パターン認識と機械学習〜指数型分布族とノンパラメトリック〜

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ʮύλʔϯೝࣝͱػցֶशʯ ྠಡษڧձ ʙࢦ਺ܕ෼෍଒ɾϊϯύϥϝτϦοΫ๏ʙ

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ࣗݾ঺հ w ໊લ w খּݪޫو .JUTVLJ0("4")"3" w ೖࣾ೥౓ w ೥౓ w ॴଐ w ג $ZCFS;։ൃΤϯδχΞ w ֶੜ࣌୅ͷݚڀ෼໺ w ࣗવݴޠॲཧɾػցֶश

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໨࣍ w ࢦ਺ܕ෼෍଒ w ࠷໬ਪఆͱे෼౷ܭྔ w ڞ໾ࣄલ෼෍ w ແ৘ใࣄલ෼෍ w ϊϯύϥϝτϦοΫ๏ w Χʔωϧີ౓ਪఆ๏ w ࠷ۙ๣๏

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ࢦ਺ܕ෼෍଒ Q w ࣜ Ͱఆٛ͞ΕΔ෼෍ͷ଒ ू߹ ! w ʮΨ΢ε෼෍ʯʮଟ߲෼෍ʯͳͲɺ  13.-ʹग़ͯ͘Δଟ͘ͷ෼෍͕ࢦ਺ܕ෼෍଒ʹؚ·ΕΔ  ˠࣜ Ͱఆٛ͠௚͢͜ͱ͕Ͱ͖Δ w ˞Y͸εΧϥʔͰ΋ϕΫτϧͰ΋ྑ͍ w ˞Y͸཭ࢄͰ΋࿈ଓͰ΋ྑ͍

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ࢦ਺ܕ෼෍଒ Q ! w Yʹؔ͢Δؔ਺ w TDBMJOHDPOTUBOUͱ΋ݺ͹Ε .-B11ΑΓ ɺ  ʮʯ͕ೖΔ͜ͱ΋͋Δ ϕϧψʔΠ෼෍ɺΨϯϚ෼෍ h ( x )

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ࢦ਺ܕ෼෍଒ Q ! w Бʹؔ͢Δؔ਺ w ֬཰ີ౓ؔ਺ͷੵ෼஋͕ʹͳΔΑ͏ʹ  ਖ਼نԽ͢ΔͨΊͷ΋ͷ g(⌘) g ( ⌘ ) Z h (x) exp ⌘T u (x) d x = 1 Z ( ⌘ ) = 1 g ( ⌘ ) = Z h (x) exp ⌘T u (x) d x

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ϕϧψʔΠ෼෍͸ࢦ਺ܕ෼෍଒͔ʁ ! w ແཧ΍ΓFYQͷதʹೖΕͯΈΔ ! ! ! w БΛࣜ ͷΑ͏ʹఆٛ͢Δ Bern ( x | µ ) = µx(1 µ )1 x Bern(x | µ) = exp { ln µx (1 µ) 1 x} = exp { x ln µ + (1 x) ln 1 µ } = exp { x(ln µ ln 1 µ) + ln 1 µ } = (1 µ) exp { ln( µ 1 µ )x } ⌘ = ln( µ 1 µ )

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ϕϧψʔΠ෼෍͸ࢦ਺ܕ෼෍଒͔ʁ ! w ࠷ऴతʹ͸ɺ ! w ͱͳΓɺࣜ ͱରԠͨ͠ Bern ( x | µ ) = µx(1 µ )1 x

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ࢀߟɿࢦ਺ܕ෼෍଒ʹؚ·Εͳ͍΋ͷ w ࠞ߹ਖ਼ن෼෍        FYQͷ࿨ʹͳͬͯ͠·͍ɺࣜ ʹ͸ͳΒͳ͍

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࠷໬ਪఆ w ࢦ਺ܕ෼෍଒ͷҰൠܗͷࣜ ͔Βɺ  ࠷໬ਪఆྔБΛٻΊΔ w ಠཱʹಉ෼෍ʹै͏σʔλू߹9ʹ͍ͭͯߟ͑Δͱɺ  ͜ͷ໬౓ؔ਺͸ ! w ର਺໬౓ؔ਺͸

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࠷໬ਪఆ w ର਺໬౓ؔ਺ͷ Бʹؔͯ͠ͷ ޯ഑͕ͱͳΔ஋Λݟͭ ͚͍ͨ

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࠷໬ਪఆ w ݪଇͱͯ͠ɺࣜ Λղ͘ͱБ͸ಘΒΕΔ ! ! w ·ͨɺ࠷໬ਪఆ஋͸ʹґଘ͢Δ े෼౷ܭྔ w ݴ͍׵͑Δͱɺ࠷໬ਪఆΛٻΊΔͨΊʹ͸ɺ  ɹɹɹͷ૯࿨ ·ͨ͸ฏۉ ͷΈ͕͋Ε͹Α͍

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࠷໬ਪఆͱਅͷύϥϝʔλ w Бͷ࠷໬ਪఆ஋͸ࣜ Λղ͘ͱಘΒΕΔ ! ! w ͷఆٛʹجͮ͘ͱɺ ! ! w ͭ·Γɺ/ˠ㱣ͷۃݶͰ͸ɺ࠷໬ਪఆ஋ʹਅͷ஋ g ( ⌘ ) Z h (x) exp ⌘T u (x) d x = 1

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ڞ໾ࣄલ෼෍ w ࢦ਺ܕ෼෍଒ͷ೚ҙͷ෼෍ʹ͍ͭͯɺ  ࣍ͷܗͰॻ͚Δڞ໾ࣄલ෼෍͕ଘࡏ͢Δ ! w ಋग़͸ॻ͍ͯͳ͍͕ɺڞ໾Ͱ͋Δ͜ͱ͕͔֬ΊΒΕΔ  ໬౓ؔ਺ ͱࣄલ෼෍ Λ͔͚ɺ  ࣄޙ෼෍ΛٻΊΔ

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ڞ໾ࣄલ෼෍ w ಋग़͸ॻ͍ͯͳ͍͕ɺڞ໾Ͱ͋Δ͜ͱ͕͔֬ΊΒΕΔ  ໬౓ؔ਺ ͱࣄલ෼෍ Λ͔͚ɺ  ࣄޙ෼෍ΛٻΊΔ

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ڞ໾ࣄલ෼෍ w ࣄલ෼෍ͷύϥϝʔλΛɺ  Ծ૝؍ଌ஋ͱͯ͠ղऍ͢Δ͜ͱ΋Ͱ͖Δ ! ! ! ! w DGQɹೋ߲෼෍ͷڞ໾ࣄલ෼෍ʮϕʔλ෼෍ʯͷ  ɹɹɹɹɹύϥϝʔλΛɺԾ૝ͷ؍ଌͱͯ͠ղऍͨ͠ Ծ૝ͷ؍ଌ਺  /ʹ૬౰ Ծ૝ͷ؍ଌ஋  V Y ʹ૬౰

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ແ৘ใࣄલ෼෍ w ࣄલ෼෍Λஔ͖͍͕ͨɺ෼෍ ΍ύϥϝʔλ ʹ͍ͭͯͷ  ஌͕ࣝͳ͍ͱ͖ w Ұ༷෼෍Λஔ͚͹ྑ͍ʁ ! w Е͕࿈ଓ͔ͭൣғ͕ܾ·ͬͯͳ͍ͱ͖ɺ  Еʹ͍ͭͯͷੵ෼͕ൃࢄͯ͠͠·͍ɺਖ਼نԽͰ͖ͳ͍  ˠมଇࣄલ෼෍

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ແ৘ใࣄલ෼෍ w ࣍ͷΑ͏ͳฏߦҠಈෆมੑΛ࣋ͬͨ෼෍Λߟ͑Δ  ྫɿਖ਼ن෼෍   w ˞ฏߦҠಈෆมੑ w YΛఆ਺෼Ҡಈͯ͠΋ɺҐஔύϥϝʔλЖΛಉ͚ͩ͡Ҡಈ͢Ε͹ɺ  ֬཰ີ౓ͷܗ͸มΘΒͳ͍ ͷͱ͖ ͱ͢Δͱɺ

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ແ৘ใࣄલ෼෍ w ฏߦҠಈෆมੑΛ࣋ͭࣄલ෼෍ʹ͍ͭͯߟ͑Δͱɺ  ੵ෼͕۠ؒฏߦҠಈͯ͠΋ɺͦͷ֬཰͸มΘΒͳ͍ ! ! w Αͬͯɺࣜ ΑΓఆ਺ͱͳΔ 

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ແ৘ใࣄલ෼෍ w Ψ΢ε෼෍ͷЖͷ৔߹ɺ  М@?ˠ㱣ͷۃݶͰແ৘ใࣄલ෼෍ͱͳΔ ! ! ! w ࣄޙ෼෍ʹɺࣄલ෼෍ͷύϥϝʔλ͕Өڹ͠ͳ͘ͳΔ

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ϊϯύϥϝτϦοΫ๏ w ύϥϝτϦοΫ w ີ౓ؔ਺ Ϟσϧ ΛબΜͰɺύϥϝʔλΛσʔλ͔Βਪఆ͢Δ  ˠϞσϧ͕σʔλΛද͢ͷʹශऑͩͱɺ༧ଌਫ਼౓͸ѱ͍ w ྫ Ψ΢ε෼෍Λσʔλʹ౰ͯ͸ΊͯɺЖɾМ?Λਪఆͨ͠  ˠσʔλ͕ଟๆੑͩͱɺΨ΢ε෼෍Ͱ͸ଊ͑ΒΕͳ͍ w ϊϯύϥϝτϦοΫ w ෼෍ͷܗঢ়ʹஔ͘Ծఆ͕গͳ͍ w ྫ ଟๆੑͩͱ͔୯ๆੑͳͲͷԾఆ͸ஔ͔ͳ͍

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ώετάϥϜີ౓ਪఆ๏ w ਅͷ֬཰ີ౓ؔ਺ ྘ઢ ͔Β  ੜ੒͞Εͨͷσʔλ఺ΑΓ  ਪఆ ੨ώετάϥϜ ͨ͠΋ͷ w YΛ෯϶ͷ۠ؒʹ۠੾Γɺ  ͦͷ۠ؒʹೖͬͨYͷ؍ଌ਺Λ  Χ΢ϯτ͢Δɻ  ͜ΕΛɺࣜ Ͱਖ਼نԽͨ͠΋ͷ

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ώετάϥϜີ౓ਪఆ๏ w ࣍ݩɾ̎࣍ݩఔ౓ͷ؆୯ͳՄࢹԽʹ͸໾ཱͭɺ  ؆ศͳํ๏ w ͜ͷΞϓϩʔν͔Βɺ࣍ͷ͕̎ͭΘ͔Δ w ͋Δ஋ͷ֬཰ີ౓Λਪఆ͢Δʹ͸ɺۙ๣ͷ؍ଌ఺ͷ஋Λߟྀ͢Δ ඞཁ͕͋Δ w ۠ؒͷ෯͸େ͖͗ͯ͢΋  খ͗ͯ͢͞΋͍͚ͳ͍ w খɿσʔλʹӨڹ͗͢͠Δ w େɿݩͷ෼෍Λશ͘࠶ݱͰ͖ͳ͍ w ˠϞσϧͷෳࡶ͞ͷબ୒ʹࣅ͍ͯΔ

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ώετάϥϜີ౓ਪఆ๏ͷ໰୊఺ w ਪఆͨ͠ີ౓͕ෆ࿈ଓͰ͋Δ ۠ؒͱ۠ؒͷؒ w ࣍ݩͷढ͍ w Yͷ࣍ݩ਺Λ%ͱ͢Δͱɺ۠ؒͷ૯਺͸.?%ݸ

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Χʔωϧີ౓ਪఆ๏ w ະ஌ͷ֬཰ີ౓Q Y ͔ΒಘΒΕͨ؍ଌू߹Λ࢖ͬͯɺ  Q Y ͷ஋Λਪఆ͍ͨ͠ w YΛؚΉখ͞ͳྖҬ3ͷ֬཰Λ1ͱ͢Δ ! w /ݸͷ؍ଌ஋͕ಘΒΕͨͱͯ͠ɺ,ݸͷ؍ଌ஋͕  3ʹؚ·ΕΔ֬཰͸ɺೋ߲෼෍ʹै͏ P = Z R p( x )d x p(K|N, P) = Bin(K|N, P)

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Χʔωϧີ౓ਪఆ๏ w ೋ߲෼෍ͷظ଴஋ɾ෼ࢄΑΓɺ࣍ͷؔ܎͕ࣜಘΒΕΔ      w /͕େ͖͍ͱ͖ɺ෼ࢄ͸খ͘͞ͳΓɺظ଴஋ͷؔ܎͔Β w ·ͨɺ3͕খ͘͞ɺQ Y ͕3಺ͰҰఆͩͱۙࣅ͢Δͱ w Ҏ্ΑΓɺ࣍ͷີ౓ਪఆͷؔ܎͕ࣜಘΒΕΔ var  K N = P(1 P) N E  K N = P K ' NP P ' p( x )V p( x ) = K NV

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Χʔωϧີ౓ਪఆ๏ w Ҏ্ΑΓɺ࣍ͷີ౓ਪఆͷؔ܎͕ࣜಘΒΕΔ ! w ֬཰ີ౓Q Y Λਪఆ͢ΔͨΊʹɺ,ͱ7Λਪఆ͢Δ w ,ΛݻఆͰ7Λਪఆ  ˠ,ۙ๣ີ౓ਪఆ๏ w 7ΛݻఆͰ,Λਪఆ  ˠΧʔωϧີ౓ਪఆ๏ p( x ) = K NV

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Χʔωϧີ౓ਪఆ๏ w 7Λݻఆ͠ɺ,Λਪఆ͍ͨ͠ w ֬཰ີ౓Q Y ΛٻΊ͍ͨ఺ΛYɺ؍ଌ఺ΛY@Oͱ͢Δ w Ұล͕IͰɺYΛத৺ͱ͢Δখ͞ͳ௒ཱํମͷ  தʹ͋Δ఺ͷ૯਺͸ ! w ҰลIͷ௒ཱํମͳͷͰɺ7͸I?%ͱͳΓɺ K = K X n=1 k ✓ x xn h ◆ p( x ) = 1 N K X n=1 1 hD k ✓ x xn h ◆

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Χʔωϧີ౓ਪఆ๏ w খ͞ͳ௒ཱํମͷҰลIͷେ͖͕͞  ฏ׈ԽͷͨΊͷύϥϝʔλʹͳ͍ͬͯΔ w I͕ݻఆʹͳͬͯ͠·͏  ˠσʔλີ౓͕ߴ͍ྖҬͱ௿͍ྖҬͰɺෆ౎߹͕͋Δ

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,ۙ๣ີ౓ਪఆ๏ w ,Λݻఆ͠ɺ7Λਪఆ͍ͨ͠ w ֬཰ີ౓Q Y ΛٻΊ͍ͨ఺ΛYɺ؍ଌ఺ΛY@Oͱ͢Δ w YΛத৺ͱͯ͠ɺ఺͕,ݸؚ·ΕΔΑ͏ͳ௒ٿΛ୳͢ͱ  7͸Ұҙʹఆ·Γɺ֬཰ີ౓͸ਪఆ͞ΕΔ ਤ͸XXXPDXUJUFDIBDKQJOEFYQIQ NPEVMF(FOFSBMBDUJPO%PXO-PBEpMF QEGUZQFDBMΑΓ p( x ) = K NV

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,ۙ๣ີ౓ਪఆ๏ w ,͕ฏ׈Խύϥϝʔλʔͱͳ͍ͬͯΔ

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·ͱΊΔͱʜ w Χʔωϧີ౓ਪఆ๏ w ྖҬͷମੵΛݻఆ͢Δ w Ұลͷ௕͕͞Iͳ௒ཱํମʹɺ؍ଌ఺YO͕Կݸ͋Δ͔ΛٻΊͨ w I͕ฏ׈Խύϥϝʔλʔ w ,ۙ๣๏ w ྖҬ಺ͷɺ؍ଌ఺YOͷݸ਺Λݻఆ͢Δ w ؍ଌ఺YO͕LݸʹͳΔΑ͏ʹɺྖҬΛ޿͛ͨ w L͕ฏ׈Խύϥϝʔλʔ

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,ۙ๣๏Λ࢖ͬͨΫϥε෼ྨ w ,ۙ๣๏ͱ."1ਪఆΛ࢖ͬͯɺΫϥε෼ྨΛߦ͏ w YͷΫϥε$@Lͷࣄޙ֬཰ΛٻΊ͍ͨ

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,ۙ๣๏Λ࢖ͬͨΫϥε෼ྨ w ϕΠζͷఆཧΑΓɺ ! w ֬཰ີ౓Q Y ͸ɺઌ΄ͲٻΊͨͱ͓Γ ! w ࣄલ෼෍͸ɺશͯͷ؍ଌ఺ͷ͏ͪΫϥεʹଐ͢Δ؍ଌ఺ ! w ໬౓͸ɺͦͷΫϥεʹଐ͢Δ؍ଌ఺Ͱͷ֬཰ີ౓ΑΓɺ p(Ck | x ) = p( x |Ck)p(Ck) p( x ) p( x ) = K NV p(Ck) = Nk N p( x |Ck) = Kk NkV

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,ۙ๣๏Λ࢖ͬͨΫϥε෼ྨ w ϕΠζͷఆཧʹ୅ೖ͢Δͱɺ ! w Αͬͯɺ,ۙ๣ͷ͏ͪɺΫϥε$@Lʹଐ͢Δ఺ͷ਺Ͱ  ଟ਺ܾΛऔΕ͹Α͍ w ಛʹɺ,ͷͱ͖࠷ۙ๣๏ͱݺ͹ΕΔ p(Ck | x ) = p( x |Ck)p(Ck) p( x ) = Kk K ˖ʹ͍ۙ̏ͭͷ఺Ͱଟ਺ܾΛऔ͍ͬͯΔ ࠷ۙ๣๏Ͱ͸ɺ ࠷ۙ๣๏Ͱ͸ɺΫϥεͷҟͳΔ఺ͷରͷ  ਨ௚ೋ౳෼ઢʹͳ͍ͬͯΔ

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໰୊఺ w ͋ΔYͷ֬཰ີ౓Q Y Λਪఆ͢Δʹ͋ͨͬͯɺ  શͯͷσʔλ఺Λอ࣋͢Δඞཁ͕͋Δ w σʔλ఺͕૿͑Δͱɺۙ๣Λ୳ࡧ͍͕ͯ࣌ؒ͘͠๲େʹ ͳΔ  ˠ୳ࡧ͢ΔͨΊͷ໦ߏ଄Λ࡞Δ ຊདྷ͸ɺ࠷΋͍ۙ఺Λશ୳ࡧ͢Δඞཁ͕͋Δ

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͓ΘΓ