ベイズで単回帰モデルを考える /bayes-simple-linear-regression

1. ϕΠζͰ୯ճؼϞσϧΛߟ͑Δ 5IJNCMFF 1
2. ໰୊ઃఆ ܇࿅σʔλͷઆ໌ม਺ ͱ໨తม਺ ͔ΒҎԼͷ ༧ଌ෼෍ΛٻΊΔ
   
   ҎԼͷ୯ճؼϞσϧΛ࢖༻͢Δ
   x = (x1 , x2 , ⋯, xN )T t = (t1 , t2 , ⋯, tN )T p(t* |x* , t, x) p(t* |x* , w, β) = 𝒩 (t* |w0 + w1 x* , β−1) 2
3. ۩ମతʹ͸ ͜͏͍͏σʔλʹର͍͍ͯ͠ײ͡ʹύϥϝʔλ Λௐ੔ͯ͠ɺ௚ઢ ΛҾ͖͍ͨɻ͜ͷσʔλ͸େମ ͱͳ͍ͬͯΔɻ
   w = (w0 , w1 )T y = w0 + w1 x t = − 2 + 2x 3
4. ϕΠζͷఆཧ p(A|B) = p(A)p(B|A) p(B) 4
5. ࣄޙ෼෍ QPTUFSJPS ͸ύϥϝʔλɺ ͸σʔλ
   
   
   ࣄޙ෼෍Λ༻͍ͨύϥϝʔλͷਪఆ͕ϕΠζਪఆͰ͢ɻ w t p(w|t) = p(w)p(t|w) p(t) ∝ p(w)p(t|w) (posterior) ∝ (prior)(likelihood) 5
6. ࣄલ෼෍ QSJPS
   p(w) = 𝒩 (w|0, α−1I), α = 0.25 6
7. ໬౓ؔ਺ MJLFMJIPPE
   L(w) = p(t|w) = 𝒩 (t|m, β−1I) where m = (w0 + w1 x1 , w0 + w1 x2 , ⋯, w0 + w1 xN )T, β = 2.0 7
8. ໬౓ؔ਺ͷྫ ͜ͷΑ͏ͳ͍͍ײ͡ͷ௚ઢͩͱͱ͍͏େ͖͍஋ΛͱΔ
   
   L((−2.1,2.2)T) = 0.39 8
9. ໬౓ؔ਺ͷྫ ͜ͷΑ͏ͳ௚ઢͩͱͱ͍͏஋ΛͱΔ
   
   L((−1.0,0.0)T) = 0.29 9
10. ໬౓ؔ਺ͷྫ ͜ͷΑ͏ͳѱ͍௚ઢͩͱͱ͍͏খ͍͞஋ΛͱΔ
    
    L((1.0, − 3.0)T) = 0.18 10
11. ࣄલ෼෍ͱ໬౓ؔ਺ʢ࠶ܝʣ ໬౓ؔ਺ MJLFMJIPPE ࣄલ෼෍ QSJPS 11
12. ࣄલ෼෍ͱ໬౓ؔ਺ͷੵ 12
13. ٻΊΒΕͨ௚ઢ ࣄޙ෼෍͸ ͷ࣌ʹ࠷େʹͳΔ
    
    w = (−1.08,0.38) L((−1.08,0.38)T) = 0.31 13
14. ໬౓ؔ਺͚ͩʢ࠷໬ਪఆʣͰ͍͍ͷͰ͸ ϕΠζʢࣄޙ෼෍ʣͩͱσʔλΛ͏·͘දݱͰ͖͍ͯͳ͍
    ࣮ࡍɺ͜ͷσʔλΛ୯ճؼϞσϧͰֶश͢Δࡍʹ͸ϕΠζ͸ඞཁͳ͍
    ʢ୯ճؼϞσϧ͕ཧղ͠΍͍͔͢Β༻͍ͨʣ
    ͔͠͠ɺҰൠʹ͸ϕΠζͰߟ͑ΔϝϦοτ͕ଟ͍ 14
15. ϕΠζͷಛ௃ w ύϥϝʔλʢ୯ճؼϞσϧͳΒ ʣʹ͍ͭͯ΋֬཰෼෍Λߟ͑ΒΕΔ
    w ࣄલ෼෍ʹʢσʔλҎ֎ͷʣط஌ͷ৘ใΛ൓өͤ͞ΒΕΔ
    w ֬཰෼෍ͷஞ࣍ߋ৽͕Ͱ͖Δ
    w աֶशΛ๷͛Δʢਖ਼ଇԽʣ
    w ʢଞʹ΋৭ʑ͋Δͱࢥ͍·͢ʣ w 15
16. ͝ਗ਼ௌ͋Γ͕ͱ͏͍͟͝·ͨ͠ 16