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20201102RLArch.pdf

Tatsuya Matsushima
November 02, 2020
150

 20201102RLArch.pdf

Tatsuya Matsushima

November 02, 2020
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Transcript

  1. %JWFSHFODFNJOJNJ[BUJPOͱͯ͠ͷ
    ߦಈͱೝࣝ

    ୈճڧԽֶशΞʔΩςΫνϟษڧձ
    ౦ژେֶେֶӃ޻ֶܥݚڀՊদඌݚڀࣨ
    ത࢜՝ఔদౢୡ໵
    NBUTVTIJNB!XFCMBCUVUPLZPBDKQ!@@UNBUT@@

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  2. ࣗݾ঺հ
    দౢୡ໵ʢ5BUTVZB.BUTVTIJNBʣ
    ޻ֶܥݚڀՊത࢜՝ఔ೥
    ʮϩϘοτֶशʯ෼໺ͷݚڀΛ͍ͯ͠·͢
    • σʔλΛ༻͍ͯϩϘοτͷ੍ޚΛֶश͢Δ෼໺
    • ڧԽֶशɾ໛฿ֶशʜ

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  3. ߦಈ΍஌֮ʹؔ͢Δ༷ʑͳཧ࿦ΛEJWFSHFODFNJOJNJ[BUJPO
    ͱͯ͠౷Ұతʹٞ࿦Ͱ͖Δ͜ͱΛࣔͨ͠࿦จΛ঺հ
    • ग़ͯ͘Δ΋ͷΛશ෦KPJOU,-ͷ࠷খԽͱͯ͠ఆࣜԽ
    • ͜Ε·Ͱʹ΋ͪΐͪ͜ΐ͜૬ޓͷؔ࿈ੑʹؔ͢Δٞ࿦͸͕͋ͬͨ
    ͜͜·Ͱ౷Ұతͳ؍఺Ͱ·ͱΊͨ΋ͷ͸ʢͨͿΜʣॳΊͯ
    ຊൃදʹ͍ͭͯ

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  4. "DUJPOBOE1FSDFQUJPOBT%JWFSHFODF.JOJNJ[BUJPO
    • 4VCNJUUFEPO4FQ W
    MBTUSFWJTFE0DU W

    • ஶऀਞ͕݁ߏ͍͢͝ʢ(PPHMF#SBJO6$-%FFQ.JOE5SPOUPʣ
    • IUUQTBSYJWPSHBCT
    • *$-3VOEFSSFWJFXIUUQTPQFOSFWJFXOFUGPSVN
    JE+C"RTGC:T+Z
    ॻࢽ৘ใ

    1MB/FU%SFBNFSʢम࢜Ϙε͕'SJTUPOʣ
    ࣗ༝ΤωϧΪʔݪཧʢ'&1ʣ
    -BZFS/PSN"EBNʜ
    %FFQ.JOEͰ͍Ζ͍Ζ

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  5. ѻ͏಺༰͕޿͗ͯ͢͜ͷൃදͰ͸·ͱΊ͖Εͳ͍ͷͰ
    ࡉ͔͍࿩͸దٓࢿྉΛࢀর͍ͯͩ͘͠͞
    • $POUSPMBT*OGFSFODFIUUQTBSYJWPSHBCT
    • ླ໦͞Μ3-ΞʔΩʢࠓ೔લ൒ʣɼླ໦͞Μ%-ྠಡձɼ
    ୩ޱ͘ΜαϚʔεΫʔϧ
    • 'SFF&OFSHZ1SJODJQMFʢ'&1ʣɾBDUJWFJOGFSFODF
    • Θ͔Γ΍͍ࣗ͢༝ΤωϧΪʔݪཧʢүଜ୓࠸͞Μʣ
    • ঺հ͞ΕΔͦΕͧΕͷݚڀͷৄࡉ͸ຊ࿦จͷSFGFSFODFࢀর
    ຊൃදʹ͍ͭͯ

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  6. +PJOU,-.JOJNJ[BUJPO

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  7. ೖྗͷ֬཰ม਺ ɼͦΕҎ֎ͷʢજࡏʣม਺
    ύϥϝʔλ
    • ࠷దԽͷաఔͰ௚઀มߋ͢Δ͜ͱ͕Ͱ͖Δ΋ͷશͯ
    • ྫʣදݱʹؔ͢Δे෼౷ܭྔɼϞσϧͷύϥϝʔλɼํࡦͷύϥϝʔλ
    ࣮ࡍͷ෼෍ ʢBDUVBMEJTUSJCVUJPOʣ
    • ੜ੒աఔ Λදݱ
    • ࣮ࡍͷ෼෍͸ະ஌ͷ৔߹΋͋Δɹྫʣσʔλ෼෍
    • ύϥϝʔλ ͷӨڹΛड͚ͯมԽ͢Δ৔߹΋͋Δ
    λʔήοτͷ෼෍
    • γεςϜͷؔ͢Δ޷·͍͠ઃఆʢQSFGFSFODFʣΛදݱ
    • ʢͪΌΜͱॻ͔Εͯͳ͍͕ʣͬͪ͜΋ύϥϝʔλ ͷӨڹΛड͚Δ
    x z
    ϕ

    x, z ∼ pϕ
    (x, z)
    ϕ
    τ(x, z)
    ϕ
    ֬཰ม਺΍෼෍ͷදه

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  8. ஫ҙ
    • Ҏ߱ग़ͯ͘Δظ଴஋͸ Ͱظ଴஋Λͱͬͨ΋ͷͱ͢Δ
    ͭ·Γɼ
    • ࣮ࡍʹܥ͔ΒಘΒΕΔαϯϓϧΛ࢖ͬͯਪఆͰ͖Δ
    • ύϥϝʔλ ʹґଘ

    E [f(x, z)] =

    f(x, z)pϕ
    (x, z)dxdz
    ϕ
    ֬཰ม਺΍෼෍ͷදه

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  9. ύϥϝʔλ ΛมԽͤͯ͞ɼશͯͷม਺ʹؔ͢Δ
    ࣮ࡍͷ෼෍ ΛͰ͖Δ͚ͩλʔήοτ ʹ͚ۙͮΔ໰୊

    ͱͯ͠ఆࣜԽ
    • SFWFSTF,-΍JOGPSNBUJPOQSPKFDUJPOͱ΋ݺ͹ΕΔ
    • ʢ༨ஊʣSFWFSTF,-WTGPSXBSE,-ͷ࿩΋େࣄ
    • ͸Ϟσϧʢ͚ͩʣͰ͸ͳ͘ੜ੒աఔΛදݱͯ͠Δ͜ͱʹ஫ҙ
    • Ұ෦͸ਅͷ෼෍Ͱະ஌ɼҰ෦͸ύϥϝʔλ ʹΑΓมԽ͠͏Δ
    • ྫʣදݱֶशͷ৔߹ɼ ͸σʔλͱ৴೦ͷಉ࣌෼෍ɼ ͸જࡏม਺Ϟσϧ
    • ྫʣ੍ޚͷ৔߹ɼ ͸ݱࡏͷํࡦͰͷيಓͷ෼෍ɼ ͸يಓͷVUJMJUZ
    ϕ

    τ
    minϕ
    KL [pϕ
    (x, z)∥τ(x, z)]

    ϕ

    τ

    τ
    +PJOU,-.JOJNJ[BUJPO

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  10. ͜ͷٞ࿦Ͱஔ͍͍ͯΔԾఆʢJOEVDUJWFCJBTʣ
    • ΤʔδΣϯτ͸ԿΒ͔ͷ໨తؔ਺Λ࠷దԽ͢Δ
    • ΤʔδΣϯτ͸֬཰ม਺Λෆ࣮֬ੑΛදݱ͢ΔͨΊʹར༻͢Δ
    Ծఆ͕͜Ε͚ͩͳͷͰQPXFSGVMͳఆࣜԽ
    • ,-ΛNFUSJDͱͯ͠ར༻͍ͯ͠Δͷ͸௥ՃతͳԾఆʢBQQFOEJY#ʣ
    +PJOU,-.JOJNJ[BUJPO

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  11. ʢิ଍ʣطଘͷ࿮૊Έͱͷൺֱ
    • "DUJWFJOGFSFODF͸જࡏม਺ͷΤϯτϩϐʔ߲͕ೖ͍ͬͯΔ
    • %JWFSHFODFNJOJNJ[BUJPO͸Ճ͑ͯೖྗͷΤϯτϩϐʔ߲͕ೖΔ
    • ͜ͷ࿦จͷߩݙͷҰͭ͸BDUJWFJOGFSFODFʹ୅ΘΔ࿮૊ΈͷఏҊ
    • 1SFGFSFODFʹදݱྗͷߴ͍ϞσϧΛ༻͍ΔͱɼλεΫʹґଘ͠ͳ͍໨
    తؔ਺͕ग़ͯ͘Δ

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  12. *OGPSNBUJPO#PVOET

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  13. +PJOU,-ͷ࠷খԽ͸ɼೖྗ ͱજࡏม਺ ͷ
    QSFGFSFODFʢ޷·͠͞ʣͱ૬ޓ৘ใྔΛ૿Ճͤ͞Δ͜ͱΛࣔ͢
    • -BUFOUQSFGFSFODF
    • 7BSJBUJPOBMGSFFFOFSHZ
    • *OQVUQSFGFSFODF
    ͜ͷੑ࣭ʹΑͬͯɼطଘͷදݱֶशͱ୳ࡧͷ໨తؔ਺Λઆ໌Մೳ
    x z
    *OGPSNBUJPO#PVOET

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  14. +PJOU,-Λ෼ղ

    • ୈ߲ɿજࡏม਺ͷ৴೦ Λ޷·͍͠෼෍ ʹ͚ۙͮΔ੍໿
    • ୈ߲ɿ૬ޓ৘ใྔ ͷม෼Լք

    • ૬ޓ৘ใྔͷԼքͷ࠷খԽ㱺૬ޓ৘ใྔͷ࠷େԽ
    • ॊೈͳ ͷ଒Λ࢖͏΄ͲλΠτͳԼք
    KL [pϕ
    (x, z)∥τ(x, z)]
    KPJOUEJWFSHFODF
    = EKL [pϕ
    (z ∣ x)∥τ(z)]
    SFBMJ[JOHMBUFOUQSFGFSFODFT
    − E [ln τ(x ∣ z) − ln pϕ
    (x)]
    JOGPSNBUJPOCPVOE

    (z|x) τ(z)
    I[x; z]
    I[x; z] ≥ I[x; z] − EKL [p(x ∣ z)∥τϕ
    (x ∣ z)] = E [ln τϕ
    (x ∣ z) − ln p(x)]
    τϕ
    (x|z)
    -BUFOU1SFGFSFODF

    *OGPSNBUJPO#PVOET

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  15. • ୈ߲ ͷղऍ
    • ͷ࠷େԽʢෛͷ࠷খԽʣ͸ೖྗΛਖ਼͘͠༧ଌͰ͖Δજࡏม਺Λ
    ୳͢໾ׂ
    • ͷ࠷খԽ͸ଟ༷ͳೖྗΛ୳͢໾ׂ
    E [ln τ(x ∣ z) − ln pϕ
    (x)]
    ln τ(x|z)

    (x)
    -BUFOU1SFGFSFODF

    *OGPSNBUJPO#PVOET

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  16. ΤʔδΣϯτ͕ೖྗ ΛมߋͰ͖ͳ͍৔߹ɼ
    MBUFOUQSFGFSFODFͷ ͷ߲͸ύϥϝʔλԽ͞Εͳ͍ʢఆ਺ʣ
    • ྫʣݻఆͷσʔληοτ͔Βͷֶश
    ͜ͷͱ͖ɼઌ΄Ͳͷ໨తؔ਺͸ม෼ਪ࿦ʹ͓͚Δࣗ༝ΤωϧΪʔ
    ʢ&-#0ʣʹରԠ

    • ͰೖྗΛ࠶ߏ੒͠ͳ͕Βɼ Λࣄલ෼෍ ʹ͚ۙͮΔ
    x

    (x)
    min
    ϕ
    {EKL [pϕ
    (z ∣ x)∥τ(z)] − E [ln τ(x ∣ z)]}
    τ(x|z) pϕ
    (z|x) τ(z)
    7BSJBUJPOBM'SFF&OFSHZ

    *OGPSNBUJPO#PVOET

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  17. +PJOU,-ͷଞͷ෼ղ

    • ୈ߲ɿೖྗͷ৚݅෇͖෼෍ Λ޷·͍͠෼෍ ʹ͚ۙͮΔ
    • ୈ߲ɿ૬ޓ৘ใྔ ͷม෼Լք
    • ࠓճ͸જࡏۭؒͰදݱʢ Ͱ৚͚͖݅ͮͯͬ͞ͱಉ༷ͳٞ࿦Ͱٻ·Δʣ
    • ೖྗΛ؍ଌͨ͠ޙͷ৴೦ ͱ؍ଌ͢Δલͷ৴೦ ͷࠩʢ৘ใήΠϯʣΛදݱ
    • Լք͸ ͕࣮ࡍͷ৚݅෇͖෼෍ ΛΑۙ͘ࣅ͍ͯ͠Δ΄ͲλΠτ
    㱺ม਺ͷؔ܎Λֶश͢Δ΄Ͳ૬ޓ৘ใྔΛ࠷େԽʢදݱྗ๛͔ͳ Λ࢖͏ͱ༗རʣ
    ૬ޓ৘ใྔͷ࠷େԽ͸ɼೖྗͷྑ͍જࡏදݱͷֶशͱɼજࡏදݱʹͱͬͯ༗ӹͳೖྗͷ
    ୳ࡧͷ྆ํΛઆ໌ʂ
    KL [pϕ
    (x, z)∥τ(x, z)]
    KPJOUEJWFSHFODF
    = EKL [pϕ
    (x ∣ z)∥τ(x)]
    SFBMJ[JOHJOQVUQSFGFSFODFT
    − E [ln τ(z ∣ x) − ln pϕ
    (z)]
    JOGPSNBUJPOCPVOE

    (x|z) τ(x)
    I[x; z]
    z
    τ(z|x) pϕ
    (z)
    τ(z|x) p(z|x)
    τ(z|x)
    *OQVU1SFGFSFODF

    *OGPSNBUJPO#PVOET

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  18. .PEFMBT1SFGFSFODF

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  19. λʔήοτͷ෼෍͸γεςϜʹؔ͢Δ޷·͠͞Λఆٛ
    • ͜ΕΛ֬཰Ϟσϧͱͯ͠ղऍ͢Δ
    • *OQVUQSFGFSFODF
    • "DUJPOQFSDFQUJPODZDMF
    • /JDIFTFFLJOH
    .PEFMBT1SFGFSFODF

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  20. ͸ΤʔδΣϯτͷ޷·͍͠ೖྗͷ෼෍Λදݱ͢Δ΋ͷ
    • ͭ·ΓΤʔδΣϯτ͸ ʹऩଋͤ͞Α͏ͱ͢Δ
    • %JWFSHFODFNJOJNJ[BUJPOʹΑͬͯΤʔδΣϯτ͸ࣗ਎ͷϞσϧΛ࢖ͬ
    ͯ༧ଌͰ͖ΔΑ͏ͳೖྗΛݟ͚ͭΑ͏ͱ͢Δ
    • ࣗ਎ͷϞσϧͷ଒Ͱ͸༧ଌͰ͖ͳ͍ೖྗΛආ͚Α͏ͱ͢Δ
    • ใु ʹରͯ͠ ͱ͢Ε͹λεΫͷใुΛಋೖՄೳ
    • ใुͱͯ͠पล໬౓Λଧͪফ߲͢໨Λ༩͑ͯɼೖྗͷ޷·͠͞ͷ߲Λ
    औΓআ͘͜ͱ΋Մೳ
    • ͜ͷลΓ͸Ͱઆ໌
    τ(x)
    τ(x)
    r(x) exp (r(x))
    *OQVUQSFGFSFODF

    .PEFMBT1SFGFSFODF

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  21. • λʔήοτΛϞσϧͱͯ͠ղऍ͢ΔͱɼEJWFSHFODFNJOJNJ[BUJPO
    ͸ϔϧϜϗϧπͷQFSDFQUJPOBTJOGFSFODFͱҰக
    • ޷·͠͞ΛϞσϧͱදݱ͢Δͷ͸ࣗ༝ΤωϧΪʔݪཧʢ'&1ʣ΍
    BDUJWFJOGFSFODFͷൃ૝͔Βண૝ʢ"QQFOEJY"

    • ஌֮ͱߦಈ͸ಉ͡KPJOU,-Λ࠷খԽ͍ͯ͠Δ͕ɼҟͳΔม਺ʹӨڹ
    • ஌֮͸ೝࣝʹج͍ͮͯɼදݱʹؔ͢Δ৴೦ʹӨڹ
    • ߦಈ͸දݱʹج͍ͮͯɼೖྗʹӨڹ
    "DUJPO1FSDFQUJPO$ZDMF

    .PEFMBT1SFGFSFODF

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  22. %JWFSHFODFNJOJNJ[BUJPOͷදݱֶश΍୳ࡧʹؔ࿈͢Δ૬ޓ৘ใྔͷԼք͸දݱྗ๛͔ͳ
    λʔήοτΛ༻͍Δ΄ͲλΠτʢʣ
    • Ϟσϧ͕༩͑ΒΕͨݩͰͷೖྗͷपล໬౓ͷNBUDIJOH
    • ͜ͷͱ͖ɼΤʔδΣϯτ͸ೖྗΛे෼ʹ༧ଌͰ͖ͯɼ
    ֎ཚʹରԠͰ͖Δࣗવͳฏߧঢ়ଶʢFDPMPHJDBMOJDIFʣʹऩଋ
    ࣅͨΑ͏ͳ֓೦ͱͯ͠TVSQSJTFNJOJNJ[BUJPO͕͋Δ
    • Ϟσϧ͕༩͑ΒΕͨݩͰͷೖྗͷपล໬౓ͷ࠷େԽ
    • ͜ͷͱ͖ɼOJDIFTFFLJOH<'SJTUPO>ͱݺ͹ΕΔ
    • ֎ཚ͕ͳ͍ͱҰ୴୳ࡧͨ͜͠ͱͷ͋Δࣗ໌ͳղʹؕΔ
    • ʢদౢղऍɿͨͿΜ͜Ε͸ʮ҉͍෦԰໰୊ʯͱݴΘΕͯΔ΋ͷʣ
    %JWFSHFODFNJOJNJ[BUJPOͩͱɼೖྗͷपล෼෍Λ߹ΘͤΑ͏ͱ͢ΔͨΊɼ
    ༧ଌͰ͖ΔશͯͷೖྗΛΧόʔ͢ΔΑ͏ͳ޿͍OJDIFʹͳΔΑ͏ʹଅ͞ΕΔ
    • දݱྗ๛͔ͳੈքϞσϧΛ༻͍Δͱ༧ଌՄೳੑ͕ߴ·Γ޿͍OJDIFʹʢλεΫใु͸௥Ճతʣ
    /JDIF4FFLJOH

    .PEFMBT1SFGFSFODF

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  23. 1BTUBOE'VUVSFʢ࣌ܥྻ΁ͷ֦ுʣ

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  24. աڈͷೖྗ Λ؍ଌ͠ɼকདྷͷೖྗ Λ
    ؍ଌ͢ΔલͷKPJOU,-ͷ࠷దԽΛߟ͑Δ
    • ͷ૬ޓ৘ใྔͷԼքΛࣜมܗ
    x<
    x>
    E [ln τ(z ∣ x) − ln pϕ
    (z)]
    JOGPSNBUJPOCPVOE
    = E [ln τ(z ∣ x) − ln pϕ (z ∣ x<) + ln pϕ (z ∣ x<) − ln pϕ
    (z)]
    ≥ E [ln τ(z ∣ x) − ln pϕ (z ∣ x<)]
    GVUVSFJOGPSNBUJPOCPVOE
    + E [ln τ (z ∣ x<) − ln pϕ
    (z)]
    QBTUJOGPSNBUJPOCPVOE
    ࣌ܥྻ΁ͷ֦ு

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  25. ૬ޓ৘ใྔͷԼքͷࣜมܗΛར༻ͯ͠
    ࠷খԽ͍ͨ͠KPJOU,- ͷ্ք͕ٻ·Δ

    • +PJOU,-͸ɼաڈͷೖྗͷ༗༻ͳදݱͷֶशͱকདྷͷ༗༻ͳೖྗͷ
    ୳ࡧͷ྆ํΛಉ࣌ʹදݱͰ͖͍ͯΔ
    KL [pϕ
    (x, z)∥τ(x, z)]
    KL [pϕ
    (x, z)∥τ(x, z)] ≤ EKL [pϕ (z ∣ x<) ∥τ(z)]
    SFBMJ[JOHQBTUMBUFOUQSFGFSFODFT
    − E [ln τ (x<
    ∣ z) − ln pϕ (x<)]
    SFQSFTFOUBUJPOMFBSOJOH
    + EKL [pϕ (x>
    ∣ x<
    , z) ∥τ (x>
    ∣ x<)]
    SFBMJ[JOHGVUVSFJOQVUQSFGFSFODFT
    − E [ln τ(z ∣ x) − ln pϕ (z ∣ x<)]
    FYQMPSBUJPO
    ࣌ܥྻ΁ͷ֦ு

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  26. ʢิ଍ʣSFWJTJPO W
    Ͱ௥Ճ͞Εͨઆ໌
    ʹ͸Ұ౓΋؍ଌ͞Εͳ͍΋ͷͱɼ࣌ؒͷܦաͱͱ΋ʹ
    ؍ଌ͞ΕΔ΋ͷ͕͋Δ
    • Ұ౓΋؍ଌ͞Εͳ͍ɿજࡏঢ়ଶͷਪఆ஋΍Ϟσϧύϥϝʔλʣ
    • ࣌ؒܦաͱͱ΋ʹ؍ଌɿ֬཰తͳߦಈ΍εΩϧ
    • ͜ΕΒͷม਺Λબ୒͢Δ͜ͱ͸ɼຊ౰͸ҼՌతͳհೖʹͳΔͷͰ
    ͸ɼະ؍ଌͷજࡏม਺ ͱɼ؍ଌ͞Εͨજࡏม਺ Ͱ৚͚݅ͮ
    ΒΕͨ Ͱ෼ׂ͢΂͖͕ͩɼ؆୯ͷͨΊʹলུ͍ͯ͠Δ
    z
    z z>
    EP(z<
    )
    z>

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  27. ʢิ଍ʣSFWJTJPO W
    Ͱ௥Ճ͞Εͨઆ໌
    ʹ͸Ұ౓΋؍ଌ͞Εͳ͍΋ͷͱɼ࣌ؒͷܦաͱͱ΋ʹ
    ؍ଌ͞ΕΔ΋ͷ͕͋Δ
    • Ұ౓΋؍ଌ͞Εͳ͍ɿજࡏঢ়ଶͷਪఆ஋΍Ϟσϧύϥϝʔλʣ
    • ࣌ؒܦաͱͱ΋ʹ؍ଌɿ֬཰తͳߦಈ΍εΩϧ
    • ͜ΕΒͷม਺Λબ୒͢Δ͜ͱ͸ɼຊ౰͸ҼՌతͳհೖʹͳΔͷͰ
    ͸ɼະ؍ଌͷજࡏม਺ ͱɼ؍ଌ͞Εͨજࡏม਺ Ͱ৚͚݅ͮ
    ΒΕͨ Ͱ෼ׂ͢΂͖͕ͩɼ؆୯ͷͨΊʹলུ͍ͯ͠Δ
    z
    z z>
    EP(z<
    )
    z>

    View full-size slide

  28. ʢิ଍ʣSFWJTJPO W
    Ͱ௥Ճ͞Εͨઆ໌
    %JWFSHFODFNJOJNJ[BUJPOͰઆ໌Ͱ͖Δݚڀͷྫ

    View full-size slide

  29. +PJOU,-ͷ࠷খԽͱͯ͠આ໌Ͱ͖Δྫ

    View full-size slide

  30. "DUVBMɿ
    5BSHFUɿ
    ྫʣ#//Λ༻͍ͨڭࢣ͋Γֶश
    • ೖྗɿը૾ ɼΫϥε
    • σʔληοτͷHMPCBMͳදݱͱͯ͠જࡏతͳύϥϝʔλ Λਪ࿦

    (x, y, w) = pϕ
    (w)

    CFMJFG

    i
    p (xi
    , yi)
    EBUB
    τ(x, y, w) ∝ τ(w)

    QSJPS

    i
    τ (yi
    ∣ xi
    , w)
    MJLFMJIPPE
    x = {xi
    } y = {yi
    }
    w
    ม෼ਪ࿦ʢ7BSJBUJPOBM*OGFSFODFʣ

    View full-size slide

  31. +PJOU,-ʹͦΕͧΕ୅ೖ͢Δͱ

    • ໨తؔ਺͕ࣗ༝ΤωϧΪʔʢ&-#0ʣʹͳΔ
    Λσϧλ෼෍ ͱ͢Δͱ

    • ͭ·Γ࠷໬ਪఆʹͳΔʢλʔήοτΛ ͱͨ͠ͱ͖ͱ౳Ձʣ
    KL [pϕ
    ∥τ] = KL [pϕ
    (w)∥τ(w)]
    DPNQMFYJUZ
    − E[ln τ(y ∣ x, w)]
    BDDVSBDZ
    + E[ln p(x, y)]
    DPOTUBOU

    (w) pϕ
    (w) → δϕ
    (w) ≐ {1JGw = ϕFMTF0}
    KL [pϕ
    (w)∥τ(w)]
    DPNQMFYJUZ
    − E[ln τ(y ∣ x, w)]
    BDDVSBDZ
    → ln τ(ϕ) − E[ln τ(y ∣ x, ϕ)]
    DPNQMFYJUZ
    ≐ E [−ln τϕ
    (y ∣ x)]
    QBSBNFUFSJ[FEUBSHFU
    τϕ
    (y|x)
    ม෼ਪ࿦ʢ7BSJBUJPOBM*OGFSFODFʣ

    View full-size slide

  32. "DUVBMɿ
    5BSHFUɿ
    ྫʣ7"&
    • ೖྗɿը૾
    • ରԠ͢Δજࡏม਺ Λਪ࿦

    (x, z) = ∏
    i
    p (xi)

    EBUB
    pϕ (zi
    ∣ xi)
    FODPEFS
    τϕ
    (x, z) = ∏
    i
    τϕ (xi
    |zi)
    EFDPEFS
    τ (zi)

    QSJPS
    x = {xi
    }
    z = {zi
    }
    ঈ٫ਪ࿦ʢ"NPSUJ[FE*OGFSFODFʣ

    View full-size slide

  33. +PJOU,-ʹͦΕͧΕ୅ೖ͢Δͱ

    • ໨తؔ਺͕ࣗ༝ΤωϧΪʔʢ&-#0ʣʹͳΔ
    • ༧ଌͷ߲ʢୈ߲ʣ͕૬ޓ৘ใྔͷԼք
    • දݱ͕ೖྗʹؔͯ͠ҙຯΛ࣋ͭΑ͏ʹଅ͢
    KL [pϕ
    ∥τϕ] = E [ln τϕ
    (x ∣ z) − ln p(x)]
    JOGPSNBUJPOCPVOEʢদౢෛ߸͕͍ΔͷͰ͸ʁʣ
    + EKL [pϕ
    (z ∣ x)∥τ(z)]
    DPNQMFYJUZ
    ঈ٫ਪ࿦ʢ"NPSUJ[FE*OGFSFODFʣ

    View full-size slide

  34. "DUVBMɿ
    5BSHFUɿ
    ྫʣ)..Ͱকདྷͷೖྗ ͕ܽଛ͍ͯ͠Δ৔߹
    • ೖྗ
    • ରԠ͢Δજࡏม਺ Λਪ࿦

    (x, z) = p (x>
    , x<)
    EBUB
    pϕ (z ∣ x<)
    CFMJFG
    τϕ
    (x, z) = τϕ (x<
    ∣ z)
    MJLFMJIPPE
    τϕ (x>
    ∣ z)
    QSFEJDUJPO
    τ(z)

    QSJPS
    x>
    x = {x<
    , x>
    }
    z = {zi
    }
    কདྷͷೖྗʢ'VUVSF*OQVUTʣ

    View full-size slide

  35. +PJOU,-ʹͦΕͧΕ୅ೖ͢Δͱ

    • ͱͯ͠কདྷͷೖྗ ͕࠷దԽʹ
    ؔΘΒͳ͍Α͏ʹ͍ͯ͠ΔʢVODPOUSPMMFEGVUVSFʣ
    KL [pϕ
    ∥τϕ] = KL [pϕ (x<
    , z) ∥τϕ (x<
    , z)]
    WBSJBUJPOBMJOGFSFODF
    + EKL [p (x>
    ∣ x<) ∥τϕ (x>
    ∣ z)]
    VODPOUSPMFEGVUVSF
    τϕ (x>
    ∣ x<
    , z) = τϕ (x>
    ∣ x<) x>
    ঈ٫ਪ࿦ʢ"NPSUJ[FE*OGFSFODFʣ

    View full-size slide

  36. "DUVBMɿ
    5BSHFUɿ
    ྫʣܾఆ࿦తͳ੍ޚΛ͢Δ৔߹
    • ೖྗͷܥྻ ͸ΤʔδΣϯτ͕ύϥϝʔλ ΛมԽͤͯ͞
    ෦෼తʹม͑ΒΕΔྫʣܾఆ࿦తͳํࡦɼ੍ޚଇɼϓϥϯʜ

    (x) = ∏
    t
    pϕ (xt
    ∣ x1:t−1)
    DPOUSPMMFEEZOBNJDT
    τ(x) = ∏
    t
    τ (xt
    ∣ x1:t−1)
    QSFGFSFODFT
    x = {xt
    } ϕ
    ʢܾఆ࿦తͳʣ੍ޚ

    View full-size slide

  37. +PJOU,-ʹͦΕͧΕ୅ೖ͢Δͱ

    • QSFGFSFODFͷର਺ͱೖྗͷΤϯτϩϐʔͷ࠷େԽ
    • $G
    .BY&OU3-
    ͸ߦಈͷΤϯτϩϐʔ࠷େԽ
    • 1SFGFSFODFͷઃఆ
    • ɿ,-DPOUSPM<,BQQFO>
    • ɿ,-SFHVMBSJ[FEDPOUSPM<5PEPSPW>
    • ɿFYQFDUFESFXBSE<4VUUPO>
    • ͜Ε͸DVSJPTJUZͷ߲͕ফ͑ΔΑ͏ͳઃఆʢ·ΕͳೖྗΛ໌ࣔతʹEJTDPVSBHFͯ͠୳ࡧΛ੍ݶʣ
    • ೖྗͷΤϯτϩϐʔͷਪఆɿٖΧ΢ϯτɼ4LFX'JUɼ3/%ɼ*$.ʜ
    KL [pϕ
    ∥τ] = − ∑
    t
    E [ln τ (xt
    ∣ x1:t−1)]
    FYQFDUFEQSFGFSFODFT
    + H [pϕ (xt
    ∣ x1:t−1)]
    DVSJPTJUZ
    τ (xt
    ∣ x1:t−1) ∝ exp (r (xt))
    τ (xt
    ∣ x1:t−1) ∝ exp (r (xt)) τ′ (xt
    ∣ x1:t−1)
    τ (xt
    ∣ x1:t−1) ∝ exp (r (xt)) pϕ
    (xt
    |x1:t−1
    )
    ʢܾఆ࿦తͳʣ੍ޚ

    View full-size slide

  38. "DUVBMɿ
    5BSHFUɿ
    ྫʣ֬཰తͳ੍ޚΛ͢Δ৔߹
    • NBY&OU3-ɾ3-BTJOGFSFODFͱ΋͍͏
    • ೖྗͷܥྻ ɼߦಈͷܥྻ
    • աڈͷೖྗߦಈ ͕؍ଌ͞Ε͍ͯΔઃఆ

    (x, a) = ∏
    t
    p (xt
    ∣ x1:t−1
    , a1:t−1)
    FOWJSPONFOU
    pϕ (at
    ∣ x1:t
    , a1:t−1)
    QPMJDZ
    τ(x, a) ∝ ∏
    t
    exp (r (xt))
    SFXBSE
    p (xt
    ∣ x1:t−1
    , a1:t−1)
    FOWJSPONFOU
    τ (at)

    BDUJPOQSJPS
    x = {xt
    } a = {at
    }
    ax<
    .BY&OUSPQZ3- .BY&OU3-


    View full-size slide

  39. +PJOU,-ʹͦΕͧΕ୅ೖ͢Δͱ

    • ߦಈۭؒͰͷ੍໿ͱظ଴ใुͷ߲ʹͳΔ
    • λʔήοτʹ؀ڥͷμΠφϛΫε͕ೖΔͷͰͷDVSJPTJUZͷ߲͸ଧͪ
    ফ͞ΕΔ
    • ߦಈͷࣄલ෼෍
    • Ұ༷෼෍ɿ42-ɼ4"$
    • աڈͷೖྗ΍ҎલͷλεΫͷ஌ࣝΛ౷߹ɿ%JTUSBM
    KL [pϕ
    ∥τ] = ∑
    t
    EKL [pϕ (at
    ∣ x1:t
    , a1:t−1) ∥τ (at)]
    DPNQMFYJUZ
    − E [r (xt)]
    FYQFDUFESFXBSE
    τ(a)
    .BY&OUSPQZ3- .BY&OU3-


    View full-size slide

  40. "DUVBMɿ
    5BSHFUɿ
    ྫʣ֬཰తͳ੍ޚʹ͓͚ΔHFOFSBMJ[FEFNQPXFSNFOU
    • աڈͷೖྗߦಈ ͕؍ଌ͞Ε͍ͯΔઃఆ
    • ೖྗ͔ΒߦಈΛ༧ଌ͢Δλʔήοτͷ෼෍ʢSFWFSTFQSFEJDUPSʣ

    (x, a) = ∏
    t
    p (xt
    ∣ x1:t−1
    , a1:t−1)
    FOWJSPONFOU
    pϕ (at
    ∣ x1:t
    , a1:t−1)
    QPMJDZ
    τ(x, a) ∝ ∏
    t
    τϕ (at
    ∣ x1:T
    , a1:t−1)
    SFWFSTFQSFEJDUPS
    ax<
    &NQPXFSNFOU

    View full-size slide

  41. +PJOU,-ʹͦΕͧΕ୅ೖ͢Δͱ

    • (FOFSBMJTFEFNQPXFSNFOU͸૬ޓ৘ใྔ
    ͷԼք
    KL [pϕ
    ∥τϕ] = EKL[p(x ∣ a)∥τ(x)]
    DPOUSPM
    − E [ln τϕ
    (a ∣ x) − ln pϕ
    (a)]
    HFOFSBMJ[FEFNQPXFSNFOU
    E [ln τϕ
    (a ∣ x) − ln pϕ
    (a)]
    HFOFSBMJ[FEFNQPXFSNFOU
    ≥ ∑
    t
    E[ln τϕ (at
    ∣ x, a1:t−1)
    EFDPEFS
    − ln pϕ (at
    ∣ a1:t−1)
    TPVSDF
    ]
    I[a; x] = E [ln pϕ
    (a|x) − ln pϕ
    (a)]
    &NQPXFSNFOU

    View full-size slide

  42. "DUVBMɿ

    5BSHFUɿ
    ྫʣ࣌ؒతʹந৅Խ͞ΕͨTLJMMͰ৚͚݅ͮΒΕͨํࡦΛֶश
    • ೖྗͷܥྻ ɼߦಈͷܥྻ ɼεΩϧͷܥྻ
    • εΩϧ͸ݻఆͷ෼෍ʢ͔աڈͷೖྗͷؔ਺ʣ͔Βબ୒
    • ߦಈ͸աڈͷೖྗͱݱࡏͷεΩϧʹैͬͯબ୒

    (x, a, z) ≐
    T/K

    k=1
    pϕ (zk)
    BCTUSBDUQPMJDZ
    T

    t=1
    pϕ (at
    ∣ x1:t
    , a1:t−1
    , z⌊t/K⌋)
    QPMJDZ
    p (xt
    ∣ x1:t−1
    , a1:t−1)
    FOWJSPONFOU
    τϕ
    (x, a, z) ∝
    T/K

    k=1
    τϕ (zk
    ∣ x)
    SFWFSTFQSFEJDUPS
    T

    t=1
    τ (at)

    BDUJPOQSJPS
    x = {xt
    } a = {at
    } z = {zt
    }
    εΩϧͷൃݟʢ4LJMM%JTDPWFSZʣ

    View full-size slide

  43. +PJOU,-ʹͦΕͧΕ୅ೖ͢Δͱ

    • 4LJMMEJTDPWFSZ͸εΩϧ ͱೖྗ ͷ૬ޓ৘ใྔ ͷԼք
    • ΤʔδΣϯτ͸ଟ༷ͳೖྗܥྻΛ࣮ݱ͠ͳ͕ΒɼҟͳΔεΩϧͰܥ
    ྻ͕ॏͳΒͳ͍Α͏ʹεΩϧΛֶश͢Δ
    • ྫʣ%*:"/<&ZTFOCBDI>ɼ%"%4<4IBSNB>
    KL [pϕ
    ∥τϕ] = EKL[p(x ∣ a)∥τ(x)]
    DPOUSPM
    + EKL [pϕ
    (a ∣ x, z)∥τ(a)]
    DPNQMFYJUZ
    − E [ln τϕ
    (z ∣ x) − ln pϕ
    (z)]
    TLJMMEJTDPWFSZ
    z x I[z; x]
    εΩϧͷൃݟʢ4LJMM%JTDPWFSZʣ

    View full-size slide

  44. "DUVBMɿ
    5BSHFUɿ
    ྫʣ&YQFDUFEJOGPSNBUJPOHBJOΛ࠷େԽ͢ΔߦಈΛબ୒
    • ֎తͳใु͕ͳ͍৔߹ʹੈքΛޮ཰తʹֶश͢ΔͨΊ
    • ೖྗɿ
    • HMPCBMͳදݱΛ࣋ͭજࡏతͳύϥϝʔλɿ

    (x, w) = pϕ
    (w)

    CFMJFG

    t
    pϕ (xt
    ∣ x1:t−1)
    DPOUSPMMFEEZOBNJDT
    τ(x, w) = τ(w)

    QSJPS

    t
    τ (xt
    ∣ x1:t−1
    , w)
    MJLFMJIPPE
    x = {xi
    }
    w
    *OGPSNBUJPO(BJO

    View full-size slide

  45. +PJOU,-ʹͦΕͧΕ୅ೖ্ͯ͠քΛऔΔ


    • ͱ͸ҟͳΓɼະདྷͷೖྗʹӨڹͰ͖Δ
    • աڈͷೖྗʹ͍ͭͯ༗ӹͳදݱΛֶश͠ɼະདྷͷදݱʹؔͯ͠༗ӹʹͳΔΑ͏ͳೖྗΛ୳ࡧ͢Δ
    • ϞσϧύϥϝʔλʹΑͬͯΤʔδΣϯτ͕؀ڥͷϧʔϧΛݟ͚ͭΔΑ͏ʹ͢Δٞ࿦
    • BDUJWFJOGFSFODF<'SJTUPO>ɼ7*.&<)PVUIPPGU>ɼ."9<4IZBN^ɼ1MBO&YQMPSF<4FLFS>
    KL [pϕ
    ∥τϕ] ≤ EKL [pϕ (w ∣ x<) ∥τ(w)]
    TJNQMJDJUZ
    − E [ln τϕ (x<
    ∣ w) − ln pϕ (x<)]
    SFQSFTFOUBUJPOMFBSOJOH
    +EKL [pϕ (x>
    ∣ x<
    , w) ∥τϕ (x>
    ∣ x<)]
    DPOUSPM
    − E [ln τϕ
    (w ∣ x) − ln pϕ (w ∣ x<)]
    JOGPSNBUJPOHBJO
    E [ln τϕ
    (w ∣ x) − ln pϕ (w ∣ x<)]
    JOGPSNBUJPOHBJO
    ≥ ∑
    t′ >t
    E [ln τϕ (w ∣ x1:t′ ) − ln pϕ (w ∣ x1:t′ −1)]
    JOUSJOTJDSFXBSE
    *OGPSNBUJPO(BJO

    View full-size slide

  46. ؔ࿈τϐοΫ

    View full-size slide

  47. ͭͷ෼෍ͷEJWFSHFODFͷ࠷খԽͱͯ͠ͷఆࣜԽ
    • දݱֶशΛ,-ͷ࠷খԽͱͯ͠ղऍɿ5IFS.-<"MFNJ>
    • ໛฿ֶशͰֶशͨ͠ํࡦͱΤΩεύʔτͷEJWFSHFODF࠷খԽ
    <(IBTFNJQPVS>ʢ$P3-ʣ
    • ຊ࿦จ͸CBDLXBSE SFWFSTF
    ,-ͷ࠷దԽͱͯ͠
    ଟ͘ͷ໰୊ͷ໨తؔ਺Λ౷Ұ
    %JWFSHFODF.JOJNJ[BUJPO

    View full-size slide

  48. ຊ࿦จͷ࿮૊Έ͸'&1ͷӨڹΛड͚͍ͯΔ
    • '&1͸ΤʔδΣϯτͱ؀ڥͷμΠφϛΫεΛ4%&ͱͯ͠දݱ
    • ಛʹBDUJWFJOGFSFODFͷղऍΛར༻
    • "DUJWFJOGFSFODFͰ͸Ϟσϧʹؔ͢Δೖྗͷ֬཰Λ࠷େԽ͢Δ͕ɼ
    ຊ࿦จͰ͸ೖྗͷEJWFSHFODFͷNJOJNJ[BUJPOʢNBUDIJOHʣ
    • "DUJWFJOGFSFODF͸ߦಈʹؔͯ͠ͷΈ୳ࡧͷ߲Λ࠷దԽ͍ͯ͠Δ
    • ͜Ε͸ຊ࿦จͷ࿮૊ΈͰߦಈͷಛఆͷࣄલ෼෍ͷઃఆʹରԠ
    • (FOFOBSBMJ[FE'SFF&OFSHZ<1BSS>ͷ໨తؔ਺͕͍͕ۙɼΤϯτ
    ϩϐʔ߲͕ώϡʔϦεςΟοΫʹಋ͔Ε͍ͯΔ
    "DUJWF*OGFSFODF

    View full-size slide

  49. 3-ͷೖྗͱߦಈͷ,-࠷খԽʹΑΔఆࣜԽ͸Α͘஌ΒΕ͍ͯΔ
    • ྫʣ$POUSPMBT*OGFSFODF<-FWJOF>
    • ຊ࿦จ͸͜ͷจ຺ʹଇͬͯજࡏදݱΛ࣋ͭΤʔδΣϯτʹ֦ு
    • աڈͷೖྗʹؔ͢Δม෼ਪ࿦ͱະདྷͷೖྗʹର͢Δ୳ࡧΛಋ͍ͨ
    %JWFSHFODFNJOJNJ[BUJPO͸ೖྗͷΤϯτϩϐʔਖ਼ଇԽͱؔ࿈
    • 4-"$<-FF>͸දݱֶशͱ੍ޚΛ૊Έ߹Θͤͨݚڀ
    • ຊ࿦จ͸Ұൠతͳ࿮૊Έ͔ΒΤϯτϩϐʔਖ਼ଇԽ߲Λಋ͍ͨ
    $POUSPMBT*OGFSFODF $B*


    View full-size slide

  50. ຊ࿦จͰ͸KPJOU,-ͷ࠷খԽͱ͍͏؍఺͔ΒΤʔδΣϯτͷ
    ߦಈͱ஌֮ʹؔ͢Δ౷Ұతͳ໨తؔ਺Λಋग़ͨ͠
    • λʔήοτͷ෼෍͸ΤʔδΣϯτ͕දݱΛਪ࿦͢Δ֬཰ϞσϧͰɼ
    ޷·͍͠γεςϜͷઃఆΛදݱ͢Δ΋ͷ
    • ͜ͷ໨తؔ਺ͷ΋ͱͰ͸ɼͲΜͳજࡏදݱ΋୳ࡧͷ߲Λ൐͏
    • +PJOU,-Λ༻͍ͯ͜Ε·Ͱͷݚڀͷ༷ʑͳ໨తؔ਺Λಋग़Ͱ͖Δ
    ౷Ұతͳ؍఺͸ΤʔδΣϯτͷઃܭࢦ਑ʹͳΓ͏Δ
    • ஌తͳΤʔδΣϯτΛ࡞੒͢Δ্Ͱɼදݱྗ๛͔ͳੈքϞσϧΛ
    λʔήοτͱͯ͠༻͍Δ͜ͱ͕༗ޮ͔΋͠Εͳ͍
    ·ͱΊ

    View full-size slide