[ly IwQ[CrjrjaÏhp[±>dY[fc[ewIgifswQdYf t [fsg t asrhp|jacYdohpaqkxl P (X|M ) ª [ewIgifswQdY1f
(43) rI° kxo2hjac[)l>hFnygm| f8° aclxhp )m| ghpackml t [fsg t asrhp|jacYdohpaqkxle~kmaclxhj[ t d m[Cw¡hj[IdY| t [Ir,nygm|pgmZ\XQhj|p[Ir θ [)h t [Ir t kmlYly )[Ir X P (X, θ |M ) orjdY|,hjkxdohj[Cr,fq[Cr,gifc[)dy|pr t [ i. θi. k. 0. i. P (X|Mi ). gMi (X, θi ). =. Z. P (X, θi |Mi )dθi =. Z. [Irvh fcg!b| giasrv[IZeYfcgmlywQ[1wQkx|j|p[IrjnTkml t gil>h gmdZ\k t X)fc[ Θi. l|p ) Iw)|jaqh w)[Qhjhj[&aclxhj Im| gifc[1rvkxdyr fcg.¥¦kx|jZ\[ . i. P (X|Mi ). =. Z. Θi. i. gMi (X, θi )P (θi |Mi )dθi , Mi. t [&nygi| giZ\X)hj|p[Ir. θi. 0. [. gMi (X, θi ) = P (X|θi , Mi ).. g(θi ). dθi ,. k. g(θi ) = log (gMi (X, θi )P (θi |Mi )).. ¾gnY|pkmygmYaqfcaqhj [Crh1ginYnT[)fc )[. 8 ! " G 99 *
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(51) 3. 1. | − L00 (u? )|− 2 + O(n−1 ).. [ |p IrjdYfÏh gh@|p[Irvhj[ gmfcgmYfc[,nTkmdy| t [Ir¥¦kmlywQhjackmlyr L ±>dYa t )nT[Il t [)l>h t [ n rjkmdBr@wQ[)|jhpgmaqlY[Crw)kml t aÏhpaqkxlyr ¨:wQ¥O« lYly[Qzo[ ¢ ©¡ª 6Fkmdyr f7° ginYnyfqas±>dYkmlyr aswQa¾®!fsg.¥¦kmlywQhjackml 0 ¨7x©. n. Ln (θi ) =. 6 kxdyr lykihjkxlyr. 1X log (P (θi |Mi )) g(θi ) = . log (gMi (Xk , θi )) + n n n k=1. f7° kmnynBk>rv t [&fcg!Z§gihj|pacw)[WY[Crjrjac[)lYlY[ t [Ir t )|paqm I[Ir rv[CwQkml t [Ir nygi|jhjac[)fcfc[Ir t [1fsg.¥¦kxlyw¡hjackxl. θi? = argmaxθi ∈Θi Ln (θi ) Aθi?. Aθi? = −. k. θij. ". ∂ 2 Ln (θi ) ∂θij ∂θil. #. j,l. Ln (θi ). [Il. θi. 0. ¨:]>©. |θi =θi? ,. [Irvh fcg j X)Z\[wQkmZ\nTk>rjgmlxhj[ t dm[Cw¡hp[)dY| t [Cr nygm|pgmZ\XQhj|p[Ir θ ª i. 6 kxdyr kxohj[IlYkmlyr P (X|Mi ). kmd[IlywQkx|j[. =. [. g(θi? ). . 2π n. Ki /2. |Aθi? |−1/2 + O(n−1 ),. 8¨ m© ª«FrjubZ\nohjkmhjas±>dY[)Z\[)l>hI θ Tn [)doh >Qhp|j[&|p[)Z\nYfsgmwQ 1nygm|. log (P (X|Mi )) = log (gMi (X, θi? )) + log (P (θi? |Mi )) +. ¾ g t a"!§w)dYfÏhp Z\gmaql>hj[Ilygil>h [Irvh f7° )igifcdyghpaqkxl t [ θ [)h t [ f8° [CrhpaqZ§ghp[)dY| t dZ\gizoaqZedYZ t [&b| giasrv[IZeYfcgmlywQ[ θˆ 0 ? i. Aθi?. Ki 1 log (2π) − log (|Aθi? |) + O(n−1 ). 2 2 ? i. i. [Qh. Aθi?. nygi|. Iθˆi. k. 1 θˆi = argmax gMi (X, θi ), n θi ∈Θi. Iθˆi. [Irh fsg!Z\gihj|paswQ[ t ° aclo¥¦kx|jZ§ghpaqkxl t [ 'asrvWY[I| nTkmdY| dylY[&kmyrj[)|pghpackml t Q¬BlYaq[&nygm|;0 Iθˆi. ". ∂ 2 log (gMi (X1 , θi )) = −E ∂θij ∂θil. #. j,l. . |θi =θˆi .. }lS[Q´µ[QhCbfqkx|prp±>dY[ n [Crhx|pgml t log(g (X, θ )P (θ |M )) rj[wQkxZ!nTkx|vhp[FwQkxZ!Z\[ log(g (X, θ )) b±>dYaµwQ|pk(shg¯m[Iw hpgml t asr.±>dY[ log(P (θ |M )) |p[Irvhj[ wQkxlyrh gil>hIª [)Z\nYfsgmw)[)| θ nBgi| θˆ [Qh A yn gm| I t gilBr ¨ 5©aclxhj|pk t dyaÏh!dYl n Mi. i. CDC "Ý EGF H.IJF. i. i. i. i. Mi. ? i. i. θi?. θˆi. i.
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(54). hj[I|jZ\[ t ° [I|j|p[)dY| [)l. n−1/2. ¨:wQ¥«FlYlY[)zb[. ©Qª 6 kxdyr kxohj[)lykmlyr 0 hj[Il t m[)| r −∞ gCm[Cw. log (P (X|Mi )). =. +. }llY Imfcacm[Igmlxh fc[Ir hj[I|jZ\[Ir t ° [I|j|p[)dy|pr. n }| { Ki ˆ log (n) log (gMi (X, θi )) − 2 90 O(1) }| { z Ki 1 ˆ log (P (θi |Mi )) + log (2π) − log (|Iθˆi |) +O(n−1/2 ). 2 2 z. O(1). ¨:>©. |p[Irvhj[1Tkm|plY . [)h. log (P (X|Mi )). O(n−1/2 ) ≈. YlYkmdyr kxohj[IlYkmlBr. Ki log (n). log (gMi (X, θˆi )) − 2. ° [Irvh t [w)[Qhjhj[ginYnY|pk¯zbacZ§ghpaqkxl1±>dY[}fq[wQ|paÏhpX)|p[}¢,£j [Crh
(55) acrprvd¾ª¾{}fcdyr¾nY|j CwQasrv IZ\[)lxhCCnTkmdY|¾fc[}Z\k t XIfq[ ®!f8° gmnYnY|pkCzoacZ\gihjackml t [ −2 log P (X|M ) [Qh [Irvh t kmlyw t Q¬ylYa¾nygi|;0. Mi. acfxw)km|p|j[CrvnTkml t. i. ¨ &m©. BICi = −2 log (gMi (X, θˆi )) + Ki log (n).. [&Z!k t X)fc[&rj )fc[Iw¡hpaqkxlYlY 1nygm| w)[&w)|jaqhjXI|j[1[Irvh. MBIC = argmin BICi . Mi. b O 6Fkmdyr\gCxkmlyre¥:giaqh\f8° Wbu>nTkmhjWYXCrv[±>dY[ fsgfckxa t [Cr!Z!k t X)fc[Ir [Crh\dYlyaÏ¥¦kx|jZ\[mª5¾gnY|jasrj[ [)l wQkxZ\nohj[ t ° dylY[Faclo¥¦km|pZ\gihjackml %# "%@rjdY|fc[IrZ!k t X)fc[Ir}[Crhhjkxdohj[)¥¦kmasr}nTkxrprjaqYfcP[mx(M kxl§do) hjacfcacrj[gifckm| r@fq[wQ|paÏhpX)|p[ Z\k t aq¬y i. 0. −2 log (gMi (X, θˆi )) + Ki log (n) − 2 log (P (Mi )). t gmlyrf8° )xgmfcaÏhp ¨ 6©[Irvh1lY )xfqacm[CgiYfc[efckm| rp±xdy[ n hj[)l t m[)| rf8° aclo¬ylya7ªe{@gi| b O ° [I|j|p[)dy|[)l ) Qw kxlxhp|j[f7° [)|p|j[IdY| t ° gmnYnY|pkCzoacZ§ghpaqO(n kxl&[)l O(1) nT[IdohnT[)|jhjdY|pT[)|
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(69) [CrhnYfcdyrFnY|jkYwpWY[ t [ ª {gm| KL M giacfcfq[IdY|prI M t kmaqh&gidyrprva >)hj|p[.ny|j )¥¦ )|p !®hjkxdyrfq[CrZ!k t MX)fc[Ir it =° kx| 1,t |j..., [!rvtdY−nT I1|jac[)dY| r M i = t + 1, ..., m nYdYasfrp±>d° acfsr rjkmlxh,nYfcdyr,wQkxZ\nYfqas±>dY Ir,±>dY[ rjgmlyrnTkmdy|,gidoh gil>h>)hj|p[FnYfcdyrny|jkow WY[Ir t [ f 0@wQ[CrZ\k t XIfq[Crrjkml>h t kxlywFrjdY| g~dyrhp IrIª 6 kxdyrgifcfckmlyrFZ!kxlxhp|j[I|±>dY[eM fc[.wQ|paqhjX)|p[e¢,£j [IrvhwQkxlyrvasrvhpgil>hFnTkxdY|w)[eZ\k t X)fc[enygm|vhpacw)dYfqac[)|C t CrvacmlY enygm|¢,dY|plYWygiZ [QhF«Fl t [)| rjkmlw)kmZ\Z\[1fc[1Z\k t XIfq[ )±>dygmrjaq²8b|pgia ª'{'kxdY| n rjdYnYnTkxrj 1m| gil t bkmlrI° aql>hj )|p[Irprj[1®!fcg t aÏ´µ )|p[)lyw)9[ 0 dKL (f, Mi ). =. inf dKL (f, gMi (., θi )). θi. i. t. t. i. t. i. t. BICi − BICt , i 6= t.. b §» y. .° gmnY|jXCr&¨ &m©¡Ykmlg i <0 t BICi − BICt. = = =. −2 log(gMi (X, θˆi )) + 2 log(gMt (X, θˆt ) + (Ki − Kt ) log(n) # " n n X 1 1X log(gMi (xk , θˆi )) + log(gMt (xk , θˆt )) + (Ki − Kt ) log(n) 2n − n n k=1 k=1 " n ! !# n 1X f (xk ) f (xk ) 1X 2n log log − + (Ki − Kt ) log(n). n n gM (xk , θˆi ) gM (xk , θˆt ) k=1. k=1. i. t. [Cr t [Idoz t [I|jlYacX)|p[Ir\rjkmZ\Z\[Ir\rjkml>h t [Cr![CrhpaqZ§gihj[)dy|pr\w)kmlyrjacrvhpgmlxh r t [Ir§±>dygil>hjaqhj Ir |p[IrjnT[Iw¡hpaqm[IZ\[)lxh¨7w¡¥@ acnYfc[)u¨v*)3)>m©j©¡ª'{kmdY| n m| gil t okmlg t kxlyw<0. dKL (f, Mi ). [)h. dKL (f, Mt ). BICi − BICt ≈ 2n[dKL (f, Mi ) − dKL (f, Mt )] + (Ki − Kt ) log(n).. [Qhjhj[ ginynY|jk¯zoaqZ§gihjackmlYac[)l ±>dY[ t )hj[I|jZ\aclYacrvhj[x rvd!\h½®[QzonYfcaswQaqhj[)| fq[ Qw kxZ!nTkx|vhp[)Z\[)l>h gmrju>Z\nohpkihpac±>dY[ t [ 0¾fq[nY|p[)Z\aq[I|¾hj[I|jZ\[ t kmZ\aclY[[)h
(70) hj[Il t m[)| r g¯m[Iw ª l[)l t t dyaÏh t kmlBw±>d° gmrju>Z\nohpkihpac±>dY[IZ![Ilxh BIC −BIC fc[Ir Z\k t X)fc[Ir M i = 1, ..., t − 1 rjkml>h t asrp±xdBgifcaϬy Cr nygi| +∞ fq[wQ|paqhjX)|p[1n¢,£j ª i. t. i. , > .-b » Y<. gilyrFwQ[ew)gmr fs®Yykmli|p[I>w)kmtlYlBg=(sh t gmlyr fc[1hp[)|pZ\[ 2 log(g / C ÛÜ Ø .Ù 0Iò¯Û1 M Û2IâÄã:Ú7Û@Ú43¡ò6543¡òCÙ7ãvÔi×Cò¯âäã70)ò¯Û@é98 3¡ÝØ,Ø òÜ3¡âäݯã CDC "Ý EGF H.IJF. ˆ. Mi (X, θi )). t. sf g§rvhpghpacrvhjas±>dY[ t dhp[Irvh t d ) ×:3¡òCÙÚ73¡òCÚ i. − 2 log(gMt (X, θˆt )). dKL (f, Mi ) ≤ dKL (f, Mm. .
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(124)
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