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ベイズで単回帰モデルを考える /bayes-simple-linear-regression
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Thimblee
November 09, 2022
Technology
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ベイズで単回帰モデルを考える /bayes-simple-linear-regression
Thimblee
November 09, 2022
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Transcript
ϕΠζͰ୯ճؼϞσϧΛߟ͑Δ 5IJNCMFF 1
ઃఆ ܇࿅σʔλͷઆ໌ม ͱతม ͔ΒҎԼͷ ༧ଌΛٻΊΔ 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
۩ମతʹ ͜͏͍͏σʔλʹର͍͍ͯ͠ײ͡ʹύϥϝʔλ Λௐͯ͠ɺઢ ΛҾ͖͍ͨɻ͜ͷσʔλେମ ͱͳ͍ͬͯΔɻ w = (w0 , w1
)T y = w0 + w1 x t = − 2 + 2x 3
ϕΠζͷఆཧ p(A|B) = p(A)p(B|A) p(B) 4
ࣄޙ QPTUFSJPS ύϥϝʔλɺ σʔλ ࣄޙΛ༻͍ͨύϥϝʔλͷਪఆ͕ϕΠζਪఆͰ͢ɻ w t p(w|t)
= p(w)p(t|w) p(t) ∝ p(w)p(t|w) (posterior) ∝ (prior)(likelihood) 5
ࣄલ QSJPS p(w) = 𝒩 (w|0, α−1I), α =
0.25 6
ؔ 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
ؔͷྫ ͜ͷΑ͏ͳ͍͍ײ͡ͷઢͩͱͱ͍͏େ͖͍ΛͱΔ L((−2.1,2.2)T) = 0.39 8
ؔͷྫ ͜ͷΑ͏ͳઢͩͱͱ͍͏ΛͱΔ L((−1.0,0.0)T) = 0.29 9
ؔͷྫ ͜ͷΑ͏ͳѱ͍ઢͩͱͱ͍͏খ͍͞ΛͱΔ L((1.0, − 3.0)T) = 0.18 10
ࣄલͱؔʢ࠶ܝʣ ؔ MJLFMJIPPE ࣄલ QSJPS 11
ࣄલͱؔͷੵ 12
ٻΊΒΕͨઢ ࣄޙ ͷ࣌ʹ࠷େʹͳΔ w = (−1.08,0.38) L((−1.08,0.38)T) = 0.31
13
͚ؔͩʢ࠷ਪఆʣͰ͍͍ͷͰ ϕΠζʢࣄޙʣͩͱσʔλΛ͏·͘දݱͰ͖͍ͯͳ͍ ࣮ࡍɺ͜ͷσʔλΛ୯ճؼϞσϧͰֶश͢ΔࡍʹϕΠζඞཁͳ͍ ʢ୯ճؼϞσϧ͕ཧղ͍͔͢͠Β༻͍ͨʣ ͔͠͠ɺҰൠʹϕΠζͰߟ͑ΔϝϦοτ͕ଟ͍ 14
ϕΠζͷಛ w ύϥϝʔλʢ୯ճؼϞσϧͳΒ ʣʹ͍ͭͯ֬Λߟ͑ΒΕΔ w ࣄલʹʢσʔλҎ֎ͷʣطͷใΛөͤ͞ΒΕΔ w ֬ͷஞ࣍ߋ৽͕Ͱ͖Δ w աֶशΛ͛Δʢਖ਼ଇԽʣ
w ʢଞʹ৭ʑ͋Δͱࢥ͍·͢ʣ w 15
͝ਗ਼ௌ͋Γ͕ͱ͏͍͟͝·ͨ͠ 16