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
2. 
   సஔߦྻ
   τϨʔε
   ߦྻࣜ
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
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࿨
5. ಺ੵ
   Λ
   
   ࣍ݩͷϕΫτϧͱ͢Δͱ͖ɺͦͷ಺ੵΛ u, v d
   u ⋅ v = d ∑ i=1 ui vi ͱॻ͘ɻ
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
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
8. ߦྻࣜ
   ߦྻ
   
   ʹ͍ͭͯ d × d A, B det(AB) = det(A) det(B) ͕੒Γཱͭɻ
9. 
   
   ରশߦྻ
   ݻ༗஋
   ਖ਼ఆ஋ߦྻ
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
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
12. δϣϧμϯඪ४ܗ ର֯ߦྻͰͳ͍ߦྻ͸Ұൠతʹ͸ର֯Խग़དྷΔͱ͸ݶΒͳ͍͕ɺ
    ՄٯߦྻΛ༻͍ͯδϣϧμϯඪ४ܗʹ͢Δ͜ͱ͕ग़དྷΔɻ δϣϧμϯࡉ๔ δϣϧμϯඪ४ܗ
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
14. 
    ੵ෼ެࣜ
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
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
17. 
    ฏۉ஋ͷఆཧ
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
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