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

Transcript

1. ϕΠζͰ୯ճؼϞσϧΛߟ͑Δ
   5IJNCMFF
   1
   

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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
   

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3. ۩ମతʹ͸
   ͜͏͍͏σʔλʹର͍͍ͯ͠ײ͡ʹύϥϝʔλ Λௐ੔ͯ͠ɺ௚ઢ
   ΛҾ͖͍ͨɻ͜ͷσʔλ͸େମ ͱͳ͍ͬͯΔɻ
   w = (w0
   , w1
   )T
   y = w0
   + w1
   x t = − 2 + 2x
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4. ϕΠζͷఆཧ
   p(A|B) =
   p(A)p(B|A)
   p(B)
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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)
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6. ࣄલ෼෍ QSJPS
   
   
   p(w) =
   𝒩
   (w|0, α−1I), α = 0.25
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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
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8. ໬౓ؔ਺ͷྫ
   ͜ͷΑ͏ͳ͍͍ײ͡ͷ௚ઢͩͱͱ͍͏େ͖͍஋ΛͱΔ
   
   L((−2.1,2.2)T) = 0.39
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9. ໬౓ؔ਺ͷྫ
   ͜ͷΑ͏ͳ௚ઢͩͱͱ͍͏஋ΛͱΔ
   
   L((−1.0,0.0)T) = 0.29
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10. ໬౓ؔ਺ͷྫ
    ͜ͷΑ͏ͳѱ͍௚ઢͩͱͱ͍͏খ͍͞஋ΛͱΔ
    
    L((1.0, − 3.0)T) = 0.18
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11. ࣄલ෼෍ͱ໬౓ؔ਺ʢ࠶ܝʣ
    ໬౓ؔ਺ MJLFMJIPPE
    
    ࣄલ෼෍ QSJPS
    
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12. ࣄલ෼෍ͱ໬౓ؔ਺ͷੵ
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13. ٻΊΒΕͨ௚ઢ
    ࣄޙ෼෍͸ ͷ࣌ʹ࠷େʹͳΔ
    
    w = (−1.08,0.38)
    L((−1.08,0.38)T) = 0.31
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14. ໬౓ؔ਺͚ͩʢ࠷໬ਪఆʣͰ͍͍ͷͰ͸
    ϕΠζʢࣄޙ෼෍ʣͩͱσʔλΛ͏·͘දݱͰ͖͍ͯͳ͍
    ࣮ࡍɺ͜ͷσʔλΛ୯ճؼϞσϧͰֶश͢Δࡍʹ͸ϕΠζ͸ඞཁͳ͍
    ʢ୯ճؼϞσϧ͕ཧղ͠΍͍͔͢Β༻͍ͨʣ
    ͔͠͠ɺҰൠʹ͸ϕΠζͰߟ͑ΔϝϦοτ͕ଟ͍
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15. ϕΠζͷಛ௃
    w ύϥϝʔλʢ୯ճؼϞσϧͳΒ ʣʹ͍ͭͯ΋֬཰෼෍Λߟ͑ΒΕΔ
    w ࣄલ෼෍ʹʢσʔλҎ֎ͷʣط஌ͷ৘ใΛ൓өͤ͞ΒΕΔ
    w ֬཰෼෍ͷஞ࣍ߋ৽͕Ͱ͖Δ
    w աֶशΛ๷͛Δʢਖ਼ଇԽʣ
    w ʢଞʹ΋৭ʑ͋Δͱࢥ͍·͢ʣ
    w
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16. ͝ਗ਼ௌ͋Γ͕ͱ͏͍͟͝·ͨ͠
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