PRML第11章

Transcript

1. PRML ୈ 11 ষ αϯϓϦϯά๏
   2019/04/01 ࡔޱ ྒี
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2. Ұൠతͳ֬཰Ϟσϧʹ͓͍ͯɺ࿈ଓతͳ֬཰ม਺Λ z ͱͨ͠ͱ͖ͷ͋Δ
   ؔ਺ f(z) ͷ֬཰෼෍ p(z) ͷԼͰͷظ଴஋
   E[f] =
   ∫
   f(z)p(z) dz (11.1)
   Λܭࢉ͍ͨ͠γνϡΤʔγϣϯ͕ଟ͘ൃੜ͢Δɻ(ྫ͑͹ɺPRML ͷ 4
   ষͷࣜ (4.145) ͳͲ)
   ଟ͘ͷ৔߹ɺ͜ͷੵ෼ (11.1) ͸ղੳతʹܭࢉͰ͖ͳ͍ɻ
   αϯϓϦϯά๏ͷҰൠతͳΞΠσΞ͸ɺ෼෍ p(z) ͔Βಠཱʹநग़͞Ε
   ͨαϯϓϧ z(l) (l = 1, · · · , L) Λಘͯɺ(11.1) ΛҎԼͷΑ͏ʹۙࣅ͢Δɻ
   f =
   1
   L
   L
   ∑
   l=1
   f(z(l)) (11.2)
   ͔͜͠͠ͷํ๏ͩͱɺ΋͠ p(z) ͕খ͍͞ྖҬͰ f(z) ͕େ͖͘ɺp(z) ͕
   େ͖͍ྖҬͰ f(z) ͕খ͔ͬͨ͞Βɺαϯϓϧ਺͕গͳ͍࣌͸ͦͷαϯ
   ϓϧͷதʹ p(z) ͕খ͍͞ྖҬ͔Βαϯϓϧ͕͋ͬͨΒɺ(11.2) ͸ p(z)
   ͕খ͍͞ྖҬ͔ΒαϯϓϧʹӨڹΛड͚͗͢Δͱ͍͏໰୊͕͋Δɻ
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3. 11.1 جຊతͳαϯϓϦϯάΞϧΰϦζϜ
   ͜͜Ͱ͸ɺ༩͑ΒΕͨ෼෍͔ΒϥϯμϜʹαϯϓϦϯάΛߦ͏ํ๏Λઆ
   ໌͢Δɻ
   αϯϓϧࣗମ͸ܭࢉػʹΑΓੜ੒͞Ε͍ͯΔͨΊɺαϯϓϧ͸ٖࣅཚ
   ਺Ͱ͋Δɻ
   ͭ·Γɺܾఆ࿦తͳܭࢉͷ૊Έ߹ΘͤʹΑΓɺϥϯμϜͳαϯϓϧΛੜ
   ੒͢Δɻ
   ·ͨԾఆͱͯ͠ɺ۠ؒ (0, 1) ͷҰ༷෼෍͔Βͷαϯϓϧੜ੒Λߦ͏Ξϧ
   ΰϦζϜ͸༩͑ΒΕ͍ͯΔͱ͢Δɻ
   ͜ͷҰ༷෼෍͔ΒͷαϯϓϦϯά͔ΒඇҰ༷෼෍ͷαϯϓϦϯάΛ
   ߦ͏ɻ
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4. 11.1.1 ඪ४తͳ෼෍
   ·ͣɺz ͕۠ؒ (0, 1) ͰҰ༷ʹ෼෍͓ͯ͠Γɺͦͷม਺ z Λ y = f(z) Ͱ
   ඇҰ༷ͳ֬཰ม਺ y ʹม׵͢Δ͜ͱΛߟ͑Δɻ
   ͜ͷͱ͖ɺPRML ͷࣜ (1.27) ΑΓɺy ͷ֬཰෼෍͸ҎԼͷΑ͏ʹͳΔɻ
   p(y) = p(z)
   dz
   dy
   (11.5)
   ͜͜Ͱɺp(z) ͸۠ؒ (0, 1) ͷҰ༷෼෍ͳͷͰɺp(z) = 1 ͱͳΔɻ
   (p(z) = c = const. ͱͯ͠ɺن֨Խ৚݅Λ՝ͤ͹ɺc = 1 ͱͳΔɻ)
   ͜͜Ͱɺy Λ z ʹม׵͢ΔҎԼͷؔ਺ h(y) Λߟ͑Δɻ
   z = h(y) ≡
   ∫
   y
   −∞
   p(ˆ
   y) dˆ
   y (11.6)
   ͜ͷΑ͏ͳؔ਺ h(y) Ͱ z ʹม׵͢Ε͹ɺp(z) = 1 ͱ h(y)′ = p(y) ≥ 0
   Λ༻͍ͯ
   p(z)
   dz
   dy
   = |h(y)′| = p(y)
   ͱͳΓɺ֬཰ม਺ y ͸ p(y) ʹै͏͜ͱ͕Θ͔Δɻ
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5. 11.1.1 ඪ४తͳ෼෍
   ͜ΕΑΓɺp(y) ʹै͏ y Λ z ͔ΒٻΊΔʹ͸ɺ(11.6) Ͱఆٛ͞Εͨؔ਺
   h ͷٯؔ਺Λ༻͍ͯɺy = h−1(z) ͱͯ͠ม׵͢Ε͹ྑ͍ɻ
   ͳͷͰɺҰ༷෼෍͔ΒಘΒΕͨαϯϓϧ zi
   ͔Βɺ͋ΔඇҰ༷෼෍ p(y)
   ͔Β༩͑ΒΕΔαϯϓϧ yi
   ΛಘΔͨΊͷखॱ͸ҎԼͷΑ͏ʹͳΔɻ
   1. (11.6) ΑΓɺp(y) Λੵ෼ͯ͠ h(y) ΛಘΔɻ
   2. h(y) ͷٯؔ਺ h−1(z) ΛٻΊΔɻ
   3. h−1 Λ༻͍ͯɺyi
   = h−1(zi
   ) Ͱαϯϓϧ yi
   ΛಘΔɻ
   ͜ͷखॱͰͷαϯϓϦϯά๏Λٯؔ਺๏ͱ͍͏ɻ
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6. 11.1.1 ඪ४తͳ෼෍
   ྫͱͯ͠ɺαϯϓϧΛҎԼͷࢦ਺෼෍͔Βಘ͍ͨͱ͢Δɻ
   p(y) = λ exp (−λy) (11.7)
   ͜͜Ͱɺλ ͸ਖ਼ͷύϥϝʔλͰɺ0 ≤ y < ∞ Ͱ͋Δɻ
   ·ͣ (11.6) ΑΓɺ0 ≤ y < ∞ Ͱ͋Δ͜ͱʹ஫ҙͯ͠ɺ
   h(y) =
   ∫
   y
   0
   λ exp (−λˆ
   y) dˆ
   y
   = −
   [
   exp (−λˆ
   y)
   ]ˆ
   y=y
   ˆ
   y=0
   = 1 − exp (−λy)
   ͱͳΓɺh(y) = z ͱ͢Δͱɺy ͸ (−λy ≤ 0 ΑΓɺ1 − exp (−λy) < 1 Ͱ
   ͋Δ͜ͱʹ஫ҙ͢Δͱ)
   z = 1 − exp (−λy)
   → − λy = ln (1 − z)
   →y = −λ−1 ln (1 − z)
   ͱͳΔɻ
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7. 11.1.1 ඪ४తͳ෼෍
   ͜ΕΑΓɺҰ༷෼෍͔Β zi
   ΛಘΔͱɺyi
   = −λ−1 ln (1 − zi
   ) ͱͯ͠ಘͨ
   yi
   ͸ࢦ਺෼෍ (11.7) ͔Βͷαϯϓϧʹͳ͍ͬͯΔɻ
   ͜ͷํ๏͕͏·͍͘͘ʹ͸ɺੵ෼ (11.6) ͕࣮ߦͰ͖ɺٯؔ਺ h−1 ͕ٻ
   ·Δඞཁ͕͋Δɻ
   ্ͷྫͷࢦ਺෼෍ͷΑ͏ͳ୯७ͳ෼෍Ͱ͋Ε͹ՄೳͰ͋Δ͕ɺҰൠతʹ
   ͸ෆՄೳͰ͋Δɻ
   ͦͷΑ͏ͳෳࡶͳ෼෍ʹ͍ͭͯ͸ผͷΞϓϩʔν͕ඞཁͰɺ࣍ʹड़΂
   Δغ٫αϯϓϦϯά΍ॏ఺αϯϓϦϯά͕༗ޮͰ͋Δɻ
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8. 11.1.2 غ٫αϯϓϦϯά
   ࣍ͷغ٫αϯϓϦϯάͷઆ໌Λߦ͏ɻ
   ͜͜Ͱ͸ɺલͱಉ͡Α͏ʹ෼෍ p(z) ͔ΒαϯϓϦϯάΛߦ͍͍ͨͱ
   ͢Δɻ
   ·ͨԾఆͱͯ͠ɺp(z) ͸ن֨Խఆ਺Λআ͍ͯΘ͔͍ͬͯΔͱ͢Δɻ
   ͭ·Γɺن֨Խఆ਺Λ Zp
   ͱͯ͠ɺp(z) Λ
   p(z) =
   1
   Zp
   p(z) (11.13)
   ͱ͢Δͱɺp(z) ͸Θ͔͍ͬͯΔ͕ɺZp
   ͷ஋͸Θ͔͍ͬͯͳ͍ͱ͢Δɻ
   غ٫αϯϓϦϯάΛߦ͏ʹ͋ͨͬͯɺΑΓαϯϓϦϯά͕؆୯ͳ
   (11.1.1 ͷٯؔ਺๏ͰαϯϓϦϯάͰ͖ΔΑ͏ͳ) ෼෍ q(z) Λ༻ҙ͢Δɻ
   (͜ͷΑ͏ͳ෼෍ΛఏҊ෼෍ͱ͍͏ɻ)
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9. 11.1.2 غ٫αϯϓϦϯά
   ࣍ʹɺਖ਼ͷఆ਺ k Λ༻ҙͯ͠ɺ͢΂ͯͷ z ʹରͯ͠ kq(z) ≥ p(z) ͱͳ
   ΔΑ͏ʹ k ͷ஋ΛܾΊΔɻ
   p(z) ͸Θ͔͍ͬͯΔ͔Βɺk ͷ஋͸ٻΊΒΕΔɻ
   ҎԼ͕ kq(z) ͱ p(z) ͷྫͰ͋Δɻ
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10. 11.1.2 غ٫αϯϓϦϯά
    ࣮ࡍͷαϯϓϦϯάͷํ๏͸ɺ·ͣఏҊ෼෍ q(z) ͔Βཚ਺ z0
    Λੜ੒
    ͢Δɻ
    ࣍ʹ۠ؒ [0, kq(z0
    )] ͷҰ༷෼෍͔Βཚ਺ u0
    Λੜ੒͢Δɻ
    ੜ੒ͨ͠ u0
    ͕ɺu0
    > p(z0
    ) Ͱ͋Ε͹غ٫͞Ε (ࣺͯΒΕ)ɺͦΕҎ֎Ͱ
    ͋Ε͹ u0
    ͸ p(z) ͔ΒͷαϯϓϦϯάͱͯ͠อ࣋͞ΕΔɻ
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11. 11.1.2 غ٫αϯϓϦϯά
    αϯϓϦϯά͸Ͱ͖Δ͚ͩޮ཰తʹߦ͍͍ͨͷͰɺغ٫͞ΕΔαϯϓϧ
    ͷ਺͸ݮΒ͍ͨ͠ɻ
    ͦ͜Ͱɺαϯϓϧ z ͕डཧ͞ΕΔ֬཰ p(accept) ΛٻΊͯΈΔɻ
    αϯϓϧ z ͸ q(z) ͔Βੜ੒͞Εɺ۠ؒ [0, kq(z)] ͷҰ༷෼෍͔Βੜ੒͞
    ΕΔαϯϓϧ͸ p(z)/kq(z) ͷ֬཰Ͱडཧ͞ΕΔͷͰɺαϯϓϧ͕डཧ
    ͞ΕΔ֬཰ p(accept) ͸
    p(accept) =
    ∫
    {p(z)/kq(z)}q(z) dz
    =
    1
    k
    ∫
    p(z) dz
    =
    ∫
    p(z) dz
    ∫
    kq(z) dz
    (11.14)
    ͱͳΔɻ
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12. 11.1.2 غ٫αϯϓϦϯά
    ͭ·Γɺ(11.14) ΑΓɺk ͸͢΂ͯͷ z ʹରͯ͠ kq(z) ≥ p(z) Λຬͨ͢
    ݶΓɺͰ͖Δ͚ͩখ͘͢͞Δඞཁ͕͋Δɻ
    ·ͨɺԼͷਤͷփ৭ͷ෦෼ΛͰ͖Δ͚ͩখ͘͢͞ΔΑ͏ͳఏҊ෼෍Λબ
    ΂͹ɺडཧ཰্͕͕Δɻ
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13. 11.1.3 దԠతغ٫αϯϓϦϯά
    غ٫αϯϓϦϯάͰ͸ɺద੾ͳఏҊ෼෍Λ༻ҙ͢Δඞཁ͕͋Δ͕ɺద੾
    ͳఏҊ෼෍͕Θ͔Βͳ͍৔߹ɺదԠతغ٫αϯϓϦϯά͕࢖͑Δɻ
    ͜ͷαϯϓϦϯάͰ͸ɺαϯϓϦϯάΛ͍ͨ͠෼෍ p(z) ͸Θ͔͍ͬͯ
    Δͱ͢Δɻ
    ͞ΒʹɺҎԼͷ੺͍ؔ਺Α͏ʹ p(z) ͸ର਺Ԝ (ର਺্͕ʹತͳؔ਺) Ͱ
    ͋Δͱ͢Δɻ
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14. 11.1.3 దԠతغ٫αϯϓϦϯά
    ·ͣ͸ॳظू߹ͱͯ͠ɺԿ఺͔αϯϓϧ {zi
    } ͕༩͑Δɻ
    ͦͯ͠ɺͦͷ఺Ͱͷ ln p(z) ͷඍ෼܎਺Λ −λi
    ͱ͢Δɻ
    d
    dz
    ln p(z)
    z=zi
    = −λi
    ͜ͷඍ෼܎਺ −λi
    Λ༻͍ͯɺఏҊ෼෍ͷର਺ ln q(z) ΛҎԼͷΑ͏ʹ
    ࡞Δɻ
    ln q(z) = −λi
    (z − zi
    ) + ln λi
    ki
    (ˆ
    zi−1,i
    < z ≤ ˆ
    zi,i+1
    )
    ͜͜Ͱɺˆ
    zi−1,i
    ͸ zi−1
    Ͱͷ઀ઢͱ zi
    Ͱͷ઀ઢͱͷަ఺ͷ z ࠲ඪɻ
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15. 11.1.3 దԠతغ٫αϯϓϦϯά
    ͜ͷΑ͏ʹͯ͠࡞ΒΕͨఏҊ෼෍͸ p(z) ͕ର਺ԜͰ͋Ε͹ɺ
    p(z) ≤ q(z) ͱͳΓɺغ٫αϯϓϦϯά͕ద༻Ͱ͖Δɻ
    ͜ΕΑΓɺq(z) ͸
    q(z) = λi
    ki
    exp {−λi
    (z − zi
    )} (ˆ
    zi−1,i
    < z ≤ ˆ
    zi,i+1
    )
    ͱͳΔɻ
    ͜ͷ q(z) ͔ΒͷαϯϓϦϯά͸؆୯Ͱɺ11.1.1 ͷٯؔ਺๏Λ༻͍ͯα
    ϯϓϦϯάͰ͖Δɻ(ԋश 11.9)
    ͦ͜Ͱɺq(z) ͔Βͷ৽ͨͳαϯϓϦϯά z ͕༩͑ΒΕͨΒɺغ٫αϯϓ
    Ϧϯάͷͱ͖ͱಉ༷ʹ [0, q(z)] ͷҰ༷෼෍͔Βαϯϓϧ u ΛಘΔɻ
    ͦͯ͠ɺu > p(u) ͳΒغ٫͠ɺͦΕҎ֎ͳΒडཧ͢Δɻ
    غ٫͞ΕͨΒɺq(z) Λ࡞੒͢ΔͨΊͷ৽͍͠఺ʹ͢Δɻ
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16. 11.1.3 దԠతغ٫αϯϓϦϯά
    ͜͜Ͱ͸ɺغ٫αϯϓϦϯά͕αϯϓϧม਺ z ͕ߴ࣍ݩϕΫτϧͷ࣌ʹ
    ͸޲͔ͳ͍͜ͱΛઆ໌͢Δɻ
    ྫͱͯ͠ɺฏۉ͕θϩͰڞ෼ࢄߦྻ͕ σ2
    p
    I Ͱ͋ΔΨ΢ε෼෍͔Βαϯϓ
    Ϧϯά͍ͨ͠ͱ͢Δɻ
    p(z) = N(z|0, σ2
    p
    I)
    ·ͨɺఏҊ෼෍ q(z) ͸ฏۉ͕θϩͰڞ෼ࢄߦྻ͕ σ2
    q
    I Ͱ͋ΔΨ΢ε෼
    ෍Ͱ͋Δͱ͢Δɻ
    q(z) = N(z|0, σ2
    q
    I)
    غ٫αϯϓϦϯάͰ͸ɺ͢΂ͯͷ z Ͱ kq(z) ≥ p(z) Ͱ͋ΔΑ͏ͳ k ͕
    ଘࡏ͠ͳͯ͘͸͍͚ͳ͍ɻ
    ͦͷΑ͏ͳ k ͕ଘࡏ͢Δʹ͸ɺq(z) ͕ p(z) ΑΓฏ͍ͨ෼෍Ͱͳ͍ͱ͍
    ͚ͳ͍ɻ
    ͭ·Γɺσ2
    q
    ≥ σ2
    p
    Ͱ͋Δඞཁ͕͋Δɻ
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17. 11.1.3 దԠతغ٫αϯϓϦϯά
    σ2
    q
    ≥ σ2
    p
    ͷ৚݅Ͱɺ͢΂ͯͷ z Ͱ kq(z) ≥ p(z) ͱͳΔͨΊʹ͸ҎԼͷਤ
    ͷΑ͏ʹɺ࠷େ஋Λ༩͑Δ఺Ͱ z = 0 Ͱ kq(z = 0) = p(z = 0) ͱͳΕ
    ͹Α͘ɺ
    N(x|µ, Σ) =
    1
    (2π)D/2
    1
    |Σ|1/2
    exp
    {
    −
    1
    2
    (x − µ)TΣ−1(x − µ)
    }
    ΑΓɺ
    k =
    (
    σq
    σp
    )D
    ͱऔΕ͹ྑ͍ɻ
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18. 11.1.3 దԠతغ٫αϯϓϦϯά
    ͜ΕΑΓɺαϯϓϦϯάͷडཧ཰ p(accept) ͸ (11.14) ΑΓ
    p(accept) =
    ∫
    p(z) dz
    ∫
    kq(z) dz
    =
    1
    k
    =
    (
    σp
    σq
    )D
    ͱͳΔɻ
    ͭ·ΓɺD = 1000 Ͱ͸ɺσq
    /σp
    = 1.01 ͷͱ͖ (σq
    ͕ͨͬͨ 1 ύʔηϯ
    τ͚ͩ σp
    ΑΓେ͖͍ͱ͖)
    p(accept) =
    (
    1
    1.01
    )1000
    ∼
    1
    20000
    ͱͳΓɺ΄ͱΜͲडཧ͞Εͳ͍ɻ
    ͭ·Γɺغ٫αϯϓϦϯά͸ߴ࣍ݩʹ͸ద͓ͯ͠Βͣɺ1 ࣍ݩ·ͨ͸ 2
    ࣍ݩ͘Β͍ͷͱ͖ʹదͨ͠αϯϓϦϯάͰ͋Δɻ
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19. 11.1.4 ॏ఺αϯϓϦϯά
    ॏ఺αϯϓϦϯάͰ͸ɺ෼෍ p(z) ͔ΒαϯϓϦϯάΛಘΔͷͰ͸ͳ͘ɺ
    (11.1) ͷΑ͏ͳظ଴஋ͷۙࣅ஋Λ௚઀ٻΊΔɻ
    (11.2) ͷΑ͏ʹۙࣅ͢Δͱɺ{z(l)} ͸ෳࡶͳ෼෍ p(z) ͔Βͷαϯϓϧͳ
    ͷͰɺαϯϓϦϯά͕೉͍͠ɻ
    ͦ͜Ͱɺ͜͜Ͱ΋αϯϓϦϯά͕ൺֱత؆୯ͳఏҊ෼෍ q(z) Λར༻
    ͢Δɻ
    ఏҊ෼෍Λར༻ͯ͠ɺ(11.1) ΛҎԼͷΑ͏ʹۙࣅ͢Δɻ
    E[f] =
    ∫
    f(z)p(z) dz
    =
    ∫
    f(z)
    p(z)
    q(z)
    q(z) dz
    ∼
    1
    L
    L
    ∑
    l=1
    p(z(l))
    q(z(l))
    f(z(l))
    (11.19)
    ͜͜Ͱɺ{z(l)} ͸ q(z) ͔ΒͷαϯϓϧͰ͋Δɻ
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20. 11.1.4 ॏ఺αϯϓϦϯά
    ͜͜Ͱɺrl
    = p(z(l))/q(z(l)) ͸ॏཁ౓ॏΈͱ͍ͬͯɺq(z) ͰαϯϓϦϯ
    άͨ͜͠ͱʹΑΔ p(z) ͔ΒͷζϨΛิਖ਼͢ΔҼࢠͰ͋Δɻ
    ͨͱ͑͹ɺ͋Δ l ʹରͯ͠ɺq(z(l)) ͕΄ͱΜͲ 1 ͩͬͨͱ͖ɺαϯϓϧ
    ͷதʹසൟʹ z(l) ͕ೖͬͯདྷΔ͜ͱ͕༧૝͞ΕΔɻ
    ͨͩ͠ɺp(z(l)) ͕খ͍͞৔߹͸ɺ͋·Γαϯϓϧ z(l) ͸ (11.19) ʹد༩
    ͢Δ΂͖Ͱ͸ͳ͍ɻ
    ͜ͷͱ͖ɺrl
    = p(z(l))/q(z(l)) ͕খ͘͞ͳͬͯɺ͔ͨ͠ʹαϯϓϧ z(l)
    ͸ (11.19) ʹد༩͠ͳ͍ɻ
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21. 11.1.4 ॏ఺αϯϓϦϯά
    ·ͨɺҎલʹ΋͋ͬͨ௨Γɺp(z) ͕ن֨Խఆ਺Λআ͍ͯΘ͔͍ͬͯΔͱ
    ͢Δɻ
    ͭ·Γɺp(z) = p(z)/Zp
    ͱͨ͠ͱ͖ʹɺp(z) ͸Θ͔͍ͬͯΔ͕ɺZp
    ͸
    Θ͔͍ͬͯͳ͍ͱԾఆ͢Δɻ·ͨɺಉ༷ʹ q(z) = q(z)/Zq
    ͱ͢Δɻ
    ͜ͷͱ͖ɺ(11.19) ͸ҎԼͷΑ͏ʹͳΔɻ
    E[f] =
    ∫
    f(z)p(z) dz
    =
    ∫
    f(z)
    p(z)
    q(z)
    q(z) dz ∼
    1
    L
    L
    ∑
    l=1
    p(z(l))
    q(z(l))
    f(z(l))
    =
    Zq
    Zp
    1
    L
    L
    ∑
    l=1
    p(z(l))
    q(z(l))
    f(z(l)) =
    Zq
    Zp
    1
    L
    L
    ∑
    l=1
    rl
    f(z(l))
    (11.20)
    ͜͜Ͱɺ{z(l)} ͸ q(z) ͔ΒαϯϓϦϯά͞Εͨαϯϓϧͷू߹Ͱ͋Γɺ
    rl
    = p(z(l))/q(z(l)) ͱఆٛͨ͠ɻ
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22. 11.1.4 ॏ఺αϯϓϦϯά
    ·ͨɺZp
    =
    ∫
    p(z) dz ͱ Zq
    = q(z)/q(z) Λ༻͍ΔͱɺZp
    /Zq
    ΋ҎԼͷ
    Α͏ʹಉ͡αϯϓϧͷू߹ {z(l)} Λ࢖ͬͯɺۙࣅతʹܭࢉͰ͖Δɻ
    Zp
    Zq
    =
    ∫
    1
    Zq
    p(z) dz =
    ∫
    p(z)
    q(z)
    q(z) dz
    ∼
    1
    L
    L
    ∑
    l=1
    rl
    (11.21)
    ΑͬͯɺE[f] ͸ҎԼͷΑ͏ʹۙࣅͰ͖Δɻ
    E[f] ∼
    Zq
    Zp
    1
    L
    L
    ∑
    l=1
    rl
    f(z(l)) ∼
    L
    ∑
    L
    m=1
    rm
    ·
    1
    L
    L
    ∑
    l=1
    rl
    f(z(l))
    =
    L
    ∑
    l=1
    rl
    ∑
    L
    m=1
    rm
    f(z(l)) =
    L
    ∑
    l=1
    wl
    f(z(l))
    (11.22)
    ͜͜Ͱɺwl
    ͸ҎԼͰఆٛ͞ΕΔɻ
    wl
    =
    rl
    ∑
    L
    m=1
    rm
    (11.23)
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23. 11.1.4 ॏ఺αϯϓϦϯά
    ͜ΕΑΓɺ͔֬ʹॏ఺αϯϓϦϯάΛ༻͍Δͱɺظ଴஋ E[f] Λۙࣅత
    ʹٻΊΔ͜ͱ͕Ͱ͖Δɻ
    ͨͩ͠ɺ͜ͷαϯϓϦϯάͰ࢖༻͢ΔఏҊ෼෍ q(z) ͸αϯϓϧΛٻΊ
    ͍ͨ෼෍ p(z) ͱ͋Δఔ౓ࣅ͍ͯΔ෼෍Λ࢖༻͢Δඞཁ͕͋Δɻ
    ͨͱ͑͹ɺp(z) ͕͋Δαϯϓϧۭؒͷখ͞ͳൣғ A Ͱ 0 Ͱͳ͍Α͏ͳ
    ෼෍Ͱ͋ΓɺఏҊ෼෍ q(z) ͕ͦͷൣғ A Ͱ 0 Ͱ͋Δͱɺαϯϓϧͷू
    ߹ {z(l)} ͷதͰൣғ A ʹೖΔ΋ͷ͸ͳ͍ͷͰɺॏཁ౓ॏΈͷू߹ {rl
    }
    ͸͸͢΂ͯ 0 ͱͳΔɻ
    ͜ͷͱ͖ɺE[f] ͷۙࣅ஋͸ 0 ͱͳͬͯ͠·͏ɻ
    ͞ΒʹɺE[f] ͕ 0 ͱͳΔͷ͕ਖ਼͍͔͠൱͔ (ਖ਼͍͠৔߹΋͋Δ) ΛଌΔ
    ࢦඪ͕ॏ఺αϯϓϦϯάͰ͸ଘࡏ͠ͳ͍ͷ͕࠷΋ਂࠁͳ఺Ͱ͋Δɻ
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24. 11.1.5 SIR
    غ٫αϯϓϦϯάͷ໰୊఺͸ɺ্͔Βԡ͑͞ΔΑ͏ͳ k ΛܾΊΔͱɺغ
    ٫཰͕ߴ͘ͳΔ͜ͱͩͬͨɻ
    ͜͜Ͱ͸ɺk ΛઃఆͤͣʹࡁΉํ๏Ͱ͋Δ
    SIR(Sampling-Importance-Resampling) ʹ͍ͭͯઆ໌͢Δɻ
    ·ͣɺఏҊ෼෍ q(z) ͔Β L ݸͷαϯϓϧू߹ {z(l)} ΛಘΔɻ
    (Sampling)
    (11.23) ΑΓɺwl
    Λܭࢉ͢Δɻ(Importance)
    wl
    ͸
    ∑
    l
    wl
    = 1 Λຬͨ͢ͷͰɺ཭ࢄతͳ֬཰෼෍ͱΈͳͤΔͷͰɺͦ
    ͷ֬཰෼෍ pl
    = wl
    ʹै͏֬཰Ͱ {z(l)} ͔Β L ݸαϯϓϦϯά͢Δɻ
    (Resampling)
    ҎԼͰɺ͜ͷΑ͏ʹ࠶αϯϓϦϯά͞Εͨαϯϓϧू߹ {z(l)} ͸
    L → ∞ ͰαϯϓϦϯά͍ͨ͠෼෍ p(z) ͔ΒͷαϯϓϦϯάʹͳ͍ͬͯ
    Δ͜ͱΛࣔ͢ɻ
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25. 11.1.5 SIR
    αϯϓϧۭ͕ؒҰม਺ͷ৔߹Ͱɺ֬཰෼෍ pl
    = wl
    ͷྦྷੵ෼෍ p(z ≤ a)
    Λܭࢉͯ͠ΈΔɻ(֬཰ม਺ z ͕ a ҎԼͱͳΔ֬཰)
    p(z ≤ a) ͸ (11.23) Λ༻͍Δͱ
    p(z ≤ a) =
    ∑
    l:z(l)≤a
    wl
    =
    ∑
    l
    I(z(l) ≤ a)p(z(l))/q(z(l))
    ∑
    m
    p(z(m))/q(z(m))
    (11.25)
    ͱͳΔɻ
    ͜͜ͰɺI(z ≤ a) ͸Ҿ਺͕ਅͷ࣌͸ 1 ͰɺͦΕҎ֎Ͱ͸ 0 ͱͳΔؔ਺Ͱ
    ͋Δɻ
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26. 11.1.5 SIR
    L → ∞ ͱ͢Δͱɺ{z(l)} ͸΋ͱ΋ͱ q(z) ͔ΒαϯϓϦϯά͞Εͨαϯ
    ϓϧͳͷͰɺ
    p(z ≤ a) =
    ∫
    I(z ≤ a){p(z)/q(z)}q(z) dz
    ∫
    {p(z)/q(z)}q(z) dz
    =
    ∫
    I(z ≤ a)p(z) dz
    ∫
    p(z) dz
    =
    ∫
    I(z ≤ a)p(z) dz
    (11.26)
    ͱͳΓɺ͔֬ʹ L → ∞ Ͱ͸ɺαϯϓϦϯά͍ͨ͠෼෍ p(z) ͷྦྷੵ෼
    ෍ʹҰக͍ͯ͠Δɻ
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27. 11.1.6 αϯϓϦϯάͱ EM ΞϧΰϦζϜ
    EM ΞϧΰϦζϜͷ E εςοϓͷۙࣅܭࢉʹαϯϓϦϯά๏͸࢖༻Ͱ
    ͖Δɻ
    EM ΞϧΰϦζϜͷ E εςοϓͰ͸ɺӅΕม਺ Z ͷࣄޙ෼෍
    p(Z|X, θold) ʹΑΔ׬શσʔλର਺໬౓ ln p(Z, X|θ) ͷظ଴஋
    Q(θ, θold) ΛٻΊΔɻ
    Q(θ, θold) =
    ∫
    p(Z|X, θold) ln p(Z, X|θ) dZ (11.28)
    ͜ͷੵ෼Λɺࣄޙ෼෍ p(Z|X, θold) ͔Βͷαϯϓϧू߹ {Z(l)} Λ࢖ͬ
    ͯɺҎԼͷ༗ݶ࿨Ͱۙࣅ͢Δɻ
    Q(θ, θold) ∼
    1
    L
    L
    ∑
    l=1
    ln p(Z(l), X|θ) (11.29)
    ͦͯ͠ɺ͜ͷ Q(θ, θold) Λ༻͍ͯ M εςοϓΛ࣮ߦ͢Δɻ
    ͜ͷΑ͏ͳ EM ΞϧΰϦζϜΛϞϯςΧϧϩ EM ΞϧΰϦζϜͱ͍͏ɻ
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28. 11.2 Ϛϧίϑ࿈࠯ϞϯςΧϧϩ
    ͜͜·Ͱղઆ͖ͯͨ͠αϯϓϦϯά๏Ͱ͸ɺαϯϓϧۭ͕ؒ௿࣍ݩͳΒ
    ༗ޮͰ͋Δ͕ɺߴ࣍ݩʹͳΔͱ্ख͍͔͘ͳ͍͜ͱ͕Θ͔͍ͬͯΔɻ
    ͜͜Ͱಋೖ͢ΔϚϧίϑ࿈࠯ϞϯςΧϧϩ (MCMC) Ͱ͸αϯϓϧۭؒ
    ͕ߴ࣍ݩͰ͋ͬͯ΋Α͘ػೳ͢Δɻ
    ɹ
    MCMC ʹ͓͍ͯ΋ɺఏҊ෼෍Λಋೖ͢Δɻ
    ͨͩ͠ɺ͜͜ͰͷఏҊ෼෍͸ݱࡏͷαϯϓϧ z(τ) ʹґଘ͠ɺq(z|z(τ))
    ͱͳΓɺ࣍ͷαϯϓϧ z(τ+1) ͸ q(z|z(τ)) ͔ΒಘΒΕΔɻ(ޙ΄Ͳઆ໌͢
    ΔϚϧίϑ࿈࠯)
    ·ͨɺ໨ඪ͸෼෍ p(z) ͔ΒαϯϓϧΛಘΔ͜ͱͰ͋Δ͕ɺ͜͜Ͱ΋
    p(z) = p(z)/Zp
    ͱ͠ɺp(z) ͸Θ͔͍ͬͯΔ͕ɺZp
    ͷ஋͸Θ͔Βͳ͍ͱ
    ͢Δɻ
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29. 11.2 Ϛϧίϑ࿈࠯ϞϯςΧϧϩ
    ·ͣ MCMC ͷҰൠ࿦ʹೖΔલʹɺ࠷΋؆୯ͳྫͰ͋Δ Metropolis Ξϧ
    ΰϦζϜΛղઆ͢Δɻ
    ఏҊ෼෍͸͢΂ͯͷαϯϓϧʹ͍ͭͯରশͰ͋Δͱ͢Δɻ
    ྫ͑͹ zA
    ͱ zB
    ʹରͯ͠ q(zA
    |zB
    ) = q(zB
    |zA
    ) (ରশͰͳ͍৔߹͸
    11.2.2 Ͱղઆ͢Δɻ)
    ·ͣɺαϯϓϧ z(τ) ͕༩͑ΒΕͯɺ࣍ͷαϯϓϧީิ z⋆ ͕ఏҊ෼෍
    q(z|z(τ)) ͔ΒಘΒΕͨͱ͢Δɻ
    ͦͷαϯϓϧީิ z⋆ ͸ҎԼͷ֬཰ A(z⋆, z(τ)) Ͱडཧ͞ΕΔɻ
    A(z⋆, z(τ)) = min
    (
    1,
    p(z⋆)
    p(z(τ))
    )
    (11.33)
    ࣮૷Ͱ͸ɺ୯Ґ۠ؒ (0, 1) ͷҰ༷෼෍͔Βཚ਺ u Λऔಘ͠ɺ
    A(z⋆, z(τ)) > u Ͱ͋Ε͹ɺαϯϓϧީิ z⋆ ͸αϯϓϧͱͯ͠डཧ͞Εɺ
    ͦΕҎ֎ͷ৔߹͸غ٫͞ΕΔΑ͏ʹ͢Δɻ
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30. 11.2 Ϛϧίϑ࿈࠯ϞϯςΧϧϩ
    ͜͜Ͱ (11.33) ΑΓɺp(z⋆) > p(z(τ)) Ͱ͋Ε͹ɺA(z⋆, z(τ)) = 1 ͱͳΓɺ
    ͲΜͳཚ਺ u Ͱ΋ඞͣ A(z⋆, z(τ)) > u ͱͳΔͷͰɺඞͣαϯϓϧީิ
    z⋆ ͸αϯϓϧͱͯ͠डཧ͞ΕΔ͜ͱʹ஫ҙɻ
    ΋͠ɺαϯϓϧީิ z⋆ ͸αϯϓϧͱͯ͠डཧ͞ΕͨΒɺz(τ+1) = z⋆ ͱ
    ͠ɺغ٫͞ΕͨΒ z(τ+1) = z(τ) ͱ͢Δɻ
    ͜ͷ఺͕ɺغ٫͞ΕͨΒ୯ʹαϯϓϧΛࣺͯΔغ٫αϯϓϦϯάͱͷҧ
    ͍Ͱ͋Δɻ
    (11.2.2 Ͱূ໌͢Δ͕ɺ) ೚ҙͷ zA
    ͱ zB
    ʹରͯ͠ q(zA
    |zB
    ) ͕ਖ਼ͳΒ
    ͹ɺz(τ) ͕ै͏෼෍͸ τ → ∞ ͰαϯϓϧΛಘ͍ͨ෼෍ p(z) ʹۙͮ͘ɻ
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31. 11.2.1 Ϛϧίϑ࿈࠯
    MCMC ͷҰൠతͳٞ࿦Λ͢ΔͨΊɺ·ͣ͸Ϛϧίϑ࿈࠯ͷٞ࿦Λߦ͏ɻ
    ͦ͜Ͱɺ·ͣ m εςοϓͰͷ֬཰ม਺Λ z(m) ͱ͢Δɻ
    ͦͯ͠ɺm = 1, · · · , M ͱͨ͠ͱ͖ͷ֬཰ม਺ͷܥྻ z(1), · · · , z(M) ʹ
    ରͯ͠ɺҎԼͷੑ࣭ (ಠཱੑ) Λຬͨ͢ͱ͖ɺz(1), · · · , z(M) ΛϚϧίϑ
    ࿈࠯ͱ͍͏ɻ
    p(z(m+1)|z(1), · · · , z(m)) = p(z(m+1)|z(m)) (11.37)
    ͭ·Γɺεςοϓ m + 1 ͷ֬཰աఔ͕Ұݸલͷεςοϓ m ΑΓ΋લͷ
    εςοϓʹ͸ґଘ͠ͳ͍ͱ͍͏͜ͱͰ͋Δɻ
    ͜ͷ༷ࢠΛάϥϑͰද͢ͱɺҎԼͷΑ͏ʹͳΔɻ
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32. 11.2.1 Ϛϧίϑ࿈࠯
    m + 1 εςοϓͰͷαϯϓϧ z(m+1) Λൃੜͤ͞Δपล෼෍ p(z(m+1))
    ͸
    p(z(m+1)) =
    ∑
    z(m)
    p(z(m+1), z(m)) =
    ∑
    z(m)
    p(z(m+1)|z(m))p(z(m)) (11.38)
    ͱͳΔɻ
    ͜͜ͰɺTm
    (z(m), z(m+1)) = p(z(m+1)|z(m)) ͸ભҠ֬཰ͱ͍͏ɻ
    ಛʹભҠ֬཰͕εςοϓ m ʹґΒͳ͍
    (Tm
    (z(m), z(m+1)) = T(z(m), z(m+1))) Ϛϧίϑ࿈࠯ΛۉҰϚϧίϑ࿈
    ࠯ͱ͍͍ɺࠓޙ͸ͦͷΑ͏ͳભҠ֬཰ʹݶఆͯ͠࿩ΛਐΊΔɻ
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33. 11.2.1 Ϛϧίϑ࿈࠯
    Ҏ্ͷٞ࿦ΑΓɺT(z(m), z(m+1)) ͕༩͑ΒΕΕ͹ɺϚϧίϑ࿈࠯ʹै
    ͏֬཰ม਺͸࣍ͷΑ͏ʹͯ͠ൃੜͤ͞Δ͜ͱ͕Ͱ͖Δɻ
    1. ॳظঢ়ଶ z(1) Λॳظ෼෍ (ྫ͑͹αϯϓϦϯά͕ՄೳͳఏҊ෼෍
    q(z)) ͔ΒαϯϓϦϯά͢Δ
    2. m = 1, · · · , M − 1 ʹରͯ͠ɺભҠ֬཰ T(z(m), z(m+1)) ΑΓ
    z(m+1) Λൃੜͤ͞Δ
    ͜ͷΑ͏ʹͯ͠ൃੜͤͨ͞αϯϓϧ z(m+1) ͸ (11.38) ΑΓɺ͔֬ʹ
    p(z(m+1)) ͔Βൃੜͤͨ͞αϯϓϧͰ͋Δɻ
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34. 11.2.1 Ϛϧίϑ࿈࠯
    ͜ͷΑ͏ʹαϯϓϧ z(m) Λੜ੒͠ଓ͚ͯɺm → ∞ ͱͨ͠ͱ͖ʹ෼෍
    p(z(m)) ͸ऩଋ͢Δ͔Ͳ͏͔͕໰୊ͱͳΔɻ
    ͜Ε͸Ϛϧίϑ࿈࠯͕ΤϧΰʔυੑΛຬ͍ͨͯ͠Ε͹ɺऩଋ͢Δ͜ͱ͕
    Θ͔͍ͬͯΔɻ
    Ϛϧίϑ࿈࠯͕ΤϧΰʔυతͰ͋Δͱ͸ɺن໿ੑʢͲͷঢ়ଶ͔ΒͰ΋೚
    ҙͷঢ়ଶ΁ભҠͰ͖Δʣͱਖ਼࠶ؼੑʢ೚ҙͷঢ়ଶ΁ԿճͰ΋ભҠͰ͖
    Δʣͱඇपظੑʢ೚ҙͷঢ়ଶ͸ҰճͷભҠͰݩʹ໭ΕΔʣΛશͯಉ࣌ʹ
    ຬͨ͢͜ͱΛݴ͏ɻ
    ͦͯ͠ΤϧΰʔυੑΛຬ͍ͨͯ͠ΔۉҰϚϧίϑ࿈࠯Ͱ͸ɺm → ∞ ͱ
    ͨ͠ͱ͖ɺ෼෍ p(z(m)) ͸ҎԼͷৄࡉ௼Γ߹͍৚݅Λຬͨ͢෼෍ p⋆(z)
    ʹऩଋ͢Δɻ
    p⋆(z)T(z, z′) = p⋆(z′)T(z′, z) (11.40)
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35. 11.2.1 Ϛϧίϑ࿈࠯
    Ҏ্Λ౿·͑ΔͱɺఏҊ෼෍ q(z) ͔ΒɺϚϧίϑ࿈࠯Λ࢖ͬͯر๬ͷ
    ෼෍ p⋆(z) ͔ΒͷαϯϓϧΛಘ͍ͨͳΒҎԼͷखॱͰαϯϓϦϯάΛߦ
    ͑͹͍͍ɻ
    1. (11.40) ͷৄࡉ௼Γ߹͍৚݅ΛΈͨ͢Α͏ͳભҠ֬཰ T(z, z′) Λ༻
    ҙ͢Δ
    2. ॳظঢ়ଶ z(1) ΛఏҊ෼෍ q(z) ͔ΒαϯϓϦϯά͢Δ
    3. m = 1, · · · , M − 1 ʹରͯ͠ɺT(z(m), z(m+1)) ͔Β z(m+1) Λൃੜ
    ͤ͞Δ
    4. m → ∞ ͷͱ͖ͷαϯϓϧ z(m) ͸ر๬ͷ෼෍ p⋆(z) ͔Βͷαϯϓ
    ϧͰ͋Δ
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36. 11.2.2 Metropolis-Hastings ΞϧΰϦζϜ
    ҎલɺMetropolis ΞϧΰϦζϜΛ MCMC ͷҰྫͱͯ͠঺հ͕ͨ͠ɺͦ
    Ε͕αϯϓϦϯά͍ͨ͠෼෍ p(z) ͔ΒͷαϯϓϦϯάʹͳ͍ͬͯΔ͜
    ͱ͸આ໌͠ͳ͔ͬͨɻ
    ͜͜Ͱ͸ɺMetropolis ΞϧΰϦζϜΛ֦ுͯ͠ɺఏҊ෼෍͕ରশͰͳ͍
    ৔߹ (zA
    ͱ zB
    ʹରͯ͠ q(zA
    |zB
    ) ̸= q(zB
    |zA
    )) ͷΞϧΰϦζϜ
    (Metropolis-Hastings ΞϧΰϦζϜ) Λಋೖ͢Δɻ
    ·ͣɺαϯϓϧ z(τ) ͕༩͑ΒΕͯɺ࣍ͷαϯϓϧީิ z⋆ ͕ఏҊ෼෍
    qk
    (z|z(τ)) ͔ΒಘΒΕͨͱ͢Δɻ(ఴ͑ࣈ k ͸ભҠઌ͕ෳ਺͋ͬͨ৔߹
    ͷͨΊʹ͚͍ͭͯΔɻ11.3 Ͱ۩ମྫΛݟΔɻ)
    ͦͷαϯϓϧީิ z⋆ ͸ҎԼͷ֬཰ Ak
    (z⋆, z(τ)) Ͱडཧ͞ΕΔɻ
    Ak
    (z⋆, z(τ)) = min
    (
    1,
    p(z⋆)qk
    (z(τ)|z⋆)
    p(z(τ))qk
    (z⋆|z(τ))
    )
    (11.44)
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37. 11.2.2 Metropolis-Hastings ΞϧΰϦζϜ
    ͜ͷ Metropolis-Hastings ΞϧΰϦζϜͰ͸ɺqk
    (z′|z) ʹΑΓαϯϓϧީ
    ิ͕બ͹ΕɺAk
    (z′, z) ʹΑΓडཧ͞ΕΔ͔Ͳ͏͔ܾ·ΔͷͰɺભҠ֬
    ཰͸ Tk
    (z, z′) = qk
    (z′|z)Ak
    (z′, z) ͱͳΔ͔Βɺ(11.44) ΑΓ
    p(z)Tk
    (z, z′) =p(z)qk
    (z′|z)Ak
    (z′, z)
    =p(z)qk
    (z′|z) · min
    (
    1,
    p(z′)qk
    (z|z′)
    p(z)qk
    (z′|z)
    )
    =p(z)qk
    (z′|z) · min
    (
    1,
    p(z′)qk
    (z|z′)
    p(z)qk
    (z′|z)
    )
    =min
    (
    p(z)qk
    (z′|z), p(z′)qk
    (z|z′)
    )
    =p(z′)qk
    (z|z′) · min
    (
    p(z)qk
    (z′|z)
    p(z′)qk
    (z|z′)
    , 1
    )
    =p(z′)qk
    (z|z′)Ak
    (z, z′) = p(z′)Tk
    (z′, z)
    (11.45)
    ͱͳΔͷͰɺαϯϓϧΛಘ͍ͨ෼෍ p(z) ͕ৄࡉ௼Γ߹͍৚݅ (11.40) Λ
    ຬͨ͢ͷͰɺMetropolis-Hastings ΞϧΰϦζϜΛ܁Γฦ͢ͱɺp(z) ͔Β
    ͷαϯϓϧΛಘΔ͜ͱ͕Ͱ͖Δ͜ͱ͕Θ͔Δɻ
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38. 11.2.2 Metropolis-Hastings ΞϧΰϦζϜ
    ͜ͷ Metropolis-Hastings ΞϧΰϦζϜʹ΋஫ҙ఺͕͋Δɻ
    ྫ͑͹ɺҎԼͷਤͷΑ͏ʹɺରশͳఏҊ෼෍ q(z) Λݱࡏͷঢ়ଶ z(τ) Λ
    த৺ʹͨ͠౳ํΨ΢ε෼෍ͱ͠ɺp(z) ͸ํ޲ʹΑͬͯେ͖͘ҟͳΔภ
    ࠩΛ࣋ͭΨ΢ε෼෍ͱ͢Δɻ(੺͕ p(z) Ͱɺ੨͕ q(z))
    ఏҊ෼෍ͷεέʔϧ ρ ͕খ͍͞ͱغ٫཰͸Լ͕Δ͕ɺz ͕ z(τ) ͔Βେ͖
    ͘มԽ͠ͳ͍ͨΊɺϥϯμϜ΢ΥʔΫΛͱΓɺܥྻ z(1), · · · ͷؒͰ௕͍
    ૬ؔΛ࣋ͭɻ
    Ұํɺεέʔϧ ρ େ͖͗͘͢͠Δͱ p(z) ͕খ͍͞αϯϓϧΛऔͬͯ͘
    ΔՄೳੑ͕ߴ͘ͳΓɺغ٫཰্͕͕Δɻ
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39. 11.3 ΪϒεαϯϓϦϯά
    ࣍͸ MCMC ͷҰྫͰ͋ΔΪϒεαϯϓϦϯάΛઆ໌͢Δɻ
    ޙͰड़΂ΔΑ͏ʹɺΪϒεαϯϓϦϯά͸ Metropolis-Hastings Ξϧΰ
    ϦζϜͷಛघͳ৔߹Ͱ͋Δ͜ͱ͕Θ͔Δɻ
    αϯϓϦϯάΛ͍ͨ͠෼෍Λ p(z) = p(z1
    , · · · , zM
    ) ͱ͠ɺ֤εςοϓͰ
    ͸ z = (z1
    , · · · , zM
    )T ͷதͷҰͭͷ੒෼ zi
    Λ p(zi
    |z\i
    ) ͔ΒαϯϓϦϯ
    ά͠ɺߋ৽͢Δɻ
    ͜͜Ͱɺz\i
    ͸ z ͔Β zi
    ΛऔΓআ͍ͨ΋ͷͰ͋Δɻ
    1. {zi
    : i = 1, · · · , M} ΛॳظԽ͢Δ
    2. τ = 1, · · · , T ʹରͯ͠ҎԼΛߦ͏ɻ
    ▶ p(z1
    |z(τ)
    2
    , · · · , z(τ)
    M
    ) ͔Β z(τ+1)
    1
    ΛαϯϓϦϯά
    ▶ p(z2
    |z(τ+1)
    1
    , z(τ)
    3
    , · · · , z(τ)
    M
    ) ͔Β z(τ+1)
    2
    ΛαϯϓϦϯά
    .
    .
    .
    ▶ p(zM
    |z(τ+1)
    1
    , z(τ+1)
    2
    , · · · , z(τ+1)
    M−1
    ) ͔Β z(τ+1)
    M
    ΛαϯϓϦϯά
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40. 11.3 ΪϒεαϯϓϦϯά
    ຊདྷαϯϓϧΛٻΊ͍ͨ෼෍ p(z) = p(z1
    , · · · , zM
    ) ͔ΒͷαϯϓϦϯά
    ͸αϯϓϧۭ͕ؒߴ࣍ݩͰ͋ΔͨΊɺغ٫αϯϓϦϯάͳͲͰ͸αϯϓ
    ϦϯάͰ͖ͳ͍ɻ
    ͔͠͠ɺ৚݅෇͖෼෍ p(zi
    |z(τ+1)
    1
    , · · · , z(τ+1)
    i−1
    , z(τ)
    i+1
    , · · · , z(τ)
    M
    ) ͸αϯϓ
    ϧۭ͕ؒҰ࣍ݩͳͷͰɺغ٫αϯϓϦϯάͳͲͰαϯϓϧ z(τ+1)
    i
    Λಘ
    Δ͜ͱ͕Ͱ͖Δɻ
    ·ͨɺΪϒεαϯϓϦϯάͰͷ֤εςοϓͰ͸ɺzi
    ͷΈ͕มߋ͞ΕΔͷ
    ͰɺఏҊ෼෍͸ qi
    (z∗|z) = p(z∗
    i
    |z\i
    ) ͱͳΔɻ
    ͜͜Ͱɺz ͸εςοϓલͷม਺Ͱ z∗ ͸εςοϓલͷม਺Ͱ͋Γɺzi
    ͷ
    Έ͕มߋ͞ΕΔͷͰ z∗
    \i
    = z\i
    Ͱ͋Δɻ
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41. 11.3 ΪϒεαϯϓϦϯά
    ·ͨɺ͜ͷఏҊ෼෍ qk
    (z∗|z) = p(z∗
    k
    |z\k
    ) ͱҰൠతͳੑ࣭
    p(z) = p(zk
    |z\k
    )p(z\k
    ) Λ༻͍Δͱ
    p(z∗)qk
    (z|z∗)
    p(z)qk
    (z∗|z)
    =
    p(z∗
    k
    |z∗
    \k
    )p(z∗
    \k
    )p(zk
    |z∗
    \k
    )
    p(zk
    |z\k
    )p(z\k
    )p(z∗
    k
    |z\k
    )
    = 1 (11.49)
    ͱͳΔɻ
    ͜͜Ͱɺz∗
    \k
    = z\k
    Λ༻͍ͨɻ
    ͭ·Γɺ(11.44) ΑΓɺ
    Ak
    (z∗, z) = min
    (
    1,
    p(z∗)qk
    (z|z∗)
    p(z)qk
    (z∗|z)
    )
    = min(1, 1) = 1
    ͱͳΓɺΪϒεαϯϓϦϯά͸ Metropolis-Hastings ΞϧΰϦζϜͷಛ
    घͳ৔߹ (ৗʹडཧ) Ͱ͋Δ͜ͱ͕Θ͔Δɻ
    ΑͬͯɺΪϒεαϯϓϦϯάΛଓ͚Ε͹ɺαϯϓϧΛಘ͍ͨ෼෍ p(z)
    ͔ΒͷαϯϓϧΛಘΔ͜ͱ͕Ͱ͖Δ͜ͱ͕Θ͔Δɻ
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42. 11.4 εϥΠεαϯϓϦϯά
    Metropolis ΞϧΰϦζϜͷ೉఺͸εςοϓ෯Λখ͘͞औΓ͗͢Δͱϥϯ
    μϜ΢ΥʔΫΛى͜͠ɺେ͖͘औΓ͗͢Δͱغ٫཰্͕͕Δ͜ͱͰ
    ͋ͬͨɻ
    εϥΠεαϯϓϦϯάͰ͸ɺ෼෍ͷಛ௃ʹ߹Θͤͯεςοϓ෯Λࣗಈత
    ʹௐઅ͢Δ͜ͱ͕Ͱ͖Δɻ
    ͜͜Ͱ΋αϯϓϧΛಘ͍ͨ෼෍ͷܗ͸ਖ਼نԽఆ਺Λআ͍ͯΘ͔͍ͬͯ
    Δͱ͢Δɻ(p(z) ͷܗ͸Θ͔͍ͬͯΔ)
    ·ͨɺαϯϓϧۭؒ͸Ұ࣍ݩͱ͢Δɻ
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43. 11.4 εϥΠεαϯϓϦϯά
    αϯϓϧऔಘͷखॱͱͯ͠͸ɺݱࡏͷ఺Λ z ͱͨ͠ͱ͖ʹൣғ
    0 ≤ u ≤ p(z) ͔ΒҰ༷ʹ u ΛαϯϓϦϯά͢Δɻ
    ͦͯ͠ u Λݻఆͯ͠ɺp(z) > u ͱͳΔΑ͏ͳ z ͷྖҬ͔Β࣍ͷ z Λந
    ग़͢Δɻ(p(z) = u ͰεϥΠε)
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44. 11.4 εϥΠεαϯϓϦϯά
    ͜ͷαϯϓϦϯά͸ಉ࣌෼෍ ˆ
    p(z, u) ͔ΒαϯϓϦϯάΛߦ͏͜ͱͱ౳
    ͍͠ɻ
    ˆ
    p(z, u) =
    {
    1/Zp
    (0 ≤ u ≤ p(z))
    0 (otherwise)
    (11.51)
    ͨͩ͠ɺ
    Zp
    =
    ∫
    p(z) dz
    Ͱ͋Δɻ
    ·ͨɺz ͷपล෼෍͸ (11.51) ΑΓ
    ∫
    ˆ
    p(z, u) du =
    ∫
    p(z)
    0
    1
    Zp
    du =
    p(z)
    Zp
    = p(z)
    ͱͳΔͷͰɺαϯϓϧ (u, z) Λ ˆ
    p(z, u) ͔Βಘͯɺu Λແࢹ͢Ε͹ر๬
    ͷ෼෍ p(z) ͔Βͷαϯϓϧ z ͕ಘΒΕΔɻ
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45. 11.4 εϥΠεαϯϓϦϯά
    ͨͩ͠ɺ࣮ࡍ໰୊ p(z) > u ͱͳΔΑ͏ͳ z ͷྖҬ͔Β࣍ͷ z Λநग़͢
    Δ͜ͱ͕ࠔ೉ͳ͜ͱ͕ଟ͍ɻ
    ͦ͜ͰɺԼͷਤͷΑ͏ʹݱࡏͷ z ͷ஋Λ z(τ) ͱ͢Δͱɺͦͷ z(τ) ΛؚΉ
    ൣғ zmin
    ≤ z(τ) ≤ zmax
    ͷҰ༷෼෍͔Β z(τ+1) Λநग़͢Δํ๏͕͋Δɻ
    Ͱ͖Δ͚ͩ p(z) > u ͱͳΔΑ͏ͳ z Λ zmin
    ≤ z(τ) ≤ zmax
    ͷதʹؚΊ
    ΔΑ͏ʹൣғΛܾΊ͍͕ͨɺ޿͛͗͢Δͱغ٫཰্͕͕ͬͯ͠·͏໰୊
    ఺͕͋Δɻ
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46. 11.4 εϥΠεαϯϓϦϯά
    ͦ͜ͰɺൣғͷܾΊํͱͯ͠ɺ·ͣ z(τ) ΛؚΉΑ͏ʹϥϯμϜʹൣғ
    w ΛܾΊΔɻ
    ͦͷ w ͕εϥΠε֎ʹग़ΔΑ͏ʹ֦ு͢Δɻ
    ͦͷ֦ு͞Εͨൣғͷத͔Βαϯϓϧ z′ Λநग़͠ɺͦΕ͕εϥΠε಺
    ʹ͋Ε͹αϯϓϧͱ͢Δɻ(z(τ+1) = z′)
    ΋͠εϥΠε֎ʹ͋Ε͹ɺൣғ w Λ (z(τ) ΛؚΉΑ͏ʹ)z′ Λ୺఺ͱ͢
    ΔΑ͏ʹॖখ͢Δɻ
    ͦͯ͠৽ͨʹ z(τ+1) ͱͳΓ͏ΔީิΛൣғ w ͷத͔ΒऔΓग़͢ɻ
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47. 11.5 ϋΠϒϦουϞϯςΧϧϩΞϧΰϦζϜ
    ͜Ε·Ͱͷٞ࿦ͷதͰΘ͔ͬͨΑ͏ʹ Metropolis-Hastings ΞϧΰϦζ
    Ϝͷ໰୊఺͸ɺখ͞ͳεςοϓΛऔΔͱϥϯμϜ΢ΥʔΫΛҾ͖ى͜
    ͠ɺεςοϓΛେ͖͘͢Δͱغ٫཰্͕͕Δ͜ͱͰ͋Δɻ
    ͜͜Ͱ͸ɺ෺ཧγεςϜΛࢀߟʹ͠ɺغ٫཰Λখ͘͞อͬͨ··ɺε
    ςοϓΛେ͖͘ͱΔํ๏Λߟ͑Δɻ
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48. 11.5.1 ྗֶܥ
    ·ͣɺϋϛϧτϯྗֶʹ͍ͭͯઆ໌͢Δɻ
    ͜Ε·Ͱ཭ࢄతͳεςοϓ τ Ͱͷ֬཰ม਺Λ z(τ) ͱ͕ͨ͠ɺτ Λ࿈ଓ
    ม਺ʹ֦ு͠ɺ֬཰ม਺Λ τ ͷؔ਺ z(τ) ͱ͢Δɻ
    ྗֶͰݴ͑͹ɺ͜ͷ τ ͕࣌ؒͰɺz(τ) ͕෺ମͷҐஔϕΫτϧͱͳΔɻ
    ·ͨɺz(τ) ͷ τ ඍ෼
    r =
    dz
    dτ
    (11.53)
    ͸ӡಈྔ (෺ମͷ଎౓ͱಉ౳ͷྔ) Ͱ͋Δɻ
    ·ͨɺܥͷϙςϯγϟϧΤωϧΪʔΛ E(z) ͱ͢Δͱɺ෺ମ͸ҎԼͷӡ
    ಈํఔࣜʹै͍ӡಈΛߦ͏͜ͱ͕஌ΒΕ͍ͯΔɻ
    dr
    dτ
    = −
    ∂E(z)
    ∂z
    (11.55)
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49. 11.5.1 ྗֶܥ
    ͜ΕΛϋϛϧτϯܗࣜͰ·ͱΊͳ͓͢ɻ
    ·ͣ͸ӡಈΤωϧΪʔ K(r) ΛҎԼͷΑ͏ʹఆٛ͢Δɻ
    K(r) =
    1
    2
    ∥r∥2 (11.56)
    ͞ΒʹɺϙςϯγϟϧΤωϧΪʔ E(z) ͱӡಈΤωϧΪʔ K(r) Λ଍͠
    ͨશΤωϧΪʔΛҎԼͰఆٛ͢Δɻ
    H(z, r) = E(z) + K(r) (11.57)
    ͜͜ͰɺશΤωϧΪʔ H(z, r) ͸ϋϛϧτϯؔ਺ͱ͍͏ɻ
    ͜ͷஈ֊Ͱ͸ɺҐஔϕΫτϧ z ͱӡಈྔϕΫτϧ r ͸ಠཱͳϕΫτϧͰ
    ͋Δ͜ͱʹ஫ҙɻ((11.53) ͱ͍͏ؔ܎͸ຬͨ͞ͳ͍ɻ)
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50. 11.5.1 ྗֶܥ
    ͜ͷϋϛϧτϯؔ਺Λ༻͍Δͱɺܥͷํఔࣜ (11.53) ͱ (11.55) ͸ҎԼ
    ͷ 2 ͭͷࣜͰදͤΔ͜ͱ͕Θ͔Δɻ
    dz
    dτ
    =
    ∂H
    ∂r
    (11.58)
    dr
    dτ
    = −
    ∂H
    ∂z
    (11.59)
    ͞Βʹɺ(11.58) ͱ (11.59) Λຬͨ࣌͢ɺϋϛϧτϯؔ਺ͷ࣌ؒඍ෼Λܭ
    ࢉ͢ΔͱҎԼͷΑ͏ʹ 0 ͱͳΔɻ
    dH(z, r)
    dτ
    = 0 (11.60)
    ͭ·Γɺ(11.58) ͱ (11.59) Λຬͨ͢ӡಈ͸ɺશΤωϧΪʔ H(z, r) Λอ
    ଘ͢Δɻ
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51. 11.5.1 ྗֶܥ
    ͞ΒʹɺԼͷਤͷΑ͏ͳ (z, r) ۭؒ (Ґ૬ۭؒ) ͷ͋ΔྖҬΛߟ͑Δɻ
    ͜ͷྖҬͷ֤఺͕ํఔࣜ (11.58) ͱ (11.59) Λຬͨ͠ͳ͕ΒҠಈͨ͠ޙ
    ͷମੵ͸ɺݩͷମੵͱಉ͡Ͱ͋Δ͜ͱ͕஌ΒΕ͍ͯΔɻ(Ϧ΢ϰΟϧͷ
    ఆཧ)
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52. 11.5.1 ྗֶܥ
    Ҏ্ͷϋϛϧτϯؔ਺ͷ࣌ؒෆมੑͱϦ΢ϰΟϧͷఆཧ͔ΒɺҎԼͷΑ
    ͏ʹఆٛ͞Εͨ֬཰෼෍͸Ґ૬্ۭؒͷ (11.58) ͱ (11.59) Λຬͨ͢ม
    Խʹରͯ͠ෆมͰ͋Δ͜ͱ͕Θ͔Δɻ
    p(z, r) =
    1
    ZH
    exp (−H(z, r)) (11.63)
    ͜͜Ͱɺ֬཰෼෍͸ม਺ม׵ޙ΋ن֨Խ͕੒ཱ͍ͯ͠ͳ͍ͱ͍͚ͳ͍
    (۩ମతʹ͸ PRML(1.27) ͷΑ͏ͳม׵Λ͢Δ) ͷͰɺϋϛϧτϯؔ਺ͷ
    ࣌ؒෆมੑ͚ͩͰ͸ p(z, r) ͸ෆมʹͳΒͳ͍͜ͱʹ஫ҙɻ
    (11.58) ͱ (11.59) Λຬͨ͢ (p(z, r) Λෆมʹ͢Δ)z ͱ r ͷ਺஋తͳ࣌
    ؒมԽ (਺஋ੵ෼) ͷ༩͑ํͱͯ͠ɺϦʔϓϑϩοά཭ࢄԽͱ͍͏ํ๏
    ͕͋Δɻ
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53. 11.5.1 ྗֶܥ
    Ϧʔϓϑϩοά཭ࢄԽͱ͸ɺҎԼͷΑ͏ͳߋ৽ํ๏Ͱ͋Δɻ
    ˆ
    r(τ + ϵ/2) =ˆ
    r(τ) −
    ϵ
    2
    ∂E
    ∂z
    (ˆ
    z(τ)) (11.64)
    ˆ
    z(τ + ϵ) =ˆ
    z(τ) + ϵˆ
    r(τ + ϵ/2) (11.65)
    ˆ
    r(τ + ϵ) =ˆ
    r(τ + ϵ/2) −
    ϵ
    2
    ∂E
    ∂z
    (ˆ
    z(τ + ϵ)) (11.66)
    ͜ͷߋ৽͸਺஋ੵ෼ʹ΋͔͔ΘΒͣϦ΢ϰΟϧͷఆཧΛຬͨ͢ߋ৽ํ
    ๏Ͱ͋Δɻ͔͠͠ɺH ͷ஋ʹ͸ ϵ ෼ͷޡ͕ࠩग़Δɻ
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54. 11.5.2 ϋΠϒϦουϞϯςΧϧϩΞϧΰϦζϜ
    Ҏ্ͷϋϛϧτϯྗֶͱ Metropolis ΞϧΰϦζϜΛ༥߹ͤͨ͞ϋΠϒ
    ϦουϞϯςΧϧϩΞϧΰϦζϜΛಋग़͢Δɻ
    ·ͣɺॳظঢ়ଶΛ (z, r) ͱ͠ɺϦʔϓϑϩοά཭ࢄԽͰߋ৽ͨ͠ม਺Λ
    (z⋆, r⋆) ͱ͠ɺҎԼͷ֬཰Ͱडཧ͢Δɻ
    min(1, exp {H(z, r) − H(z⋆, r⋆)}) (11.67)
    ΋͠ϋϛϧτϯྗֶΛ׬શʹγϛϡϨʔτͨ͠ΒɺH ͸ෆมͳͷͰඞ
    ͣडཧ͞ΕΔ͕ɺϦʔϓϑϩοά཭ࢄԽΛ༻͍ͨΒɺ਺஋ੵ෼ʹΑΔޡ
    ࠩʹΑΓɺH ͕มԽ͢Δ৔߹͕͋ΔͷͰɺغ٫͞ΕΔ͜ͱ͕͋Δɻ
    ·ͨɺৄࡉ௼Γ߹͍৚͕݅੒ཱ͢Δ͔Ͳ͏͔͸ԋश 11.17 Ͱٞ࿦ͯ͠
    ͍Δɻ
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55. 11.6 ෼഑ؔ਺ͷਪఆ
    ຊষͷ΄ͱΜͲͷ৔߹ͰɺαϯϓϧΛٻΊ͍ͨ෼෍͸ن֨Խఆ਺Λআ͍
    ͯΘ͔͍ͬͯΔͱԾఆͨ͠ɻ
    ͭ·ΓɺαϯϓϧΛٻΊ͍ͨ෼෍ pE
    (z) Λ
    pE
    (z) =
    1
    ZE
    exp (−E(z)) (11.71)
    ͱͨ࣌͠ʹ E(z) ͷؔ਺ܗ͸Θ͔͍ͬͯΔ͕ɺZE
    ͷ஋͸Θ͔Βͳ͍ͱ
    ͍͏͜ͱͰ͋Δɻ
    ͞Βʹɺ͜Ε·Ͱͷ͍ΖΜͳαϯϓϦϯά๏Ͱ͸ ZE
    ͸Θ͔Βͳͯ͘
    ΋ɺۙࣅతʹ pE
    (z) ͔ΒαϯϓϦϯάͰ͖͍ͯͨɻ
    ͨͩ͠ɺZE
    ͷ஋͕Θ͔ΔͱϞσϧͷൺֱ͕Ͱ͖ɺศརͳ͕࣌͋Δɻ
    ͜͜Ͱ͸ɺZE
    ͷ஋Λ஌Δํ๏Λઆ໌͢Δɻ
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56. 11.6 ෼഑ؔ਺ͷਪఆ
    ϞσϧൺֱͰ஌Γ͍ͨͷ͸ɺ2 ͭͷϞσϧͷ෼഑ؔ਺ͷൺͰ͋Δɻ
    ͦ͜ͰɺҎԼͷ෼෍ pG
    (z) Λߟ͑Δɻ
    pG
    (z) =
    1
    ZG
    exp (−G(z))
    ͜͜ͰɺpG
    (z) ͔ΒͷαϯϓϦϯά͸ՄೳͰ͋ΔͱԾఆ͢Δɻ
    ͜ͷ࣌ɺൺ ZE
    /ZG
    ͸ pG
    (z) ͔Βͷαϯϓϧͷू߹ {z(l)} Λ༻͍ͯҎԼ
    ͷΑ͏ʹܭࢉͰ͖Δɻ
    ZE
    ZG
    =
    ∑
    z
    exp (−E(z))
    ∑
    z
    exp (−G(z))
    =
    ∑
    z
    exp (−E(z) + G(z)) exp (−G(z))
    ∑
    z
    exp (−G(z))
    =
    ∑
    z
    exp (−E(z) + G(z))pG
    (z)
    =EpG
    [exp (−E + G)]
    ∼
    1
    L
    ∑
    l
    exp (−E(z(l)) + G(z(l)))
    (11.72)
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