ػցֶशΞϧΰϦζϜʹજΉ ෆެฏͳόΠΞεͱͦͷཧ࿦ ೥ιαΠΤςΟେձ݄೔ ෱஍Ұే ஜ೾େֶཧݚ"*1

ࣗݾ঺հ w ໊લ෱஍Ұే 'VLVDIJ,B[VUP w ॴଐஜ೾େֶγεςϜ৘ใܥॿڭ w ܦྺ w ஜ೾େֶγεςϜ৘ใ޻ֶઐ߈Պത࢜ޙظ՝ఔमྃ w ཧݚ"*1ಛผݚڀһ w ݱࡏஜ೾େֶγεςϜ৘ใܥॿڭ w ݱࡏཧݚ"*1٬һݚڀһ w ݚڀڵຯ w ػցֶशʹ͓͚ΔެฏੑɼϓϥΠόγʔͷཧ࿦໘ w ਺ཧ౷ܭɼಛʹɼ൚ؔ਺ਪఆ

ຊ೔ͷ಺༰ wެฏͳֶशͷجૅ wػցֶशʹ͓͚Δෆެฏͱ͸ʁ wػցֶशͷෆެฏͷݪҼ͸ʁ wެฏͳֶशΛߦ͏ํ๏͸ʁ w ެฏͳֶशʹର͢Δ#BZFT࠷ద༧ଌํ๏ w ެฏͳֶशʹର͢ΔNJOJNBY࠷దͳֶशํ๏

ެฏͳֶशͷجૅ

ػցֶशʹ͓͚Δෆެฏ Hiring [Dastin’18] Criminal risk assessment [Angwin+’16] Machine translation 
 [Şarbak’s facebook post] Face recognition [Buolamwini+’18] 5

ެฏੑ 'BJSOFTT w ػցֶश͕ҙࢥܾఆʹ༻͍ΒΕ͍ͯΔ w ҙࢥܾఆ͕ࠩผΛੜΉՄೳੑ͕͋Δ 6 ࠾༻ ৴༻είΞ ೖࢼ อݥྉ཰ ݸਓ உੑ ঁੑ

"NB[POͷޏ༻γεςϜʹ͓͚Δࠩผ <%BTUJO> w "NB[POͰ։ൃதͷ"*ޏ༻γεςϜ w աڈͷޏ༻ཤྺ͔ΒԠืऀͷείΞΛܭࢉ w ҎԼͷΑ͏ͳࠩผతͳείΞ෇͕ൃݟ͞Εͨ w lঁੑͷzͱ͍͏୯ޠͷ͸͍ͬͨཤྺॻʹേଇΛ͔͚Δ w ͭͷঁࢠେ BMMXPNFO`TDPMMFHFT ʹରͯ͠ධՁΛԼ͛Δ ରॲ͕Ͱ͖ͣγεςϜͷಋೖΛஅ೦

(FOEFS4IBEFT<#VPMBNXJOJ> w ֤ࣾͷإೝূ"1*Λௐࠪ w ഽͷ৭ͱੑผͷࠩʹΑΔೝࣝਫ਼౓Λൺֱ w ࠇਓঁੑͷೝࣝਫ਼౓͕ଞͱൺ΂େ͖͘ྼΔ http://gendershades.org/overview.html

.BDIJOF#JBT<"OHXJO> w $0.1"4ΞϧΰϦζϜ w नਓͷྦྷ൜ʹର͢ΔείΞΛ༩͑ͯ͘ΕΔ w είΞ͕ߴ͍΄Ͳྦྷ൜ϦεΫ͕ߴ͍ 9 ϦεΫ͸௿͍͕ ճͷྦྷ൜ ϦεΫ͸ߴ͍͕ ྦྷ൜͍ͯ͠ͳ͍

.BDIJOF#JBT<"OHXJO> 10 είΞͷਖ਼֬ੑͱਓछͷؔ܎ੑ w ޡͬͯࠇਓͷείΞΛߴ͘ਪఆ w ޡͬͯനਓͷείΞΛ௿͘ਪఆ നਓ͕༏۰͞ΕͨείΞ෇͕͞Ε͍ͯΔ ޡͬͯϦεΫ͕௿͍ͱਪఆ ޡͬͯϦεΫ͕ߴ͍ͱਪఆ

ެฏੑͷࣾձతཁٻ w ػցֶशͷެฏੑ͸ࣾձ͔Βͷཁٻ΋ڧ͍ w #JH%BUBʹؔ͢Δ8IJUF)PVTF3FQPSU w 8IJUF)PVTF3FQPSU<1PEFTUB> 
 lΞϧΰϦζϜʹΑΔજࡏతͳࠩผΛ؂ࢹ͠ͳ͚Ε ͹ͳΒͳ͍z w ಉ༷ͳ಺༰͕8IJUF)PVTF3FQPSU<.VOP[> 11

ެฏੑͷ๏తཁٻ w 5JUMF7** w ਓछɼഽͷ৭ɼफڭɼੑผɼग़਎ࠃʹΑΔޏ༻ࠩผͷېࢭ w உঁޏ༻ػձۉ౳๏ w ৬৔ʹ͓͚Δஉঁࠩผͷېࢭ w (%13 w "SUJDMFݸਓσʔλॲཧʹؔ͢Δنఆ w lద๏ɺެฏ͔ͭಁ໌ੑͷ͋ΔखஈͰॲཧ͠ͳ͚Ε͹ͳΒͳ͍z 12 ޏ༻ʹػցֶशΛ࢖͏৔߹͸ରॲ͕ඞཁෆՄܽ (%13͸ͲΜͳλεΫͰ΋ެฏੑ͕ཁٻ͞ΕΔՄೳੑ͕͋Δ

"*ྙཧΨΠυϥΠϯʹ͓͚Δެฏੑ w"*ͷࣾձར༻ʹ͓͚Δྙཧͷ໰୊ͷରԠͱͯ͠ΨΠυϥΠϯ੔ උ͕ਐΜͰ͍Δ wΨΠυϥΠϯͷҰ෦ͱͯ͠ެฏੑ͕ݴٴ͞Ε͍ͯΔ w*&&&&UIJDBMMZ"MJHOFE%FTJHO wͭͷݪଇͷ͏ͪͷҰͭʮਓݖʯʹ͓͍ͯެฏੑͷݴٴ͋Γ w૯຿ল"*ར׆༻ΨΠυϥΠϯ wͷݪଇͷҰͭͱͯ͠ʮެฏੑʯ͕ڍ͛ΒΕ͍ͯΔ w("*ݪଇ wͭͷݪଇͷҰͭͱͯ͠ʮਓؒத৺ͷՁ஋؍ٴͼެฏੑʯ͕ ڍ͛ΒΕ͍ͯΔ

ෆެฏͷݪҼ ͳͥػցֶश͕ෆެฏͳग़ྗΛ͢Δͷ͔ w σʔλऩू͔Βֶशͷաఔʹ͓͍ͯόΠΞε͕৐Δ͜ͱ͕ݪҼ w ༷ʑͳόΠΞε͕ͷΔݪҼ͕ٞ࿦͞Ε͍ͯΔ<#BSPDBT> w େ͖ͭ͘ʹΘ͚Δ 14 ෆެฏ σʔλऩूʹ͓͚Δ 
 όΠΞε w ࠩผతϥϕϧ෇͚ w αϯϓϦϯάόΠΞε ֶशʹ͓͚Δ 
 όΠΞε w ֶशϞσϧͷઃܭ w গ਺άϧʔϓͷແࢹ σʔλ ֶश

σʔλऩूʹ͓͚ΔόΠΞε w ֶशऀ͕ಘΒΕΔαϯϓϧ͕ෆެฏ 15 ࠩผతͳϥϕϧ wϥϕϧ͸ਓͷखʹΑͬͯ෇͚ΒΕΔ wྫ ޏ༻ͷ࠾൱͸աڈਓ͕൑அͨ͠ wϥϕϧ෇͚ʹࠩผతͳࢥ͍ࠐΈΛ൓ө͢ΔՄೳੑ͋Γ wҙࣝతແҙࣝతʹؔΘΒͣ

σʔλऩूʹ͓͚ΔόΠΞε w ֶशऀ͕ಘΒΕΔαϯϓϧ͕ෆެฏ 16 αϯϓϦϯάόΠΞε wภͬͨαϯϓϧ͔͠ಘΒΕͳ͍͕࣌͋Δ wྫ ͓ۚͷି෇Λͨ͠ਓ͕࠴຿ෆཤߦʹؕΔ͔Ͳ͏͔Θ͔Βͳ͍ wաڈ͓ۚΛି෇ͨ͠ਓΛࠩผతʹબ୒͍ͯ͠ΔՄೳੑ͋Γ

"EVMUEBUB<$BMEFST> w 64ͷશ݅ௐࠪσʔλ w ݸਓͷऩೖ͕LҎ্͔ҎԼ͔༧ଌ͢Δ໰୊ 17 .BMF 'FNBMF )JHIJODPNF -PXJODPNF ͕ ߴऩೖ ͕ ߴऩೖ σʔλʹஉঁؒͷόΠΞε͕͋Δ

ֶशʹ͓͚ΔόΠΞε w ֶशʹΑͬͯෆެฏͳ෼ྨث͕ಘΒΕΔ 18 গ਺άϧʔϓͷແࢹ wσʔλͷஉঁ͕ͻͲ͘ภ͍ͬͯΔͱ͢Δ wྫ ΄ͱΜͲͷσʔλ͕உੑͷ΋ͷͰঁੑͷσʔλ͕΄ͱΜͲͳ͍ w༧ଌੑೳΛ্͛ΔͨΊগ਺೿ΛϊΠζͱΈͳ͢Մೳੑ͕͋Δ

"EVMUEBUB<$BMEFST> w /BJWF#BZFTͰֶश͠෼ྨ͢Δ 19 .BMF 'FNBMF )JHIJODPNF -PXJODPNF ͕ ߴऩೖ ͕ ߴऩೖ ΑΓࠩผతͳ༧ଌ݁ՌʹͳΔ

ֶशʹ͓͚ΔόΠΞε w ֶशʹΑͬͯෆެฏͳ෼ྨث͕ಘΒΕΔ 20 ֶशͷ༧ଌϞσϧͷઃܭʹΑΔࠩผ wػցֶशͰ͸ϞσϧઃܭΛͦ͠ͷޙϞσϧͷύϥϝʔλΛσʔλ͔ΒಘΔ wϞσϧઃܭͷ࢓ํʹΑͬͯ͸ࠩผΛট͘ wྫ உঁͷ஋Λ࢖ͬͯ༧ଌΛߦ͏ϞσϧΛઃܭ w௚઀ࠩผ EJTQBSBUFUSFBUNFOU CMJOEOFTT Ϟσϧઃܭ ͳΜͷಛ௃ྔΛ࢖͏͔ ༧ଌؔ਺͸ ઢܗɼଟ߲ࣜɼ3),4ɼ%FFQ//

3FEMJOJOHF ff FDU<$BMEFST> w௚઀உঁͷ஋Λ࢖Θͳͯ͘΋ࠩผ͕ى͜Δ w உঁ΍ਓछͳͲͱڧ͘ґଘͨ͠σʔλ Λ࢖͏͜ͱͰؒ઀తʹࠩผ͕ൃੜ w ྫ ֶྺΛ࢖ͬͯ࠾൱ΛܾΊΔͱ 
 ੑ͕ࠩੜ·ΕΔՄೳੑ͋Γ w ྫ ॅॴΛ࢖ͬͯ࠾൱ΛܾΊΔͱ 
 ਓछͷ͕ࠩੜ·ΕΔՄೳੑ͋Γ 21 'SPNXJLJQFEJB

"EVMUEBUB<$BMEFST> w ੑผΛऔΓআ͍ͯ࠶౓/BJWF#BZFTͰֶश͠෼ྨ͢Δ 22 .BMF 'FNBMF )JHIJODPNF -PXJODPNF ͕ ߴऩೖ ͕ ߴऩೖ ΑΓࠩผతͳ༧ଌ݁ՌʹͳΔ

෼ྨֶशͷ࢓૊Έ ϥϕϧ෇͖σʔλ ֶशΞϧΰϦζϜ ෼ྨث f *NBHFTGSPN<#VPMBNXJOJ>

෼ྨֶशͷ࢓૊Έ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ *NBHFTGSPN<#VPMBNXJOJ>

෼ྨֶशͷ࢓૊Έ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ ͳΔ΂͘౰ͨΔΑ͏ 
 f Λબ୒͍ͨ͠ *NBHFTGSPN<#VPMBNXJOJ>

෼ྨֶशͷ࢓૊Έ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ f ʹΑΔ෼ྨޡ͕ࠩ 
 ࠷খʹͳΔΑ͏ʹ͢Δ *NBHFTGSPN<#VPMBNXJOJ>

ެฏͳ෼ྨֶशͷ࢓૊Έ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ ͳΔ΂͘ 
 ౰ͨΔΑ͏ʹf Λબ୒ ެฏͳf Λબ୒ *NBHFTGSPN<#VPMBNXJOJ>

ެฏͳ෼ྨֶशͷ࢓૊Έ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ lެฏͳf zͱ͸ʁ f ʹΑΔ෼ྨޡ͕ࠩ 
 ࠷খʹͳΔΑ͏ʹ͢Δ *NBHFTGSPN<#VPMBNXJOJ>

ઃఆ w ؆୯ͷͨΊʹڭࢣ͋Γ෼ྨ໰୊ͷΈΛߟ͑Δ w ɹɹɹɹɹɹɹɹɹɹֶྺɼ৬ྺɼࢿ֨ͳͲ w ɹɹɹɹɹɹɹɹɹɹੑผɼਓछɼफڭɼ੓࣏ࢤ޲ɼ೥ྸͳͲ w ɹɹɹɹɹɹɹɹɹɹ༧ଌ͍ͨ͠΋ͷ FH ࠾൱ w ɹɹɹɹɹɹɹɹɹɹΞϧΰϦζϜʹΑͬͯ༧ଌ͞Εͨϥϕϧ ೖྗ X ϥϕϧ Y ༧ଌϥϕϧ ̂ Y ผͷೖྗ X S = உੑ S = ঁੑ ೖྗ X ϥϕϧ Y ηϯγςΟϒଐੑ S ༧ଌϥϕϧ ̂ Y ֶश

ެฏੑج४ w ༷ʑͳެฏੑج४ w %FNPHSBQIJDQBSJUZ w &RVBMJ[FEPEET<)BSEU> w *OEJWJEVBMGBJSOFTT<%XPSL> w $BMJCSBUJPO<1MFJTT> w $PVOUFSGBDUVBMGBJSOFTT<,VTOFS> w 1BUITQFDJ fi DDPVOUFSGBDUVBMGBJSOFTT <$IJBQQB>

%FNPHSBQIJDQBSJUZ w ηϯγςΟϒଐੑͰ৚͚݅ͮΒΕͨ༧ଌϥϕϧͷ෼෍͕Ұ க %FNPHSBQIJDQBSJUZ ℙ{ ̂ Y ∈ 𝒜 |S = s} = ℙ{ ̂ Y ∈ 𝒜 |S = s′  } ೚ҙͷ 𝒜 , s, s′  ʹ͍ͭͯ ࠾༻ ඇ࠾༻ ࠾༻ ඇ࠾༻ உੑ ঁੑ = ̂ Y|S = உੑ ̂ Y|S = ঁੑ

&RVBMJ[FEPEET<)BSEU> w ِཅੑͷΈͷҰகͷ৔߹&RVBM PQQPSUVOJUZͱݺͿ &RVBMJ[FEPEET ℙ{ ̂ Y ∈ 𝒜 |Y = y, S = s} = ℙ{ ̂ Y ∈ 𝒜 |Y = y, S = s′  } ೚ҙͷ 𝒜 , y, s, s′  ʹ͍ͭͯ ࠾༻ ඇ࠾༻ உੑ ࠾༻ ඇ࠾༻ ঁੑ ਅͷ࠾༻෼෍ ֶश ࠾༻ ඇ࠾༻ ࠾༻ ඇ࠾༻ ެฏ ֶशͷ݁ՌมԽͨ͠෦෼ ྘ͷ෦෼ ͷҰக

*OEJWJEVBMGBJSOFTT<%XPSL> w ੑผҎ֎શ͘ಉ͡ਓ͕͍Ε͹࠾൱΋ಉ͡ʹ͢Δ΂͖ w ࣅͨΑ͏ͳਓ͸ࣅͨ݁ՌΛड͚औΔ΂͖ w ֬཰త༧ଌؔ਺ w -JQTDIJU[QSPQFSUZ ೚ҙͷx, x′  ʹ͍ͭͯ D( f(x), f(x′  )) ≤ d(x, x′  ) ≈ ⟹ f : 𝒳 → Δ( 𝒴 ) ݁Ռͷ෼෍ؒͷ ڑ཭

ެฏͳ෼ྨֶशͷ࢓૊Έ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ ެฏੑج४Λୡ੒ f ʹΑΔ෼ྨޡ͕ࠩ 
 ࠷খʹͳΔΑ͏ʹ͢Δ *NBHFTGSPN<#VPMBNXJOJ>

ެฏͳֶशͷجຊઓུ w ࠷దԽ໰୊ͱͯ͠ఆࣜԽ w േଇɾ੍໿Λઃ͚Δ͜ͱͰެฏੑΛ੍ޚ minf Err(f ) + ηUnfair(f ) േଇϕʔε minf Err(f ) TVCUP Unfair(f ) ≤ η ੍໿ϕʔε

ެฏͳֶशख๏ w1SPYZ"QQSPBDI<,BNJTIJNB 'VLVDIJ ;BGBS> wඍ෼ෆՄೳ߲Λඍ෼Մೳͳؔ਺Ͱۙࣅ w3FEVDUJPO"QQSPBDI<"HBSXBM $PUUFS "HBSXBM> wެฏͳֶशͷ࠷దԽ໰୊Λஞ࣍తͳ੍໿ͳ͠࠷దԽʹॻ͖׵͑ Δ w'BJS3FQSFTFOUBUJPO<;BGBS 9JF .PZFS ;IBP $SFBHFS> wηϯγςΟϒଐੑʹґଘ͠ͳ͍ਂ૚දݱΛֶश͠ɼͦͷදݱΛ ࢖ͬͯ༧ଌΛߦ͏

ެฏͳֶशʹର͢Δ#BZFT࠷ ద༧ଌํ๏

෼ྨʹ͓͚Δ#BZFTPQUJNBMDMBTTJ fi FS w ෼ྨ໰୊ w ɹɹɹɹɹɹɹɹɹɹɹɹֶྺɼ৬ྺɼࢿ֨ͳͲ w ɹɹɹɹɹɹɹɹɹɹɹɹੑผɼਓछɼफڭɼ੓࣏ࢤ޲ɼ೥ྸͳͲ w ɹɹɹɹɹɹɹɹɹɹɹɹ༧ଌ͍ͨ͠΋ͷ FH ࠾൱ w ɹɹɹɹɹɹɹɹɹɹɹɹΞϧΰϦζϜʹΑͬͯ༧ଌ͞Εͨϥϕϧ w ֬཰తͳ༧ଌث w േଇϕʔεͷެฏͳֶश w ֶशͷจ຺Ͱ͸໨తؔ਺ɾެฏੑ੍໿Λ௚઀ධՁͰ͖ͳ͍ w ͷ෼෍͕ಘΒΕΔ৔߹ͷ࠷దԽ໰୊ͷղ͸Կ͔ʁ w #BZFTPQUJNBMDMBTTJ fi FS f : ℝd × {0,1} → [0,1] minf 𝔼 [Y(1 − f(X, S)) + (1 − Y)f(X, S)] + λUnfair( f ) (Y, X, S) ೖྗ X ∈ ℝd ϥϕϧ Y ∈ {0,1} ηϯγςΟϒଐੑ S ∈ {0,1} ༧ଌϥϕϧ ̂ Y ∈ {0,1}

ී௨ͷ෼ྨ໰୊ʹ͓͚Δ#BZFTPQUJNBM DMBTTJ fi FS<&MLBO> wެฏੑ੍໿͕ͳ͍৔߹ͷ#BZFTPQUJNBMDMBTTJ fi FS͸ʁ ଛࣦΛ࠷খʹ͢Δ֬཰తͳ#BZFTPQUJNBMDMBTTJ fi FS ͸೚ҙͷ ʹ͍ͭͯҎԼͷࣜͰ༩͑ΒΕΔɽ f* α ∈ [0,1] f*(x) = Hα (η(x) − 1 2 ) ఆཧʢ؆қ൛ʣ<&MLBO> w w ͸ࢦࣔؔ਺ w Hα (z) = 1{z > 0} + α1{z = 0} 1 η(x) = 𝔼 [Y|X = x]

%FNPHSBQIJDQBSJUZ੍໿ʹ͓͚Δ#BZFT PQUJNBMDMBTTJ fi FS<.FOPO> wެฏੑΛ%FNPHSBQIJDQBSJUZͰଌΔͱ͢Δ w w Unfair(f ) = ℙ{ ̂ Y = 1|S = 0} − ℙ{ ̂ Y = 1|S = 1} η(x, s) = 𝔼 [Y = 1|X = x, S = s] π = ℙ{S = 0} %FNPHSBQIJDQBSJUZʹର͢Δേଇ෇͖ͷଛࣦΛ࠷খ ʹ͢Δ֬཰తͳ#BZFTPQUJNBMDMBTTJ fi FS ͸೚ҙͷ ʹ͍ͭͯҎԼͷࣜͰ༩͑ΒΕΔɽ f* α ∈ [0,1] f*(x, s) = Hα (η(x, s) − (−1)s λ 2s(1 − π) + 2(1 − s)π − 1 2 ) ఆཧʢ؆қ൛ʣ<.FOPO>

%FNPHSBQIJDQBSJUZ੍໿ʹ͓͚Δ#BZFT PQUJNBMDMBTTJ fi FS<.FOPO> wެฏੑΛ%FNPHSBQIJDQBSJUZͰଌΔͱ͢Δ w w Unfair(f ) = ℙ{ ̂ Y = 1|S = 0} − ℙ{ ̂ Y = 1|S = 1} η(x, s) = 𝔼 [Y = 1|X = x, S = s] π = ℙ{S = 0} %FNPHSBQIJDQBSJUZʹର͢Δേଇ෇͖ͷଛࣦΛ࠷খ ʹ͢Δ֬཰తͳ#BZFTPQUJNBMDMBTTJ fi FS ͸೚ҙͷ ʹ͍ͭͯҎԼͷࣜͰ༩͑ΒΕΔɽ f* α ∈ [0,1] f*(x, s) = Hα (η(x, s) − (−1)s λ 2s(1 − π) + 2(1 − s)π − 1 2 ) ఆཧʢ؆қ൛ʣ<.FOPO> ͰͷΈมΘΔఆ਺ s

%FNPHSBQIJDQBSJUZ੍໿ʹ͓͚Δ#BZFT PQUJNBMDMBTTJ fi FS<.FOPO> w#BZFTPQUJNBMDMBTTJ fi FS͸ηϯγςΟϒଐੑ͝ͱʹᮢ஋ Λม͍͑ͯΔ S = 0 S = 1 η(x,0) η(x,1) ͷׂ߹͕ಉ͡ʹͳΔΑ͏ʹη ϯγςΟϒଐੑ͝ͱʹม͑Δ ̂ Y = 1

&RVBMPQQPSUVOJUZ੍໿ʹ͓͚Δ#BZFT PQUJNBMDMBTTJ fi FS<.FOPO> wެฏੑΛ&RVBMPQQPSUVOJUZͰଌΔͱ͢Δ wUnfair(f ) = ℙ{ ̂ Y = 1|Y = 1,S = 0} − ℙ{ ̂ Y = 1|Y = 1,S = 1} &RVBMPQQPSUVOJUZʹର͢Δേଇ෇͖ͷଛࣦΛ࠷খʹ ͢Δ֬཰తͳ#BZFTPQUJNBMDMBTTJ fi FS ͸೚ҙͷ ʹ͍ͭͯҎԼͷࣜͰ༩͑ΒΕΔɽ f* α ∈ [0,1] f*(x, s) = Hα (η(x, s) − 1 2 ) ఆཧʢ؆қ൛ʣ<.FOPO> ੍໿ͳ͠ͱಉ͡ʂ

&RVBMPQQPSUVOJUZ੍໿ʹ͓͚Δܾఆత #BZFTPQUJNBMDMBTTJ fi FS<$I[IFO> w׬શͳ&RVBMPQQPSUVOJUZ੍໿Λߟ͑Δ wേଇͰ͸ͳ͘ ͷ੍໿ͱ͢Δ w w Unfair(f ) = 0 Unfair(f ) = ℙ{ ̂ Y = 1|Y = 1,S = 0} − ℙ{ ̂ Y = 1|Y = 1,S = 1} πs = ℙ{Y = 1,S = s} &RVBMPQQPSUVOJUZʹର͢Δേଇ෇͖ͷଛࣦΛ࠷খʹ ͢Δܾఆతͳ#BZFTPQUJNBMDMBTTJ fi FS ͸೚ҙͷ ʹ͍ͭͯҎԼͷࣜͰ༩͑ΒΕΔɽ f* α ∈ [0,1] f*(x, s) = 1 {η(x, s)(1 + (−1)s θ πs ) ≥ 1 2 } ఆཧʢ؆қ൛ʣ<$I[IFO> ຖʹᮢ஋Λม͍͑ͯΔ s

ճؼʹ͓͚Δ#BZFTPQUJNBMSFHSFTTPS w ύϥϝτϦοΫճؼϞσϧ w ͸ฏۉͷϊΠζ w ܾఆతͳ༧ଌث w %FNPHSBQIJDQBSJUZ੍໿ͷެฏͳֶश Y = f*(X, S) + ξ ξ f : ℝd × [M] → ℝ minf 𝔼 [(Y − f(X, S))2] sub to ∀s ∈ [M], E, ℙ{f(X, S) ∈ E|S = s} = ℙ{f(X, S) ∈ E}

8BTTFSTUFJOCBSZDFOUFSʹΑΔ 
 #BZFTPQUJNBMFSSPSͷಛ௃͚ͮ<$I[IFO> w %FNPHSBQIJDQBSJUZ੍໿ͷެฏͳֶश͸8BTTFSTUFJO CBSZDFOUFSͰಛ௃͚ͮͰ͖Δ w 8BTTFSTUFJOڑ཭ w ͷ Ͱ৚͚݅ͭͨ෼෍ w W2 (ν0 , ν1 ) = infπ∈Π(ν0 ,ν1 ) ∫ |x − y|2 π(dx, dy) f*(X, S) S = s νf*|s ps = ℙ{S = s} minf:DP 𝔼 [(f*(X, S) − f(X, S))2] = minν ∑ s∈[M] ps W2 2 (νf*|s , ν) ఆཧ<$I[IFO> 8BTTFSTUFJOCBSZDFOUFS

ճؼʹ͓͚Δ#BZFTPQUJNBMSFHSFTTPS <$I[IFO> w %FNPHSBQIJDQBSJUZ੍໿ͷ#BZFTPQUJNBMSFHSFTTPS͸ҎԼͷఆཧͰٻ·Δ w ͸ ͷ Ͱ৚݅෇͚ͨ࣌͠ͷ$%'ͱٖࣅ*$%' Fs , F−1 s f*(X, S) S = s g*(x, s) = ∑ s∈[M] ps F−1 s (Fs (f*(x, s))) ఆཧ<$I[IFO>

ճؼʹ͓͚Δ#BZFTPQUJNBMSFHSFTTPS <$I[IFO> w %FNPHSBQIJDQBSJUZ੍໿ͷ#BZFTPQUJNBMSFHSFTTPS͸ҎԼͷఆཧͰٻ·Δ w ͸ ͷ Ͱ৚݅෇͚ͨ࣌͠ͷ$%'ͱٖࣅ*$%' Fs , F−1 s f*(X, S) S = s g*(x, s) = ∑ s∈[M] ps F−1 s (Fs (f*(x, s))) ఆཧ<$I[IFO> ੨ͱ྘ͷ໘ੵ͸౳͍͠ ੺͸੨ͱ྘ͷฏۉ

ճؼʹ͓͚Δ#BZFTPQUJNBMSFHSFTTPS <$I[IFOBOE4DISFVEFS> w%FNPHSBQIJDQBSJUZͷެฏੑ੍໿Λ؇࿨ͨ࣌͠ͷ#BZFT PQUJNBMSFHSFTTPS w w U(f ) = infν ∑ s∈[M] ps W2 2 (νf|s , ν) U(f ) ≤ αU(f*) g* α (x, s) = (1 − α)g* 0 (x, s) + αf*(x, s) ఆཧ<$I[IFOBOE4DISFVEFS> લϖʔδͷ#BZFTPQUJNBM SFHSFTTPS

ެฏͳֶशʹର͢ΔNJOJNBY ࠷దͳֶशํ๏

.JOJNBY࠷దੑ w#BZFTPQUJNBM͸ ͷ෼෍ͷ࠷దͳ༧ଌ wݱ࣮తʹ͸ ͷ෼෍͸ಘΒΕͳ͍ w༗ݶαϯϓϧ͔͠ಘΒΕͳ͍ঢ়گͰެฏͰ͔ͭ࠷΋ޡ͕ࠩখ͍͞ ֶशΞϧΰϦζϜ͸Կ͔ʁ࠷దੑ wͭ࠷దੑͷج४NJOJNBY࠷దੑ (Y, X, S) (Y, X, S) inf ̂ f:Fair supP Err(f; P) NJOJNBY࠷దଛࣦ શͯͷެฏͳΞϧΰϦζϜʹ ͍ͭͯ࠷খԽ ͷ෼෍ ʹ͍ͭͯ 
 ࠷ѱέʔε (Y, X, S) P

Ճ๏తόΠΞεઢܗϞσϧʹ͓͚Δճؼͷ NJOJNBY࠷దੑ<$I[IFOBOE4DISFVEFS> wଛࣦ wެฏͳΞϧΰϦζϜ೚ҙͷαϯϓϧαΠζ ʹ͍ͭͯ w ͸ΞϧΰϦζϜ͕ग़ྗ͢Δ༧ଌث w ͜ͷ໰୊ʹ͓͚ΔNJOJNBY࠷దͳΞϧΰϦζϜͱ͸ʁ Err( f; P) = 𝔼 P [( f*(X, S) − f(X, S))2] n ℙ{U( ̂ f ) ≤ αU( f*)} ≥ 1 − δ ̂ f w w ͸ฏۉͷ෼ࢄ ͷਖ਼ن෼෍ʹै͏ w ͸ฏۉ ͷڞ෼ࢄߦྻ ͷଟมྔਖ਼ن෼෍ʹै͏ Y = f*(X, S) + ξ = ⟨β, X⟩ + bS + ξ ξ σξ X μ Σ Ճ๏తόΠΞεઢܗϞσϧ

Ճ๏తόΠΞεઢܗϞσϧʹ͓͚Δճؼͷ NJOJNBY࠷దੑ<$I[IFOBOE4DISFVEFS> wଛࣦ wެฏͳΞϧΰϦζϜ೚ҙͷαϯϓϧαΠζ ʹ͍ͭͯ w ͸ΞϧΰϦζϜ͕ग़ྗ͢Δ༧ଌث w ͜ͷ໰୊ʹ͓͚ΔNJOJNBY࠷దͳΞϧΰϦζϜͱ͸ʁ Err( f; P) = 𝔼 P [( f*(X, S) − f(X, S))2] n ℙ{U( ̂ f ) ≤ αU( f*)} ≥ 1 − δ ̂ f w w ͸ฏۉͷ෼ࢄ ͷਖ਼ن෼෍ʹै͏ w ͸ฏۉ ͷڞ෼ࢄߦྻ ͷଟมྔਖ਼ن෼෍ʹै͏ Y = f*(X, S) + ξ = ⟨β, X⟩ + bS + ξ ξ σξ X μ Σ Ճ๏తόΠΞεઢܗϞσϧ ʹґଘ͢Δ߲͸Ճ๏తͳఆ਺ S

Ճ๏తόΠΞεઢܗϞσϧʹ͓͚Δճؼͷ NJOJNBY࠷దੑ<$I[IFOBOE4DISFVEFS> w͜ͷઃఆͩͱҰகੑΛ࣋ͨͳ͍ͨΊجຊతʹ͸ఆ਺ wୈ߲͕࣮࣭తͳ͜ͷ໰୊ʹ͓͚ΔNJOJNBY࠷దଛࣦʹ ͳ͍ͬͯΔ w͜ͷ߲͸ެฏੑ੍໿͕ͳ͍NJOJNBY࠷దଛࣦͱಉ͡ Ճ๏తόΠΞεઢܗϞσϧʹ͓͍ͯNJOJNBY࠷దଛࣦ͸Ҏ Լͷఆ਺ഒͰ্Լ͔Β཈͑ΒΕΔ {σ2 ξ ( d + M n + ln(1/δ) n )} ∨ {(1 − α)2U(f*)} ఆཧ

Ճ๏తόΠΞεઢܗϞσϧͷNJOJNBY࠷దΞ ϧΰϦζϜ<$I[IFOBOE4DISFVEFS> wͲͷΑ͏ͳΞϧΰϦζϜͳΒNJOJNBY࠷దʹͳΔͷ͔ʁ w1MVHJOਪఆ๏ w͜ͷϞσϧͷ#BZFTPQUJNBMSFHSFTTPS͕ҎԼͰ΋ͱ·Δ w1MVHJOਪఆ๏ະ஌ύϥϝʔλͰ͋Δ ͱ Λαϯϓϧ͔Β ਪఆͯ͠ɼ্هؔ਺ʹ୅ೖͨ͠΋ͷΛ༧ଌثͱ͢Δ f* DP,α (x, s) = ⟨β, x⟩ + αbs + (1 − α)∑ s∈[M] ps bs β bs

ઢܗϞσϧʹ͓͚ΔNJOJNBY࠷దੑ <'VLVDIJ> w EFNPHSBQIJDQBSJUZ੍໿ԼͰͷ#BZFTPQUJNBMSFHSFTTPS w%1੍໿ wଛࣦ wެฏͳΞϧΰϦζϜ೚ҙͷαϯϓϧαΠζ ʹ͍ͭͯ w͜ͷϞσϧͰͷNJOJNBY࠷దͳΞϧΰϦζϜ͸ʁ f* DP ∀s ∈ [M], E, ℙ{f(X, S) ∈ E|S = s} = ℙ{f(X, S) ∈ E} Err( f; P) = 𝔼 P [( f* DP (X, S) − f(X, S))2] n ℙ{maxs,s′  ∈[M] W2 (ν ̂ f|s , ν ̂ f|s′  ) ≤ Cn−α} ≥ 1 − δ w w ͸ฏۉͷ෼ࢄ ͷਖ਼ن෼෍ʹै͏ w ͸ฏۉ ͷڞ෼ࢄߦྻ ͷଟมྔਖ਼ن෼෍ʹै͏ Y = f*(X, S) + ξ = ⟨βs , X⟩ + ξ ξ σξ X μS σX I ઢܗϞσϧ

ઢܗϞσϧʹ͓͚ΔNJOJNBY࠷దੑ <'VLVDIJ> w EFNPHSBQIJDQBSJUZ੍໿ԼͰͷ#BZFTPQUJNBMSFHSFTTPS w%1੍໿ wଛࣦ wެฏͳΞϧΰϦζϜ೚ҙͷαϯϓϧαΠζ ʹ͍ͭͯ w͜ͷϞσϧͰͷNJOJNBY࠷దͳΞϧΰϦζϜ͸ʁ f* DP ∀s ∈ [M], E, ℙ{f(X, S) ∈ E|S = s} = ℙ{f(X, S) ∈ E} Err( f; P) = 𝔼 P [( f* DP (X, S) − f(X, S))2] n ℙ{maxs,s′  ∈[M] W2 (ν ̂ f|s , ν ̂ f|s′  ) ≤ Cn−α} ≥ 1 − δ w w ͸ฏۉͷ෼ࢄ ͷਖ਼ن෼෍ʹै͏ w ͸ฏۉ ͷڞ෼ࢄߦྻ ͷଟมྔਖ਼ن෼෍ʹै͏ Y = f*(X, S) + ξ = ⟨βs , X⟩ + ξ ξ σξ X μS σX I ઢܗϞσϧ ઢܗύϥϝʔλͱ ͷฏۉ͕ ʹΑΓมΘΔ X S ଛࣦΛެฏͰ࠷దͳ༧ଌثͱͷ 
 ޡࠩͰఆٛʢҰகੑ͕੒Γཱͭʣ ࠷ѱέʔεͷެฏੑΛධՁ ʢ ͸ฏۉతͳެฏੑʣ U

ઢܗϞσϧʹ͓͚ΔNJOJNBY࠷దੑ <'VLVDIJ> w ͱ Λఆ਺ͱݟΕ͹࠷దଛࣦ͸ w ͸ฏۉ ͷ࠷େϊϧϜ w ͸Ϟσϧͷύϥϝʔλ਺ʹରԠ͢ΔͨΊطଘͷཧ࿦ͱ΋੔߹͢Δ w ͸ެฏੑಛ༗ʹݱΕΔ߲ w ͸ ʹର͢Δ ͷϊϧϜͷෆҰக౓ͷΑ͏ͳ஋ w࣮࣭తʹ͸ ͷ෼ࢄͷෆҰக౓Λද͢ w࠷దͳΞϧΰϦζϜ͸1MVHJOਪఆྔʹΑͬͯٻ·Δ σX U σξ B2dM n U μs dM B B S βs Y ઢܗϞσϧʹ͓͍ͯNJOJNBY࠷దଛࣦ ͸ҎԼͷΑ͏ʹ཈ ͑ΒΕΔ R* σ2 ξ B2dM n ≲ R* ≲ σ2 X σ2 ξ B2dM ∨ U2σ2 ξ M ∨ U2σX ∨ U2B2 n ఆཧ

·ͱΊ wެฏੑͷ஫໨ͱࣾձతͳඞཁੑ wݱ࣮ͷΞϓϦέʔγϣϯͰ໰୊ʹͳ͓ͬͯΓɼࣾձతɾ๏తʹඞཁੑ͕ग़͖ͯ ͍ͯΔ wػցֶशʹ͓͚ΔෆެฏͷݪҼ wσʔλऩूʹ͓͚ΔόΠΞεɾֶशʹ͓͚ΔόΠΞε wެฏͳֶशͷ࢓૊Έ wެฏੑج४EFNPHSBQIJDQBSJUZ FRVBMJ[FEPEET ʜ wެฏੑͷ੍໿ɾേଇ෇͖ͷଛࣦ࠷খԽ wެฏͳֶशͷ#BZFTPQUJNBMͳ෼ྨɾճؼ wᮢ஋Λม͑ΔΑ͏ͳΠϝʔδͰா৲߹ΘͤΛߦ͏ wެฏͳֶशͷNJOJNBY࠷దͳճؼ wՃ๏తϞσϧͰͷ࠷దੑ<$I[IFOBOE4DISFVEFS>ɼઢܗϞσϧͰͷ࠷ద ੑ<'VLVDIJ>

ࢀߟจݙ w<#VPMBNXJOJ`>#VPMBNXJOJ +PZ BOE5JNOJU(FCSV(FOEFSTIBEFT*OUFSTFDUJPOBM BDDVSBDZEJTQBSJUJFTJODPNNFSDJBMHFOEFSDMBTTJ fi DBUJPO$POGFSFODFPOGBJSOFTT BDDPVOUBCJMJUZBOEUSBOTQBSFODZ1.-3 w<%BTUJO`>+F ff SFZ%BTUJO"NB[POTDSBQTTFDSFU"*SFDSVJUJOHUPPMUIBUTIPXFECJBT BHBJOTUXPNFO 63-IUUQTXXXSFVUFSTDPNBSUJDMFVTBNB[PODPNKPCT BVUPNBUJPOJOTJHIUBNB[POTDSBQTTFDSFUBJSFDSVJUJOHUPPMUIBUTIPXFECJBTBHBJOTU XPNFOJE64,$/.,( w<㵽BSCBL`TGBDFCPPLQPTU>&㵽BSCBLVSMIUUQTXXXGBDFCPPLDPNQIPUPQIQ GCJETFUBUZQF UIFBUFS w<"OHXJO`>"OHXJO +VMJB +F ff -BSTPO 4VSZB.BUUV BOE-BVSFO,JSDIOFS .BDIJOF#JBTVSMIUUQTXXXQSPQVCMJDBPSHBSUJDMFNBDIJOFCJBTSJTLBTTFTTNFOUT JODSJNJOBMTFOUFODJOH w<1PEFTUB>&9&$0''*$&0'5)&13&4*%&/5 #*(%"5"4&*;*/( 0110356/*5*&4 13&4&37*/(7"-6&4 w<.VOP[>&9&$0''*$&0'5)&13&4*%&/5 #JH%BUB"3FQPSUPO"MHPSJUINJD 4ZTUFNT 0QQPSUVOJUZ BOE$JWJM3JHIUT

ࢀߟจݙ w<#BSPDBT>#BSPDBT 4PMPO BOE"OESFX%4FMCTU#JHEBUBTEJTQBSBUFJNQBDU$BMJG- 3FW w<$BMEFST>$BMEFST 5PPOBOE4JDDP7FSXFSl5ISFFOBJWF#BZFTBQQSPBDIFTGPSEJTDSJNJOBUJPO GSFFDMBTTJ fi DBUJPOz*O%BUB.JOJOHBOE,OPXMFEHF%JTDPWFSZ QQr w<)BSEU>)BSEU .PSJU[ &SJD1SJDF BOE/BUIBO4SFCSP %FD l&RVBMJUZPG0QQPSUVOJUZJO 4VQFSWJTFE-FBSOJOHz*O"EWBODFTJO/FVSBM*OGPSNBUJPO1SPDFTTJOH4ZTUFNT#BSDFMPOB 4QBJO$VSSBO"TTPDJBUFT *OD QQr w<%XPSL>%XPSL $ZOUIJB .PSJU[)BSEU 5POJBOO1JUBTTJ 0NFS3FJOHPME BOE3JDIBSE;FNFM l'BJSOFTTUISPVHIBXBSFOFTTz*O1SPDFFEJOHTPGUIFSE*OOPWBUJPOTJO5IFPSFUJDBM$PNQVUFS 4DJFODF$POGFSFODF&ECZ4IB fi (PMEXBTTFS"$.$BNCSJEHF ." 64""$. QQr w<1MFJTT>1MFJTT (FP ff .BOJTI3BHIBWBO 'FMJY8V +PO,MFJOCFSH BOE,JMJBO28FJOCFSHFSl0O 'BJSOFTTBOE$BMJCSBUJPOz*O"EWBODFTJO/FVSBM*OGPSNBUJPO1SPDFTTJOH4ZTUFNT"TTPDJBUFT *OD QQr w<,VTOFS>.BUU+,VTOFS +PTIVB-PGUVT $ISJT3VTTFMM 3JDBSEP4JMWB$PVOUFSGBDUVBMGBJSOFTT "EWBODFTJOOFVSBMJOGPSNBUJPOQSPDFTTJOHTZTUFNT w<$IJBQQB>$IJBQQB 41BUI4QFDJ fi D$PVOUFSGBDUVBM'BJSOFTT1SPDFFEJOHTPGUIF"""* $POGFSFODFPO"SUJ fi DJBM*OUFMMJHFODF EPJ BBBJWJ

ࢀߟจݙ w<,BNJTIJNB>,BNJTIJNB 5PTIJIJSP 4IPUBSP"LBIP )JEFLJ"TPI BOE+VO4BLVNBl'BJSOFTT"XBSF $MBTTJ fi FSXJUI1SFKVEJDF3FNPWFS3FHVMBSJ[FSz*O.BDIJOF-FBSOJOHBOE,OPXMFEHF%JTDPWFSZJO %BUBCBTFT&VSPQFBO$POGFSFODF &$.-1,%% 1BSU**7PM-FDUVSF/PUFTJO$PNQVUFS4DJFODF #SJTUPM 6,4QSJOHFS QQr w<'VLVDIJ>'VLVDIJ ,B[VUP +VO4BLVNB BOE5PTIJIJSP,BNJTIJNBl1SFEJDUJPOXJUI.PEFM#BTFE /FVUSBMJUZz*O.BDIJOF-FBSOJOHBOE,OPXMFEHF%JTDPWFSZJO%BUBCBTFT&VSPQFBO$POGFSFODF &$.- 1,%% 1BSU**7PM-FDUVSF/PUFTJO$PNQVUFS4DJFODF1SBHVF $[FDI3FQVCMJD4QSJOHFS QQr JTCOEPJ\a@^ w<;BGBS>.VIBNNBE#JMBM;BGBS *TBCFM7BMFSB .BOVFM(PNF[3PESJHVF[ BOE,SJTIOB1(VNNBEJ 'BJSOFTTDPOTUSBJOUT" fl FYJCMFBQQSPBDIGPSGBJSDMBTTJ fi DBUJPO5IF+PVSOBMPG.BDIJOF-FBSOJOH 3FTFBSDI w<"HBSXBM>"HBSXBM "MFLI "MJOB#FZHFM[JNFS .JSPTMBW%VEÍL +PIO-BOHGPSE BOE)BOOB8BMMBDIl" 3FEVDUJPOT"QQSPBDIUP'BJS$MBTTJ fi DBUJPOz*OUIFUI*OUFSOBUJPOBM$POGFSFODFPO.BDIJOF-FBSOJOH QQr w<$PUUFS>"OESFX$PUUFS )FJOSJDI+JBOH ,BSUIJL4SJEIBSBO5XP1MBZFS(BNFTGPS& ffi DJFOU/PO $POWFY$POTUSBJOFE0QUJNJ[BUJPO1SPDFFEJOHTPGUIFUI*OUFSOBUJPOBM$POGFSFODFPO"MHPSJUINJD -FBSOJOH5IFPSZ 1.-3 w<"HBSXBM>"HBSXBM "MFLI .JSPTMBW%VEÍL BOE;IJXFJ4UFWFO8Vl'BJS3FHSFTTJPO2VBOUJUBUJWF %F fi OJUJPOTBOE3FEVDUJPOCBTFE"MHPSJUINTz*OUI*OUFSOBUJPOBM$POGFSFODFPO.BDIJOF-FBSOJOH *$.-7PM+VOF*OUFSOBUJPOBM.BDIJOF-FBSOJOH4PDJFUZ *.-4 QQr

ࢀߟจݙ w<;BGBS>3JDI;FNFM :V8V ,FWJO4XFSTLZ 5POJ1JUBTTJ $ZOUIJB%XPSL -FBSOJOH'BJS3FQSFTFOUBUJPOT1SPDFFEJOHTPGUIFUI*OUFSOBUJPOBM $POGFSFODFPO.BDIJOF-FBSOJOH 1.-3 w<9JF>2J[IF9JF ;JIBOH%BJ :VMVO%V &EVBSE)PWZ (SBIBN/FVCJH $POUSPMMBCMF*OWBSJBODFUISPVHI"EWFSTBSJBM'FBUVSF-FBSOJOH"EWBODFTJO /FVSBM*OGPSNBUJPO1SPDFTTJOH4ZTUFNT w<.PZFS>%BOJFM.PZFS 4IVZBOH(BP 3PC#SFLFMNBOT "SBN(BMTUZBO (SFH 7FS4UFFH*OWBSJBOU3FQSFTFOUBUJPOTXJUIPVU"EWFSTBSJBM5SBJOJOH"EWBODFTJO /FVSBM*OGPSNBUJPO1SPDFTTJOH4ZTUFNT w<;IBP>)BO;IBP 3FNJ5BDIFU%FT$PNCFT ,VO;IBOH (FP ff SFZ(PSEPO0O -FBSOJOH*OWBSJBOU3FQSFTFOUBUJPOTGPS%PNBJO"EBQUBUJPO1SPDFFEJOHTPGUIF UI*OUFSOBUJPOBM$POGFSFODFPO.BDIJOF-FBSOJOH 1.-3 w<$SFBHFS>&MMJPU$SFBHFS %BWJE.BESBT +PFSO)FOSJL+BDPCTFO .BSJTTB8FJT ,FWJO4XFSTLZ 5POJBOO1JUBTTJ 3JDIBSE;FNFM'MFYJCMZ'BJS3FQSFTFOUBUJPO -FBSOJOHCZ%JTFOUBOHMFNFOU1SPDFFEJOHTPGUIFUI*OUFSOBUJPOBM$POGFSFODF PO.BDIJOF-FBSOJOH 1.-3

ࢀߟจݙ w<&MLBO>$IBSMFT&MLBO5IFGPVOEBUJPOTPGDPTUTFOTJUJWFMFBSOJOH*O*OUFSOBUJPOBM KPJOUDPOGFSFODFPO"SUJGJDJBM*OUFMMJHFODF *+$"* QBHFTr w<.FOPO>"EJUZB,SJTIOB.FOPO 3PCFSU$8JMMJBNTPO5IFDPTUPGGBJSOFTTJO CJOBSZDMBTTJ fi DBUJPO1SPDFFEJOHTPGUIFTU$POGFSFODFPO'BJSOFTT "DDPVOUBCJMJUZ BOE5SBOTQBSFODZ 1.-3 w<$I[IFO>&WHFOJJ$I[IFO $ISJTUPQIF%FOJT .PIBNFE)FCJSJ -VDB0OFUP BOE .BTTJNJMJBOP1POUJM-FWFSBHJOHMBCFMFEBOEVOMBCFMFEEBUBGPSDPOTJTUFOUGBJSCJOBSZ DMBTTJ fi DBUJPO"EWBODFTJO/FVSBM*OGPSNBUJPO1SPDFTTJOH4ZTUFNT w<$I[IFO>&WHFOJJ$I[IFO $ISJTUPQIF%FOJT .PIBNFE)FCJSJ BOE.BTTJNJMJBOP 1POUJM'BJS3FHSFTTJPOXJUI8BTTFSTUFJO#BSZDFOUFST*O"EWBODFTJO/FVSBM *OGPSNBUJPO1SPDFTTJOH4ZTUFNT WPMVNF QBHFTr$VSSBO"TTPDJBUFT *OD w<$I[IFOBOE4DISFVEFS>&WHFOJJ$I[IFOBOE/JDPMBT4DISFVEFS"NJOJNBY GSBNFXPSLGPSRVBOUJGZJOHSJTLGBJSOFTTUSBEFP ff JOSFHSFTTJPOEPJ BSYJW w<'VLVDIJ>'VLVDIJ ,B[VUP BOE+VO4BLVNB.JOJNBY0QUJNBM'BJS3FHSFTTJPO VOEFS-JOFBS.PEFMBS9JWQSFQSJOUBS9JW