PRML(分類編)

Transcript

1. PRML Λ୊ࡐʹػցֶशΛਂ͘ཧղ͢Δηϛφʔ ʲ෼ྨ໰୊ฤʳ ࡔޱ ྒี 1 / 47
2. 0. ࠓճͷηϛφʔʹ͍ͭͯ ࠓճͷηϛφʔͰ͸ɺPRML ͷୈ 4 ষͷઢܗࣝผϞσϧΛத৺ʹ͓࿩ ͍ͨ͠͠ͱࢥ͍·͢ɻ ͳ͓஫ҙ఺ͱͯ͠ɺຊεϥΠυͷࣜ൪߸ͱ PRML ͷࣜ൪߸͸ҟͳΓ· ͢ͷͰɺ͝஫ҙ͍ͩ͘͞ɻ 2 / 47
3. ໨࣍ 1. ༧උ஌ࣝ 1-1. ෼ྨ໰୊ͷΞϓϩʔν 2. ϑΟογϟʔͷઢܗ൑ผ (PRML 4.1.4) 3. ϩδεςΟοΫճؼ (PRML 4.3, 4.4, 4.5) 3-1. ϩδεςΟοΫճؼ 3-2. ϩδεςΟοΫճؼͷ࠷໬ਪఆ 3-3. ϕΠζϩδεςΟοΫճؼ 3 / 47
4. 1. ༧උ஌ࣝ ػցֶशɺಛʹͦͷதͰ΋ڭࢣ͋ΓֶशͰ͸ɺ·ͣೖྗσʔλͷू߹ {x1 , x2 , · · · , xN } ͱͦΕͧΕʹରԠ͢Δ໨ඪϕΫτϧͷू߹ {t1 , t2 , · · · , tN } Λ༻ҙ͢Δɻ(܇࿅σʔλɺ·ͨ͸ڭࢣσʔλ) ༻ҙͨ͠܇࿅σʔλΛ༻͍ͯɺೖྗσʔλ͔Β໨ඪϕΫτϧΛ༧ଌ͢Δ ؔ਺ y(x) Λ࡞Δɻ(ֶश) ֶशऴྃޙɺະ஌ͷσʔλ x ͷ໨ඪϕΫτϧΛ y(x) Ͱ༧ଌ͢Δ ֤ೖྗϕΫτϧΛ༗ݶݸͷ཭ࢄతͳΧςΰϦʹׂΓ౰ͯΔ৔߹ (ྫ͑ ͹ɺखॻ͖਺ࣈͷೝࣝ) ΛΫϥε෼ྨͱ͍͍ɺग़ྗ͕࿈ଓม਺ͷ৔߹ Λճؼͱ͍͏ɻ ࠓճ͸Ϋϥε෼ྨͷ໰୊Λߟ͑Δɻ 4 / 47
5. 1-1. ෼ྨ໰୊ͷΞϓϩʔν ·ͣ͸ɺ܇࿅σʔλ͕༩͑ΒΕͨ࣌ͷ෼ྨ໰୊ͷΞϓϩʔνΛ 2 ͭ঺հ ͢Δɻ 1 ͭ໨͸ɺ܇࿅σʔλ͔Βࣝผؔ਺ y(x) Λ࡞Δํ๏Ͱ͋Δɻ ͜͜Ͱ x ͸ D ࣍ݩͷೖྗϕΫτϧ (ྫ͑͹ɺn ݸ໨ͷ܇࿅σʔλͷೖྗ Ͱ͋Δ xn ) Ͱ͋Δɻ ͜ͷΞϓϩʔνͰ͸ɺy(x) ͷ஋Λ༻͍ͯɺ܇࿅σʔλʹͳ͍ະ஌ͷ σʔλ x ͕ͲͷΫϥεʹ෼ྨ͞ΕΔ΂͖ͳͷ͔Λ༧ଌ͢Δɻ(ޙͰྫΛ ͋͛Δ͕ɺྫ͑͹ x ͸ y(x) ≥ 0 ͳΒΫϥε 1, y(x) < 0 ͳΒΫϥε 2 ʹଐ͢ΔͳͲ) 5 / 47
6. 1-1. ෼ྨ໰୊ͷΞϓϩʔν ࣮ͨͩ͠ࡍʹ෼ྨ໰୊Λղ͘ͱ͖͸ɺ܇࿅σʔλΛ࢖ͬͯؔ਺ y(x) Λ Ұ͔Β࡞Γ্͛Δ͜ͱ͸͠ͳ͍ɻ Α͘ߦ͏ํ๏͸ɺೖྗϕΫτϧͱಉ࣍͡ݩͷύϥϝʔλϕΫτϧ (ॏ Έ)w = (w1 , w2 , · · · , wD )T ͱεΧϥʔύϥϝʔλ (όΠΞε)w0 Λ༻ҙ ͠ɺؔ਺ y(x, w, w0 ) y(x, w, w0 ) = f ( wTx + w0 ) (1.1) Λ༻ҙ͢Δɻ ͜͜Ͱɺf ͸ඇઢܗؔ਺Ͱ͋Γɺ׆ੑԽؔ਺ͱݺ͹ΕΔɻ(ྫ͑͹ϩδ εςΟοΫγάϞΠυؔ਺) ͦͯ͠ɺڭࢣσʔλʹΑΓؔ਺ y(x, w, w0 ) ͷύϥϝʔλ w, w0 Λ఺ਪ ఆ͠ɺਪఆ͞Εͨύϥϝʔλͷ஋Λ w⋆, w⋆ 0 ͱ͢Δͱɺؔ਺ y(x, w = w⋆, w0 = w⋆ 0 ) ͕ࣝผؔ਺Ͱ͋Δɻ(ύϥϝʔλͷௐ੔ํ๏͸ ͨ͘͞Μ͋Δɻ) ͜Ε͕ࣝผؔ਺Λ࡞੒͢Δํ๏Ͱ͋Γɺࠓճ͸ͦͷதͰ΋ϑΟογϟʔ ͷઢܗ൑ผͱ͍͏ํ๏Λ঺հ͢Δɻ 6 / 47
7. 1-1. ෼ྨ໰୊ͷΞϓϩʔν ΋͏ҰͭͷΞϓϩʔνͱͯ͠ɺࣝผؔ਺ y(x) ͷ࡞੒Ͱͳ͘ɺ৚݅෇͖ ֬཰ p(Ck |x) Λ܇࿅σʔλ͔Βܾఆ͢Δํ๏͕͋Δɻ ͜͜Ͱɺ෼ྨΫϥε͸ K ݸ (Ϋϥε 1, Ϋϥε 2, · · · , Ϋϥε K) ͋Δ ͱͯ͠ɺCk ͸ k ݸ໨ͷΫϥεΛද͢ɻ ͳͷͰɺ৚݅෇͖֬཰ p(Ck |x) ͸ɺԿ͔͋Δೖྗ x ͕༩͑ΒΕͨͱ͖ʹ ͦͷೖྗ͕ k ݸ໨ͷΫϥεʹଐ͢Δ֬཰Λ༩͑Δɻ ͜Ε΋࣮ࡍ͸ɺ܇࿅σʔλ͔Β p(Ck |x) ͷܗΛҰ͔ΒܾΊΔͷͰ͸ͳ ͘ɺܗ͸ܾΊ͓͍ͯͯύϥϝʔλΛ܇࿅σʔλ͔ΒܾΊΔɻ(࠷໬ਪఆ Λ͢Δɻ) 7 / 47
8. 1-1. ෼ྨ໰୊ͷΞϓϩʔν ͨͩ͠৚݅෇͖֬཰ͷࣝผؔ਺ͱҟͳΔͱ͜Ζ͸ɺ৚݅෇͖֬཰ͱͯ͠ Ծఆ͢Δؔ਺ͷܗͱͯ͠ɺن֨Խ৚݅ ∑ k p(Ck |x) = 1 (1.2) Λຬͨ͢Α͏ͳؔ਺ΛԾఆ͠ͳͯ͘͸͍͚ͳ͍ɻ ͜ͷΑ͏ʹ৚݅෇͖֬཰ΛϞσϧԽ͢Δ͜ͱʹΑͬͯɺ࠷໬ਪఆ͔Βϕ ΠζਪఆʹࣗવʹҠߦͰ͖Δɻ ࠓճ͸۩ମྫͱͯ͠ɺK = 2 ͷ࣌ͷϩδεςΟοΫճؼͷ࠷໬ਪఆͱ ϕΠζਪఆΛऔΓѻ͏ɻ 8 / 47
9. 2. ϑΟογϟʔͷઢܗ൑ผ ͦΕͰ͸ɺࣝผؔ਺Λ܇࿅σʔλ͔ΒܾΊΔํ๏ͷ۩ମྫͱͯ͠ϑΟο γϟʔͷઢܗ൑ผΛઆ໌͢Δɻ ·ͣ͸ 2 Ϋϥε෼ྨΛߟ͑Δɻ(ΫϥεͷϥϕϧΛ C1 ͱ C2 ͱ͢Δɻ) ࣝผؔ਺ͱͯ͠ɺ(1.1) ͷ׆ੑؔ਺ f Λ߃౳ؔ਺ͱͨ͠΋ͷΛߟ͑Δɻ y(x, w, w0 ) = wTx + w0 (2.1) ͦͯ͠ɺN ݸͷ܇࿅σʔλͷೖྗσʔλͱͯ͠ {x1 , x2 , · · · , xN } ͱ͠ɺ ͦΕΒʹରԠ͢Δ໨ඪม਺ͱͯ͠ {t1 , t2 , · · · , tN } ͱ͠ɺศ্ٓΫϥε C1 ͕ t = 1 ʹରԠ͠ɺΫϥε C2 ͕ t = 0 ʹରԠ͢Δͱ͢Δɻ ·ͨɺೖྗϕΫτϧ͕ॴଐ͢ΔΫϥεͷ൑ఆํ๏Ͱ͋Δ͕ɺ͋Δೖྗϕ Ϋτϧ xi ͕༩͑ΒΕͨ࣌ʹ y(xi , w, w0 ) ≥ 0 Ͱ͋Ε͹ xi ͸Ϋϥε C1 ʹॴଐ͠ (ͭ·Γ ti = 1)ɺy(xi , w, w0 ) < 0 Ͱ͋Ε͹ xi ͸Ϋϥε C2 ʹ ॴଐ͢Δ (ͭ·Γ ti = 0) ͱ͢Δɻ 9 / 47
10. 2. ϑΟογϟʔͷઢܗ൑ผ ͦ΋ͦ΋ɺD ࣍ݩͷೖྗσʔλ x Λ 1 ࣍ݩͷؔ਺ y(x, w, w0 ) ʹࣹӨ͠ ͯ y ͷਖ਼ෛͰॴଐ͢ΔΫϥεΛܾΊΔͷͰɺ࣍ݩͷݮগʹΑΓ൑அ͢Δ ͨΊͷ৘ใ͕େ͖͘ݮͬͨྔ (ͭ·Γ y) ͰΫϥεΛ൑அ͢Δ͜ͱʹ ͳΔɻ ͭ·ΓɺD ۭؒͰ͸ೖྗσʔλ͕ଐ͢ΔΫϥεͷྖҬಉ͕࢜Α͘෼཭ ͞Ε͍͕ͯͨɺ1 ࣍ݩ΁ͷࣹӨͷ࢓ํʹΑͬͯ͸ɺͦͷ෼཭͕ͳ͘ͳΔ (ॏͳͬͯ͠·͏) ͜ͱ͕͋Δɻ D = 2 ͷ࣌ͷ۩ମྫΛ࣍ͷεϥΠυͰࣔ͢ɻ 10 / 47
11. 2. ϑΟογϟʔͷઢܗ൑ผ ੨ͷϓϩοτ͕Ϋϥε C1 ʹॴଐ͢Δ܇࿅σʔλɺ੺ͷϓϩοτ͕Ϋϥ ε C2 ʹॴଐ͢Δ܇࿅σʔλͱ͠ɺը૾ 2 ຕͱ΋܇࿅σʔλ͸શ͘ಉ͡ Ͱ͋Δɻ y(x, w, w0 ) = 0 ͷ௚ઢ͸ύϥϝʔλ w, w0 ΛมԽͤ͞Δ͜ͱʹΑͬͯɺ D = 2 ͷೖྗۭؒΛॎԣແਚʹҠಈ͢Δɻ ੨ͷϓϩοτ͕ଟ͍ํΛ y(x, w, w0 ) ≥ 0 ͷྖҬͱͯ͠ɺ੺ͷϓϩοτ ͕ଟ͍ํΛ y(x, w, w0 ) < 0 ͷྖҬͱ͢Δɻ ྆ํͷΫϥϑͰڭࢣσʔλ͸౳͘͠ɺೖྗۭؒͰ͸Α͘෼཭͞Ε͍ͯΔ ͕ɺࠨଆͷࣹӨ͸ͦͷ෼཭ΛҰ෦ফͯ͠͠·͍ͬͯΔ͜ͱ͕Θ͔Δɻ 11 / 47
12. 2. ϑΟογϟʔͷઢܗ൑ผ ϑΟογϟʔͷઢܗ൑ผͰ͸ɺ܇࿅σʔλΛ࠷΋෼཭͢ΔΑ͏ͳࣹӨɺ ͭ·Γύϥϝʔλ w, w0 ΛٻΊΔ͜ͱΛߟ͑Δɻ ͦ͜Ͱ·ͣɺ܇࿅σʔλͷೖྗ {x1 , x2 , · · · , xN } ͷΫϥεผͷฏۉϕ ΫτϧΛ m1 , m2 ͱ͠ɺ m1 = 1 N1 ∑ n∈C1 xn , m2 = 1 N2 ∑ n∈C2 xn (2.2) ͱͳΔɻ ͜͜ͰɺN1 ͱ N2 ͸ͦΕͧΕɺΫϥε C1 ·ͨ͸ C2 ʹଐ͍ͯ͠Δ܇࿅ σʔλͷ਺Ͱ͋Δɻ(΋ͪΖΜɺN1 + N2 = N Λຬͨ͢ɻ) 12 / 47
13. 2. ϑΟογϟʔͷઢܗ൑ผ ҰํɺࣹӨޙͷΫϥεผͷฏۉ͸ y(x, w, w0 )(2.1) ͷઢܗੑΑΓɺҎԼ ͷΑ͏ʹͳΔɻ m1 = 1 N1 ∑ n∈C1 y(xn , w, w0 ) = wTm1 + w0 m2 = 1 N2 ∑ n∈C2 y(xn , w, w0 ) = wTm2 + w0 (2.3) Α͘෼཭͞ΕࣹͨӨͰ͸ɺ͜ΕΒͷࣹӨޙͷΫϥεผͷฏۉ͸େ͖͘ҟ ͳ͍ͬͯΔͱߟ͑ΒΕΔɻ ͭ·Γɺ m2 − m1 = wT(m2 − m1 ) (2.4) ͷ஋͕େ͖͍΄ͲɺࣹӨޙ΋ྑ͘෼཭͞Ε͍ͯΔͩΖ͏ɻ 13 / 47
14. 2. ϑΟογϟʔͷઢܗ൑ผ ͨͩ͠ɺࣹӨޙͷΫϥεؒͷฏۉ஋ͷ͕ࠩେ͖ͯ͘΋ɺࣹӨޙͷΫϥε ผͷ෼ࢄ͕େ͖͔ͬͨΒɺ̍࣍ݩʹࣹӨ͞ΕͨΫϥεͷྖҬ͕ॏͳΓ ߹ͬͯ͠·͏ɻ ͦ͜ͰɺҎԼͷΫϥεผͷ෼ࢄͷ஋͸খ͘͞ͳ͍ͬͯͯ΄͍͠ɻ s2 1 = ∑ n∈C1 {y(xn , w, w0 ) − m1 }2 s2 2 = ∑ n∈C2 {y(xn , w, w0 ) − m2 }2 (2.5) ΑͬͯɺҎԼͷϑΟογϟʔͷ൑ผج४Λ࠷େʹ͢ΔΑ͏ͳύϥϝʔλ w, w0 ͕ࣹӨޙ΋෼཭Λ࠷େʹอͭΑ͏ͳύϥϝʔλͰ͋Δ͜ͱ͕Θ ͔Δɻ J(w, w0 ) = (m2 − m1 )2 s2 1 + s2 2 (2.6) 14 / 47
15. 2. ϑΟογϟʔͷઢܗ൑ผ ͦΕͰ͸ɺ(2.6) ͷӈลΛ w, w0 Ͱॻ͖Լͦ͏ɻ ·ͣ෼ࢠ͸ (2.4) ΑΓɺ (m2 − m1 )2 = { wT(m2 − m1 ) }2 = wT(m2 − m1 )(m2 − m1 )Tw = wTSB w (2.7) ͱͳΔɻ ͜͜ͰɺSB ͸Ϋϥεؒڞ෼ࢄߦྻͱ͍͍ɺ SB = (m2 − m1 )(m2 − m1 )T (2.8) Ͱఆٛ͞ΕΔɻ 15 / 47
16. 2. ϑΟογϟʔͷઢܗ൑ผ ࣍ʹɺ(2.6) ͷ෼฼͸ s2 1 + s2 2 Ͱ͋Δ͕ɺͱΓ͋͑ͣ s2 1 ͚ͩ w, w0 Ͱॻ ͍ͯΈΔͱɺ(2.5) ΑΓɺ s2 1 = ∑ n∈C1 {wTxn + w0 − m1 }2 = ∑ n∈C1 {wTxn − wTm1 }2 = ∑ n∈C1 {wT(xn − m1 )}2 = ∑ n∈C1 wT(xn − m1 )(xn − m1 )Tw =wT [ ∑ n∈C1 (xn − m1 )(xn − m1 )T ] w (2.9) ͱͳΔɻ 16 / 47
17. 2. ϑΟογϟʔͷઢܗ൑ผ ಉ༷ͷมܗΛ s2 2 ʹ͍ͭͯ΋ߦ͏ͱɺ(2.6) ͷ෼฼ s2 1 + s2 2 ͸ҎԼͷΑ͏ ʹͳΔɻ s2 1 + s2 2 = wTSW w (2.10) ͜͜ͰɺSW ͸૯Ϋϥε಺ڞ෼ࢄߦྻͱ͍͍ɺҎԼͰఆٛ͞ΕΔɻ SW = ∑ n∈C1 (xn − m1 )(xn − m1 )T + ∑ n∈C2 (xn − m2 )(xn − m2 )T (2.11) ͜ΕΑΓɺϑΟογϟʔͷ൑ผج४ (2.6) ͸ w Λ༻͍ͯҎԼͷΑ͏ʹॻ ͚Δɻ J(w, w0 ) = J(w) = wTSB w wTSW w (2.12) ͜͜Ͱɺ݁ՌతʹϑΟογϟʔͷ൑ผج४͸ w ʹͷΈґଘ͠ɺw0 ʹ͸ ґଘ͠ͳ͍͜ͱ͕Θ͔Δɻ 17 / 47
18. 2. ϑΟογϟʔͷઢܗ൑ผ ͜Ε͔ΒϑΟογϟʔͷ൑ผج४ J(w) Λ࠷େʹ͢Δ w ΛٻΊ͍ͯ͘ ͜ͱʹͳΔ͕ɺ͜͜Ͱ 1 ͭ J(w) ʹ͸ॏཁͳੑ࣭͕͋Δɻ θϩͰ͸ͳ͍ఆ਺ α Λ༻ҙ͠ɺJ(αw) Λܭࢉ͢Δͱɺ J(αw) = (αw)TSB (αw) (αw)TSW (αw) = wTSB w wTSW w = J(w) (2.13) ͱͳΓɺϑΟογϟʔͷ൑ผج४ J(w) ͸ w ͷఆ਺ഒʹରͯ͠ෆมͰ ͋Δɻ(εέʔϧෆมੑ) ͜ͷੑ࣭ΑΓɺJ(w) Λ࠷େʹ͢Δ w(= w⋆) ͕ݟ͔ͭΔͱɺͦͷఆ਺ ഒͷ αw⋆ ΋ J(w) Λ࠷େʹ͢Δͱ͍͏͜ͱ͕Θ͔Δɻ ͜ΕΑΓɺզʑ͸ J(w) Λ࠷େʹ͢Δ w ΛٻΊΔࡍɺ࠷େʹ͢Δ w ͸ Ͳͷํ޲Λ޲͍͍ͯΔ͔͚ͩΛ஌Ε͹ྑ͍ࣄʹͳΔɻ 18 / 47
19. 2. ϑΟογϟʔͷઢܗ൑ผ ͦΕͰ͸࣮ࡍʹ J(w) Λ࠷େʹ͢Δ w ΛٻΊΔҝʹɺJ(w) Λ w Ͱඍ ෼͢Δͱɺ ∂J(w) ∂w = 2 (wTSW w)2 { SB w(wTSW w) − SW w(wTSB w) } (2.14) ͱͳΓɺ͜ΕΛθϩʹ͢Δ w ͸ҎԼͷࣜΛຬͨ͢ɻ SB w(wTSW w) − SW w(wTSB w) = 0 (2.15) ྆ลɺࠨ͔Β S−1 W Λ͔͚ͯɺগࣜ͠มܗΛ͢Δͱɺ S−1 W SB w(wTSW w) − w(wTSB w) = 0 →w = (wTSW w) (wTSB w) S−1 W SB w w ∝ S−1 W SB w (2.16) ͱͳΔ͜ͱ͕Θ͔Δɻ 19 / 47
20. 2. ϑΟογϟʔͷઢܗ൑ผ ͜͜ͰɺSB w ͸ SB w = (m2 − m1 )(m2 − m1 )Tw ∝ m2 − m1 (2.17) ͱͳΔ͜ͱΛ༻͍ΔͱɺϑΟογϟʔͷ൑ผج४ J(w) Λ࠷େʹ͢Δϕ Ϋτϧ w ͸ w ∝ S−1 W (m2 − m1 ) (2.18) ͱͳΓɺS−1 W (m2 − m1 ) ͱಉ͡ํ޲Λ޲͍͍ͯΔϕΫτϧͰ͋Δ͜ͱ͕ Θ͔Δɻ Αͬͯɺߴ࣍ݩͷೖྗσʔλΛ (2.1) Λ࢖ͬͯ̍࣍ݩʹࣹӨ͠ɺΫϥε Λ൑அ͢ΔࡍɺϑΟογϟʔͷઢܗ൑ผΛ༻͍ΔͱɺS−1 W (m2 − m1 ) ํ ޲ͷύϥϝʔλ w Λ༻͍ࣹͯӨΛߦ͏ͱɺೖྗσʔλͷ෼཭Λ࠷େݶ ʹอͪͳ͕ΒࣹӨ͢Δ͜ͱ͕Ͱ͖Δ͜ͱ͕Θ͔ͬͨɻ 20 / 47
21. 3. ϩδεςΟοΫճؼ ͜Ε·Ͱ͸ࣝผؔ਺ y(x) Λڭࢣσʔλ͔Βܾఆ͢Δํ๏ΛऔΓѻͬͨ ͕ɺ͔͜͜Β͸৚݅෇͖֬཰ p(Ck |x) Λܾఆ͢Δํ๏ΛऔΓѻ͏ɻ ࣝผؔ਺ͷ࣌ͱಉ͡Α͏ʹɺ܇࿅σʔλͱͯ͠ೖྗσʔλͷू߹ {x1 , x2 , · · · , xN } ͱͦΕͧΕʹରԠ͢Δ໨ඪม਺ͷू߹ {t1 , t2 , · · · , tN } Λ༻ҙ͢Δɻ ࠓճ͸Ϋϥε਺ K ͕ 2 ͷ࣌ (ೋ஋෼ྨ) Λѻ͏ͨΊɺ໨తม਺ tn ͸ 0 ͔ 1 ͷ཭ࢄతͳ஋ΛͱΔɻ ࠓճ͸෼ྨͷϞσϧͱͯ͠ɺϩδεςΟοΫճؼϞσϧΛ঺հ͢Δɻ (ʮճؼʯͱ෇͍͍ͯΔ͕ɺ෼ྨͷϞσϧͰ͋Δɻ) 21 / 47
22. 3-1. ϩδεςΟοΫճؼ ࣝผؔ਺ͷ࣌ͱಉ༷ʹɺ܇࿅σʔλΛ༻͍ͯ৚݅෇͖֬཰ p(Ck |x) ΛҰ ͔Β࡞Δ͜ͱ͸ͤͣɺ৚݅෇͖֬཰ p(Ck |x) ʹύϥϝʔλ w Λಋೖ͠ ͯɺp(Ck |x, w) Λߟ͑Δɻ ࠓճͷϩδεςΟοΫճؼͰ͸ɺΫϥε 1 ʹରͯ͠ɺҎԼͷΑ͏ͳؔ਺ p(C1 |x, w) Λߟ͑Δɻ p(C1 |x, w) = σ(wTϕ(x)) (3.1) 22 / 47
23. 3-1. ϩδεςΟοΫճؼ ·ͣɺ(3.1) ͷӈลͷҾ਺ͷதʹ͋ΔϕΫτϧؔ਺ ϕ(·) ͸ඇઢܗͳؔ਺ ϕj (x) (j = 0, · · · , M − 1) Λॎʹฒ΂ͨϕΫτϧؔ਺ ϕ(x) = (ϕ0 (x), ϕ1 (x), · · · , ϕM−1 (x))T Ͱ͋Δɻ ྫ͑͹ɺجఈؔ਺ ϕj (x) ͱͯ͠ҎԼͷΨ΢εجఈؔ਺͕͋Δɻ ϕj (x) = exp { − (x − µj )2 2s2 } (3.2) ͜ͷجఈؔ਺͸ x = µj Λத৺ʹͯ͠ɺ෼ࢄ s2 ʹΑͬͯࢧ഑͞ΕΔ޿͕ ΓΛ࣋ͭΨ΢εجఈؔ਺Ͱ͋Δɻ Ҏ߱͸Ұൠͷجఈؔ਺ ϕj (x) Λ༻͍ͯٞ࿦͢Δɻ 23 / 47
24. 3-1. ϩδεςΟοΫճؼ ·ͣɺؔ਺ σ(·) ͸ϩδεςΟοΫγάϞΠυؔ਺ͱݺ͹ΕɺҎԼͰఆ ٛ͞ΕΔɻ σ(x) = 1 1 + e−x (3.3) ਤͰॻ͘ͱҎԼͷΑ͏ʹͳΔɻ ϩδεςΟοΫγάϞΠυؔ਺͸ఆٛʹΑΓɺ0 ͔Β 1 ·Ͱͷ஋ΛͱΔ ͷͰɺ(3.1) ΑΓ p(C1 |x, w) ͸ҎԼͷΑ͏ʹ֬཰ͷຬͨ͢΂͖஋ͷൣғ ಺ʹ஋ΛͱΔɻ 0 < p(C1 |x, w) < 1 (3.4) 24 / 47
25. 3-1. ϩδεςΟοΫճؼ ҰํɺΫϥε 2 ʹରͯ͠ɺp(C2 |x, w) ͸ҎԼͷΑ͏ʹԾఆ͢Δɻ p(C2 |x, w) = 1 − p(C1 |x, w) = 1 − σ(wTϕ(x)) (3.5) (3.4) ΑΓɺp(C2 |x, w) ΋ҎԼͷΑ͏ʹ֬཰ͷຬͨ͢΂͖஋ͷൣғ಺ʹ ஋ΛͱΔɻ 0 < p(C2 |x, w) < 1 (3.6) ͜ΕΒͷఆٛʹΑΓɺԾఆͨؔ͠਺ p(Ck |x, w) ͸ن֨Խ৚݅ (1.2) Λຬ ͨ͢ɻ ∑ k p(Ck |x, w) = p(C1 |x, w) + 1 − p(C1 |x, w) = 1 (3.7) t = 1 ͷΫϥεΛ C1 ͱ͠ɺt = 0 ͷΫϥεΛ C2 ͱ͍ͯ͠ΔͷͰɺ໬౓ؔ ਺ p(t|x, w) ͸ p(t|x, w) = σ(wTϕ(x))t(1 − σ(y(wTϕ(x))))1−t (3.8) ͱͳΔɻ (͜ͷΑ͏ͳ෼෍ΛϕϧψʔΠ෼෍ͱ͍͏) 25 / 47
26. 3-2. ϩδεςΟοΫճؼͷ࠷໬ਪఆ ࣍ʹ࠷໬ਪఆΛߦ͏͜ͱΛߟ͑Δɻ ͍ͭ΋ͷΑ͏ʹɺ܇࿅σʔλͱͯ͠ೖྗσʔλͷू߹ X = {x1 , x2 , · · · , xN } ͱͦΕͧΕʹରԠ͢Δ໨ඪม਺ͷू߹ {t1 , t2 , · · · , tN } Λ༻ҙ͢Δɻ ڭࢣσʔλҰͭҰ͕ͭ෼෍ (3.8) ͔Βಠཱʹੜ੒͞Ε͍ͯΔͱ͢Δͱɺ ໬౓ؔ਺ p(t|X, w) ͸ҎԼͷΑ͏ʹͳΔɻ p(t|X, w) = N ∏ n=1 ytn n (1 − yn )1−tn (3.9) ͱͳΔɻ ͜͜Ͱ yn ͸ҎԼͰఆٛ͞ΕΔɻ yn = σ(wTϕ(xn )) (3.10) 26 / 47
27. 3-2. ϩδεςΟοΫճؼͷ࠷໬ਪఆ ໬౓ؔ਺ (3.9) Λ࠷େʹ͢Δ w ΛٻΊΔ͜ͱ͸ҎԼͷෛͷର਺໬౓Λ ࠷খʹ͢Δ w ΛٻΊΔ͜ͱͱ౳ՁͰ͋Δɻ E(w) = − ln p(t|X, w) = − N ∑ n=1 ln { ytn n (1 − yn )1−tn } = − N ∑ n=1 { tn ln yn + (1 − tn ) ln (1 − yn ) } (3.11) ͜Ε͸ަࠩΤϯτϩϐʔޡࠩͱݺ͹ΕΔޡࠩؔ਺Ͱɺ෼ྨ໰୊ͰΑ͘࢖ ΘΕΔޡࠩؔ਺Ͱ͋Δɻ ෼ྨ໰୊ʹ͓͚ΔަࠩΤϯτϩϐʔޡࠩͷ࠷খԽ͸ɺ֬཰࿦Λ༻͍Δͱ ໬౓ؔ਺ΛϕϧψʔΠ෼෍ (3.8) ͱԾఆͨ͠ͱ͖ͷ࠷໬ਪఆͷ݁ՌͰ͋ Δࣄ͕Θ͔Δɻ 27 / 47
28. 3-2. ϩδεςΟοΫճؼͷ࠷໬ਪఆ ࣍ʹෛͷର਺໬౓ (3.11) Λ࠷খʹ͢Δ w ΛٻΊΔͨΊʹ (3.11) ͷ w ʹର͢Δޯ഑ΛٻΊΔͱҎԼͷΑ͏ʹͳΔɻ(PRML ͷԋश 4.13 ࢀর) ∇E(w) = N ∑ n=1 (yn − tn )ϕ(xn ) (3.12) ͜ͷޯ഑ͷܗ͸ਖ਼ղϥϕϧ tn ͱ༧ଌ஋ yn ͷࠩ (ͭ·Γޡࠩ) ͱجఈؔ ਺ϕΫτϧ ϕ(xn ) ͷ࿨ͷܗΛ͍ͯ͠Δɻ 28 / 47
29. 3-2. ϩδεςΟοΫճؼͷ࠷໬ਪఆ ͜ͷޯ഑ ∇E(w) Λθϩʹ͢Δ w ΛղੳతʹٻΊΔ͜ͱ͸Ͱ͖ͳ͍ɻ ͦͷཧ༝͸༧ଌ஋ y = σ(wTϕ(x)) ͕ϩδεςΟοΫؔ਺Λ׆ੑԽؔ਺ ʹ͔࣋ͭΒͰ͋Δɻ ͜ͷΑ͏ʹޯ഑ ∇E(w) Λθϩʹ͢Δ w ΛղੳతʹٻΊΔ͜ͱ͕Ͱ͖ ͳ͍࣌͸ޯ഑߱Լ๏Λ༻͍Δ͜ͱ͕͋Δɻ(χϡʔϥϧωοτͰ΋͜ͷ ํ๏͕Α͘༻͍ΒΕΔɻ) ޯ഑߱Լ๏Ͱ͸ɺ·ͣϥϯμϜʹܾΊͨύϥϝʔλͷॳظ஋Λ w(0) ͱ ͠ɺޡࠩؔ਺ͷޯ഑Λ༻͍ͯύϥϝʔλΛҎԼͷΑ͏ʹߋ৽͢Δɻ w(1) = w(0) − η∇E(w(0)) (3.13) ͜͜Ͱ η > 0 ͸ֶशύϥϝʔλͱݺͿɻ ͜ΕΛ܁Γฦ͢͜ͱͰύϥϝʔλ͕ޯ഑ ∇E(w) ͕খ͘͞ͳΔํ޲ʹߋ ৽͞ΕɺE(w) Λ࠷খʹ͢Δύϥϝʔλʹऩଋ͢Δɻ 29 / 47
30. ϕΠζਪఆʹ͍ͭͯ ͜Ε·Ͱ (࠷໬ਪఆ) Ͱ͸ɺ໬౓ؔ਺Λ࠷େʹ͢ΔΑ͏ͳύϥϝʔλ w Λ఺ਪఆ͖ͯͨ͠ɻ ϕΠζਪఆͰ͸ɺڭࢣσʔλΛ༻͍ͯύϥϝʔλ w ͷ֬཰෼෍ (఺Ͱͳ ͘෯Λ΋ͭɺࣄޙ෼෍ͱݺ͹ΕΔ) ΛٻΊΔɻ ͦͷࣄޙ෼෍Λ༻͍ͯɺະ஌ͷσʔλͷೖྗ x ͕༩͑ΒΕͨ࣌ͷग़ྗ t ͷ༧ଌ෼෍ p(t|x, t, X) ΛٻΊΔɻ ͜ͷ༧ଌ෼෍ p(t|x, t, X) ͸ɺະ஌ͷσʔλʹର͢Δ৚݅෇͖֬཰ p(t|x, w) ͱࣄޙ෼෍ p(w|t, X) Λ༻͍ͯ p(t|x, t, X) = ∫ p(t|x, w)p(w|t, X) dw (3.14) ͱॻ͚Δɻ(PRML 1.68 ࣜࢀর) ϕΠζਪఆʹ͍ͭͯ͸ɺPRML 1.2.3 ࢀরɻ 30 / 47
31. 3-3. ϕΠζϩδεςΟοΫճؼ ࣍͸ϩδεςΟοΫճؼΛϕΠζతʹѻ͏͜ͱΛߟ͑Δɻ ϕΠζਪఆͰ͸ະ஌ͷೖྗ x ʹର͢Δग़ྗ t ͷ༧ଌ෼෍ p(t|x, t, X) Λ ٻΊΔ͜ͱ͕໨తͱͳΔɻ (3.14) ΑΓɺͦͷ༧ଌ෼෍ p(t|x, t, X) ͸ॏΈ w ͷੵ෼ͰҎԼͷΑ͏ʹ ͔͚Δɻ p(t|x, t, X) = ∫ p(t|x, w)p(w|t, X) dw (3.15) ͜͜Ͱɺp(t|x, w) ͸໬౓ؔ਺Ͱ͋Γɺp(w|t, X) ͸ύϥϝʔλͷࣄޙ෼ ෍Ͱ͋Δɻ ಛʹࠓճ͸ೋ஋෼ྨΛߟ͍͑ͯΔͷͰɺ֬཰ p(C1 |x, t, X) = ∫ p(C1 |x, w)p(w|t, X) dw (3.16) ͚ͩΛੵ෼ͯ͠ٻΊͯɺp(C2 |x, t, X) ͸ p(C2 |x, t, X) = 1 − p(C1 |x, t, X) (3.17) ͷΑ͏ʹن֨Խ৚͔݅ΒٻΊΔ͜ͱΛߟ͑Δɻ 31 / 47
32. 3-3. ϕΠζϩδεςΟοΫճؼ ͨͩ͠ɺੵ෼ (3.16) Λղੳతʹղ͘ͷ͸ෆՄೳͰ͋Δɻ(͜Ε͸ϩδε ςΟοΫγάϞΠυؔ਺ͷӨڹͰ͋Δ) ͕ͨͬͯ͠ɺੵ෼ΛۙࣅతʹٻΊΔ͜ͱΛߟ͑Δɻ ࠓճ͸ (PRML 4 ষͰ͸) ϥϓϥεۙࣅΛ༻͍ͯੵ෼ΛۙࣅతʹٻΊͯ ͍Δɻ ۩ମతʹ͸ (3.16) ͷύϥϝʔλͷࣄޙ෼෍ p(w|t, X) ʹϥϓϥεۙࣅ Λద༻ͯ͠ɺΨ΢ε෼෍ʹۙࣅ͢Δɻ ͜͜Ͱϥϓϥεۙࣅͷઆ໌Λগ͠ߦ͏ɻ 32 / 47
33. 3-3. ϕΠζϩδεςΟοΫճؼ ·ͣ͸֬཰ม਺͕Ұ࣍ݩͷม਺ z ͷ৔߹Λߟ͑ɺҎԼͷΑ͏ͳ֬཰෼෍ p(z) Λߟ͑Δɻ p(z) = 1 Z f(z) (3.18) ͜͜ͰɺZ ͸ҎԼͰఆٛ͞ΕΔن֨Խఆ਺Ͱ͋Δɻ Z = ∫ f(z) dz (3.19) ϥϓϥεۙࣅͷ໨త͸෼෍ p(z) ΛϞʔυ (dp(z)/dz = 0 ͱͳΔ z) Λத ৺ͱ͢ΔΨ΢ε෼෍ʹۙࣅ͢Δ͜ͱͰ͋Δɻ ·ͣ͸Ϟʔυ z = z0 Λݟ͚ͭΔɻϞʔυ͸ (3.18) ΑΓ df(z) dz z=z0 = 0 (3.20) ͳΔ z0 Ͱ͋Δɻ 33 / 47
34. 3-3. ϕΠζϩδεςΟοΫճؼ Ϟʔυ͕ٻ·ͬͨΒɺؔ਺ ln f(z) Λ z = z0 पΓͰҎԼͷΑ͏ʹςΠ ϥʔల։ͷ 2 ࣍·ͰͰۙࣅ͢Δɻ ln f(z) ∼ ln f(z0 ) − 1 2 A(z − z0 )2 (3.21) ͜͜Ͱɺ A = − d2 dz2 ln f(z) z=z0 (3.22) Ͱ͋Δɻ ͜͜Ͱɺ(3.20) ʹΑΓ (3.21) ͷӈลͰҰ࣍ͷ߲͕ଘࡏ͠ͳ͍ɻ (3.21) ͷ྆ลͷࢦ਺ΛͱΔͱ f(z) ∼ f(z0 ) exp { − 1 2 A(z − z0 )2 } (3.23) ͱͳΔɻ 34 / 47
35. 3-3. ϕΠζϩδεςΟοΫճؼ ن֨ԽΛ͢Δͱɺ෼෍ p(z) ͸ p(z) ∼ ( A 2π )1/2 exp { − 1 2 A(z − z0 )2 } (3.24) ͱۙࣅͰ͖Δɻ͜Ε͕ϥϓϥεۙࣅͰ͋Δɻ ͨͩ͠஫ҙ఺ͱͯ͠ɺA > 0 Ͱͳ͍ͱΨ΢ε෼෍͕ఆٛͰ͖ͳ͍ɻ 35 / 47
36. 3-3. ϕΠζϩδεςΟοΫճؼ ࣍͸Ұ࣍ݩͷ֬཰ม਺͔ΒɺϕΫτϧʹ֦ு͠Α͏ɻ ͭ·ΓɺҎԼͷ֬཰෼෍ p(z) Λఆٛ͢Δɻ p(z) = 1 Z f(z) (3.25) ͜͜Ͱɺ Z = ∫ f(z) dz (3.26) Ͱ͋Δɻ Ұ࣍ݩͷ֬཰ม਺ͱಉ͡Α͏ʹޯ഑ ∇f(z) ͕θϩʹͳΔ఺ z0 Λٻ ΊΔɻ 36 / 47
37. 3-3. ϕΠζϩδεςΟοΫճؼ Ϟʔυ͕ٻ·ͬͨΒɺln f(z) Λ z0 पΓͰςΠϥʔల։Ͱۙࣅ͢Δɻ ln f(z) ∼ ln f(z0 ) − 1 2 (z − z0 )TA(z − z0 ) (3.27) ͜͜ͰɺA ͸ҎԼͰఆٛ͞ΕΔ M × M ͷϔοηߦྻͰ͋Δɻ A = −∇∇ ln f(z) z=z0 (3.28) ࣍ʹ (3.27) ͷ྆ลͷࢦ਺ΛͱΔͱҎԼͷΑ͏ʹͳΔɻ f(z) ∼ f(z0 ) exp { − 1 2 (z − z0 )TA(z − z0 ) } (3.29) ͜ΕΑΓن֨ԽΛ͢Δͱɺ෼෍ p(z) ͸ p(z) ∼ |A|1/2 (2π)M/2 exp { − 1 2 (z − z0 )TA(z − z0 ) } = N(z|z0 , A−1) (3.30) ͱΨ΢ε෼෍ʹۙࣅͰ͖Δɻ 37 / 47
38. 3-3. ϕΠζϩδεςΟοΫճؼ Ҏ্Ͱઆ໌ͨ͠ϥϓϥεۙࣅΛ༻͍ͯҎԼͷੵ෼ (3.16) Λۙࣅ͍ͨ͠ɻ p(C1 |x, t, X) = ∫ p(C1 |x, w)p(w|t, X) dw (3.31) ·ͣɺࣄޙ෼෍ p(w|t, X) ΛٻΊΔͨΊʹࣄલ෼෍Λಋೖ͢Δɻ p(w) = N(w|m0 , S0 ) (3.32) (3.18) ΑΓɺ໬౓ؔ਺͸ p(t|X, w) ͸ p(t|X, w) = N ∏ n=1 ytn n (1 − yn )1−tn (3.33) Ͱ͋ͬͨɻ 38 / 47
39. 3-3. ϕΠζϩδεςΟοΫճؼ ͜ΕΑΓɺࣄޙ෼෍ p(w|t, X) ͸ϕΠζͷఆཧΑΓɺҎԼͰ͋Δɻ p(w|t, X) ∝ p(w)p(t|X, w) (3.34) ͱͳΔͷͰɺln p(w|t, X) ͸ҎԼͱͳΔɻ ln p(w|t, X) = − 1 2 (w − m0 )TS−1 0 (w − m0 ) + N ∑ n=1 { tn ln yn + (1 − tn ) ln (1 − yn ) } + const. (3.35) ͜ͷࣄޙ෼෍ͷର਺ ln p(w|t, X) Λ࠷େʹ͢Δύϥϝʔλ wMAP Λ (ͨ ͱ͑͹ޯ഑߱Լ๏ͳͲͰ) ٻΊͯɺͦͷ఺ wMAP ͰͷϔοηߦྻΛٻΊ ΔͱɺҎԼͷΑ͏ʹͳΔɻ S−1 N = − ∇∇ ln p(w|t, X) w=wMAP =S−1 0 + N ∑ n=1 yn (1 − yn )ϕn ϕT n w=wMAP (3.36) 39 / 47
40. 3-3. ϕΠζϩδεςΟοΫճؼ ΑͬͯɺϥϓϥεۙࣅΛ༻͍Δͱࣄޙ෼෍ p(w|t, X) ͸ҎԼͷΑ͏ʹۙ ࣅͰ͖Δɻ p(w|t, X) ∼ N(w|wMAP , SN ) (3.37) ͜ΕΑΓɺ(3.31) ͷੵ෼͸ҎԼͷΑ͏ʹۙࣅͰ͖Δɻ p(C1 |x, t, X) ∼ ∫ σ(wTϕ) N(w|wMAP , SN ) dw (3.38) ͜͜Ͱɺp(C1 |x, w) = σ(wTϕ) Λར༻ͨ͠ɻ ࣍ʹɺϩδεςΟοΫγάϞΠυؔ਺ΛҎԼͷΑ͏ʹॻ͖௚͢ɻ σ(wTϕ) = ∫ δ(a − wTϕ)σ(a) da (3.39) ͜͜Ͱɺδ(·) ͸σϡϥοΫͷσϧλؔ਺Ͱ͋Δɻ 40 / 47
41. 3-3. ϕΠζϩδεςΟοΫճؼ ͜ΕΑΓɺ(3.38) ͸ҎԼͷΑ͏ʹॻ͖௚ͤΔɻ ∫ σ(wTϕ) N(w|wMAP , SN ) dw = ∫ ∫ δ(a − wTϕ)σ(a) N(w|wMAP , SN ) da dw = ∫ ∫ δ(a − wTϕ) N(w|wMAP , SN ) dw σ(a) da = ∫ p(a) σ(a) da (3.40) ͜͜Ͱɺ p(a) = ∫ δ(a − wTϕ) N(w|wMAP , SN ) dw (3.41) Ͱ͋Δɻ 41 / 47
42. 3-3. ϕΠζϩδεςΟοΫճؼ ੵ෼ (3.41) ʹ͓͍ͯɺϕ ʹฏߦͳ͢΂ͯͷํ޲ͷ w ੵ෼͸ͦΕΒͷύ ϥϝʔλʹઢܗ੍໿Λ༩͑ɺ·ͨ ϕ ʹ௚ߦ͢Δ͢΂ͯͷํ޲ͷ w ੵ෼ ͸Ψ΢ε෼෍ N(w|wMAP , SN ) ͷपลԽΛ༩͑Δɻ ͨͱ͑͹ɺw = (w1 , w2 )T ͱ͠ɺϕ = (ϕ, 0)T Ͱ͋Δͱ͖Λߟ͑Δͱɺ ੵ෼ (3.41) ͸ҎԼͷΑ͏ʹ͔͚Δɻ p(a) = ∫ ∫ δ(a − w1 ϕ) N(w|wMAP , SN ) dw1 dw2 = ∫ δ(a − w1 ϕ) [ ∫ N(w|wMAP , SN ) dw2 ] dw1 (3.42) (??) ΑΓɺΨ΢ε෼෍ΛपลԽͨ͠पล෼෍͸࠶ͼΨ΢ε෼෍Ͱ͋Δ ͜ͱ͕Θ͔͍ͬͯΔͷͰɺϕ ʹ௚ߦ͢Δ w2 ํ޲ͷੵ෼͸Ψ΢ε෼෍ͷ पลԽΛ༩͑ɺͦͷपลԽ͞ΕͨΨ΢ε෼෍͸ N(w1 |(wMAP )1 , (SN )11 ) ͱͳΔɻ 42 / 47
43. 3-3. ϕΠζϩδεςΟοΫճؼ ·ͨɺw1 ͷੵ෼Λ͢Δͱɺੵ෼ (3.41) ͸ҎԼͷΑ͏ʹͳΔɻ p(a) = ∫ δ(a − w1 ϕ) N(w1 |(wMAP )1 , (SN )11 ) dw1 = 1 |ϕ| N(a/ϕ|(wMAP )1 , (SN )11 ) =N(a|(ϕwMAP )1 , (ϕ2SN )11 ) (3.43) ͭ·Γɺϕ ʹฏߦͳ w1 ͷํ޲ͷੵ෼͸ w1 ʹ w1 = a/ϕ ͳΔઢܗ੍໿ Λ༩͑Δ͜ͱ͕Θ͔Δɻ ͜ΕΑΓɺp(a) ͸֬཰ม਺͕ a ͷΨ΢ε෼෍ʹͳΔ͜ͱ͕Θ͔Δɻ 43 / 47
44. 3-3. ϕΠζϩδεςΟοΫճؼ Ψ΢ε෼෍͸ฏۉͱ෼ࢄ͕ܾ·Ε͹ɺܗ͕Ұҙʹఆ·Γɺฏۉ µa ͱ෼ ࢄ σ2 a ͸ҎԼͷΑ͏ʹͳΔɻ µa = ∫ p(a)a da = ∫ ∫ aδ(a − wTϕ) N(w|wMAP , SN ) dwda = ∫ wTϕ N(w|wMAP , SN ) dw = wT MAP ϕ (3.44) σ2 a = ∫ p(a)(a2 − µ2 a ) da = ∫ ∫ (a2 − µ2 a )δ(a − wTϕ) N(w|wMAP , SN ) dwda = ∫ ((wTϕ)2 − (wT MAP ϕ)2) N(w|wMAP , SN ) dw =ϕT [ ∫ (wwT − wMAP wT MAP ) N(w|wMAP , SN ) dw ] ϕ =ϕTSN ϕ (3.45) 44 / 47
45. 3-3. ϕΠζϩδεςΟοΫճؼ ͢Δͱɺ༧ଌ෼෍ p(C1 |x, t, X) ͸ (3.40) ΑΓɺҎԼͷΑ͏ʹͳΔ͜ͱ ͕Θ͔Δɻ p(C1 |x, t, X) ∼ ∫ σ(a)N(a|µa , σ2 a ) da (3.46) ͜͜Ͱɺµa ͱ σ2 a ͸ (3.44) ͱ (3.45) Ͱܭࢉͨ͠ฏۉͱ෼ࢄͷύϥϝʔ λͰ͋Δɻ ͜ͷੵ෼ (3.46) ΋·ͨղੳతʹੵ෼Ͱ͖ͳ͍ɻ ͦ͜ͰҎԼͷϓϩϏοτؔ਺ͷٯؔ਺ Φ(a) Λಋೖ͢Δɻ Φ(a) = 1 2 { 1 + erf ( a √ 2 )} (3.47) ͜͜Ͱɺޡࠩؔ਺ erf(a) ͸ҎԼͰఆٛ͞ΕΔɻ erf(a) = 2 √ π ∫ a 0 exp (−θ2) dθ (3.48) 45 / 47
46. 3-3. ϕΠζϩδεςΟοΫճؼ ϓϩϏοτؔ਺ͷٯؔ਺ Φ (√ π 8 a ) ʹΑͬͯϩδεςΟοΫγάϞΠ υؔ਺ σ(a) Λۙࣅ͢Δ͜ͱ͕Ͱ͖Δɻ ҎԼ͸ϩδεςΟοΫγάϞΠυؔ਺ σ(a)(੺ͷ࣮ઢ) ͱϓϩϏοτؔ ਺ͷٯؔ਺ Φ (√ π 8 a ) (੨ͷ఺ઢ) Λൺֱͨ͠ਤͰ͋Δɻ 46 / 47
47. 3-3. ϕΠζϩδεςΟοΫճؼ ͞ΒʹϓϩϏοτؔ਺ͷٯؔ਺ʹ͸ҎԼͷੑ࣭͕͋Δɻ(PRML ͷԋश 4.26 ࢀর) ∫ Φ(λa)N(a|µ, σ2) da = Φ ( µ (λ−2 + σ2)1/2 ) (3.49) ͜ΕΒͷੑ࣭Λ༻͍ͯɺੵ෼ (3.46) ΛҎԼͷΑ͏ʹۙࣅͯ͠ٻΊΔɻ p(C1 |x, t, X) ∼ ∫ σ(a)N(a|µa , σ2 a ) da ∼ ∫ Φ (√ π 8 a ) N(a|µa , σ2 a ) da =Φ ( µa (8/π + σ2 a )1/2 ) ∼ σ (√ 8 π µa (8/π + σ2 a )1/2 ) =σ ( µa (1 + πσ2 a /8)1/2 ) (3.50) ͜͜Ͱɺµa ͱ σ2 a ͸ (3.44) ͱ (3.45) Ͱ͋Δɻ 47 / 47