基礎数学の公式

「ベイズ統計の理論と方法」勉強会の資料

Transcript

1. جૅ਺ֶͷެࣜ
   ϕΠζ౷ܭͷཧ࿦ͱํ๏ษڧձ
   

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2. సஔߦྻ τϨʔε ߦྻࣜ
   

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3. సஔߦྻɺϕΫτϧ
   Ұൠʹ ߦྻ ʹ͍ͭͯɺ
   ͦͷసஔߦྻ ͱ͸ ߦྻͰ ͷ͜ͱͰ͋Δɻ
   k × d A = (Aij
   )
   AT d × k A = (Aji
   )
   ࣍ݩϕΫτϧ Λ ߦྻͱߟ͑Δɻ͜ΕΛॎϕΫτϧͱ͍͏ɻ
   ·ͨɺ͜ͷͱ͖ԣϕΫτϧ ͸ ߦྻͰ͋Δɻ
   d v d × 1
   vT 1 × d
   

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4. τϨʔε
   ߦྻ ͷτϨʔεΛ
   d × d A
   
   tr(A) =
   d
   ∑
   i=1
   Aii
   ͱॻ͘ɻ
   tr(AB) = tr(BA)
   ͕੒Γཱͭɻ
   ߦྻ ʹ͍ͭͯҰൠʹ ͕ͩɺҰൠʹ
   d × d A, B AB ≠ BA
   ର֯੒෼ͷ࿨
   

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5. ಺ੵ
   Λ ࣍ݩͷϕΫτϧͱ͢Δͱ͖ɺͦͷ಺ੵΛ
   u, v d
   u ⋅ v =
   d
   ∑
   i=1
   ui
   vi
   ͱॻ͘ɻ
   

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6. ϊϧϜ
   ϕΫτϧ ͷϊϧϜΛ Ͱද͢ɻ Ͱ͋Δɻ
   u ∥u∥ ∥u∥ = u ⋅ u
   u ⋅ v = uTv = tr(v ⋅ uT)
   ͕੒Γཱͭɻ͜ΕΑΓ
   (u ⋅ v)2 = vTuuTv = tr(uuTvvT)
   ͕ಘΒΕΔɻ·ͨ
   u ⋅ Av = (ATu) ⋅ v = tr(AvuT)
   ͕੒Γཱͭɻ
   

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7. ߦྻࣜ
   ߦྻ ͷߦྻࣜΛ
   d × d A
   det(A) = ∑
   σ
   sgn(σ)A1σ(1)
   A2σ(2)
   ⋯Adσ(d)
   ͱॻ͘ɻ
   ͜͜Ͱ ͸ཁૉͷ਺͕ ݸͷू߹͔Βࣗ෼ࣗ਎΁ͷશ୯ࣹ ஔ׵
   Λද͢ɻ
   ஔ׵ͷݸ਺͸શ෦Ͱ ݸͰ͋Δ͕ɺ ͸ஔ׵શମͷू߹ʹର͢Δ࿨Ͱ͋Γɺ
   σ d
   d! ∑
   σ
   ྫ
   ͸حஔ׵ͷͱ͖ ɺۮஔ׵ͷͱ͖ Ͱ͋Δɻ
   sgn(σ) −1 1
   IUUQTPHVFNPODPNTUVEZMJOFBSBMHFCSBEFUXIBU
   

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8. ߦྻࣜ
   ߦྻ ʹ͍ͭͯ
   d × d A, B
   det(AB) = det(A) det(B)
   ͕੒Γཱͭɻ
   

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9. 
   ରশߦྻ ݻ༗஋ ਖ਼ఆ஋ߦྻ
   

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10. ਖ਼ଇɾٯߦྻɾରশߦྻɾ௚ߦߦྻ
    ߦྻ ͕Մٯ͋Δ͍͸ਖ਼ଇͰ͋Δͱ͸
    ͕୯ҐߦྻͱͳΔΑ͏ͳߦྻ ͕ଘࡏ͢Δ͜ͱͰ͋Δɻ
    d × d A
    A−1A A−1
    ͕ՄٯͰ͋Δͱ͖ Λ ͷٯߦྻͱ͍͏ɻ
    A A−1 A
    ࣮਺Λཁૉʹ࣋ͭ ߦྻ ͕ରশߦྻͰ͋Δͱ͸ɺ
    ͕੒Γཱͭ͜ͱͰ͋Δɻ
    d × d A = (Aij
    )
    A = AT
    ࣮਺Λཁૉʹ࣋ͭ ߦྻ ͕௚ߦߦྻͰ͋Δͱ͸ɺ
    ͕୯ҐߦྻͰ͋Δ͜ͱͰ͋Δɻ
    d × d R = (Rij
    )
    RTR
    

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11. ର֯ߦྻɾର֯Խ
    ߦྻ ͕ର֯ߦྻͰ͋Δͱ͸ ͕੒Γཱͭ͜ͱͰ͋Δɻ
    d × d A i ≠ j ⇒ Aij
    = 0
    ೚ҙͷରশߦྻ ʹରͯ͠ Λର֯ߦྻʹ͢ΔΑ͏ͳ௚ߦߦྻ ͕ଘࡏ͢Δɻ
    ͢ͳΘͪ
    A R−1AR R
    ͱग़དྷΔɻ͜ͷͱ͖ର֯ߦྻ ΛٻΊΔ͜ͱΛʮ Λର֯Խ͢Δʯͱ͍͏ɻ
    R−1AR A
    

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12. δϣϧμϯඪ४ܗ
    ର֯ߦྻͰͳ͍ߦྻ͸Ұൠతʹ͸ର֯Խग़དྷΔͱ͸ݶΒͳ͍͕ɺ
    ՄٯߦྻΛ༻͍ͯδϣϧμϯඪ४ܗʹ͢Δ͜ͱ͕ग़དྷΔɻ
    δϣϧμϯࡉ๔ δϣϧμϯඪ४ܗ
    

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13. ݻ༗஋ɾਖ਼ఆ஋ߦྻ
    ߦྻ ʹ͍ͭͯɺෳૉ਺ ͱෳૉ਺Λཁૉͱ͢ΔϕΫτϧ ͕
    ଘࡏͯ͠
    d × d A λ v ≠ 0
    ͕੒Γཱͭͱ͖ɺ Λ ͷݻ༗஋ͱ͍͍ɺ Λ ͷݻ༗ϕΫτϧͱ͍͏ɻ
    ରশߦྻ ͷݻ༗஋͕શͯ ΑΓେ͖͍ͱ͖ɺ Λਖ਼ఆ஋ߦྻͰ͋Δͱ͍͏ɻ
    λ A v A
    A 0 A
    Av = λv
    ͕ਖ਼ఆ஋ߦྻͰ͋Ε͹ೋ࣍ࣜʹ͍ͭͯฏํ׬੒͕ग़དྷΔɻ
    A
    1
    2
    (u ⋅ Au) − u ⋅ v =
    1
    2
    ∥A1/2(u − A−1v)∥2 −
    1
    2
    ∥A−1/2v∥2
    

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14. ੵ෼ެࣜ
    

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15. ਖ਼ن෼෍
    ͱ͠ɺ Λ ͷਖ਼ఆ஋ߦྻͰ͋Δͱ͢Δɻ͜ͷͱ͖
    w ∈ ℝd A d × d
    ∫
    exp(−
    n
    2
    w ⋅ A−1w)dw =
    (2π)d/2 det(A)1/2
    nd/2
    ฏۉ͕ Ͱ෼ࢄڞ෼ࢄߦྻ͕ ͷਖ਼ن෼෍͸
    a ∈ ℝd A
    (a, A) = p(w) =
    1
    (2π)d/2 det(A)1/2
    exp(−
    1
    2
    (w − a) ⋅ A−1(w − a))
    ͱ͍͏ࣜͰද͞ΕΔɻ
    

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16. ਖ਼ن෼෍
    ͜ͷͱ͖
    ∫
    wp(w)dw = a
    Ͱ͋Γɺ೚ҙͷ ߦྻ ʹ͍ͭͯ
    d × d B
    ∫
    (w − a) ⋅ B(w − a)p(w)dw = tr(BA)
    ͷ֬཰෼෍͸ Ͱ͋Δɻ
    (Ba + b, BABT)
    Ͱ͋Δɻ Λ ͷՄٯߦྻͱ͢Δɻ
    ֬཰ม਺ ͷ֬཰෼෍͕ Ͱ͋Δͱ͖֬཰ม਺
    B d × d
    X (a, A)
    Y = BX + b
    

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17. ฏۉ஋ͷఆཧ
    

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18. ଟม਺͔ΒͳΔؔ਺ͷඍ෼ͷه๏
    Λඇෛͷ੔਺ͱ͢Δɻଟॏࢦ਺ ʹ͍ͭͯ࣍ͷΑ͏ʹఆΊΔɻ
    k1
    , …, kd
    k = (k1
    , …, kd
    )
    |k| = k1
    + k2
    + ⋯ + kd
    , k! = (k1
    )!(k2
    )!⋯(kd
    )!
    ͷ ΛؚΉ։ू߹্Ͱఆٛ͞Εͨ
    ճ࿈ଓඍ෼Մೳͳؔ਺ ʹ͍ͭͯ ͷͱ͖
    ℝd w = w0
    (r + 1) g(w) |k| ≤ r + 1
    ∂kg
    ∂wk
    (w) = (
    d
    ∏
    j=1
    ∂kj
    ∂wkj
    j
    )g(w),
    ͱఆٛ͢Δɻ
    ͜ΕΒ͸ଟม਺ͷؔ਺ͷඍ෼ʹ͓͍ͯҰൠతʹར༻͞ΕΔه๏Ͱ͋Δɻ
    (w − w0
    )k =
    d
    ∏
    j=1
    (wj
    − (w0
    )j
    )kj
    

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19. ฏۉ஋ͷఆཧ
    ೚ҙͷ ʹରͯ͠ɺ ʹΑΔఆ·Δ ͕ଘࡏͯ͠
    w w w*
    g(w) = ∑
    |k|≤r
    ∂kg
    ∂wk
    (w0
    )
    (w − w0
    )k
    k!
    + ∑
    |k|=r+1
    ∂kg
    ∂wk
    (w*)
    (w − w0
    )k
    k!
    ͕੒ཱ͢Δɻ
    ͜Ε͕ฏۉ஋ͷఆཧͰ͋Γɺؔ਺ͷมԽͷฏۉతͳڍಈʹ͍ͭͯͷఆཧͰ͋Δɻ
    ·ͨ͜͜Ͱ ͸͋Δ ͕ଘࡏͯ͠
    w* 0 < θ < 1
    w* = w0
    + θ(w − w0
    )
    ͱද͞ΕΔɻ
    ಛʹ Ͱ͋Δ͔Βɺ ͷͱ͖ ͕੒Γཱͭɻ
    ∥w* − w0
    ∥ ≤ ∥w − w0
    ∥ w → w0
    w* → w0
    

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