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ベイズ統計モデリング 10 // Doing Bayesian Data Analysis Chapter 10

todesking
August 24, 2018

ベイズ統計モデリング 10 // Doing Bayesian Data Analysis Chapter 10

todesking

August 24, 2018
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  1. ϕΠζ౷ܭϞσϦϯά
    Chapter 10
    @todesking

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  2. ϕΠδΞϯϞσϧൺֱ
    • ؍ଌ͞ΕͨσʔλΛઆ໌͢ΔͨΊͷɺෳ਺ͷϞσϧ͕ߟ͑
    ΒΕΔ

    • ͲͷϞσϧ͕ΑΓ΋ͬͱ΋Β͍͔͠?

    • ࠓ·ͰֶΜͰ͖ͨϕΠζϞσϦϯάʹΑͬͯɺʮϞσϧΛ
    ൺֱ͢ΔϞσϧʯΛߏ੒͢Δ͜ͱ͕Ͱ͖Δ

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  3. 10.1 ҰൠࣜͱϕΠζϑΝΫλʔ
    • ؍ଌ͞Εͨσʔλ(D)Λઆ໌͢Δ2ͭͷϞσϧΛߟ͑Δ

    • ֤Ϟσϧ͸ɺࣄલ෼෍P(θ)ɺ໬౓P(D|θ)͓Αͼύϥϝʔλθ͔ΒͳΔ
    D
    θ1
    θ1
    ∼ P1
    (θ1
    )
    D ∼ P1
    (D|θ1
    )
    D
    θ2
    θ2
    ∼ P2
    (θ2
    )
    D ∼ P2
    (D|θ2
    )

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  4. Ϟσϧͷந৅දݱ
    • Ϟσϧ͕ෳ਺ͷม਺͔Βߏ੒͞Ε͍ͯͯ΋ɺந৅Խ͢Ε͹
    θ, P(θ), P(D|θ) ͰදݱͰ͖Δ(ಠࣗݚڀ)
    D
    θ1
    θ = (φ1
    , φ2
    , φ3
    )
    D = (X, Y)
    P(θ) = P(φ1
    , φ2
    , φ3
    )
    P(D|θ) = P(X, Y|φ1
    , φ2
    , φ3
    )
    X
    φ1
    φ3
    Y
    φ2

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  5. ϞσϧൺֱͷͨΊͷϞσϧ
    • Ϟσϧબ୒ม਺mΛಋೖͯ͠ɺ2ͭͷϞσϧΛ·ͱΊΔ

    • ϞσϧൺֱͷͨΊͷϞσϧͱͳΔ

    • fig 10.1Ͱ͸ɺ໬౓ؔ਺͕ผʑͷέʔε(தԝ)ɺڞ௨͍ͯ͠Δέʔε(ӈ)ɺͦ
    ͷҰൠԽ͍Δͷ͔?ͷέʔε(ࠨ)͕දݱ͞Ε͍ͯΔ

    • Լਤ͸தԝͷέʔεʹ૬౰
    D
    θ1
    θ1
    ∼ P1
    (θ1
    ) θ2
    θ2
    ∼ P2
    (θ2
    ) m
    m ∼ P(m)
    P(D|θ1
    , m = 1) = P1
    (D|θ1
    )
    P(D|θ2
    , m = 2) = P2
    (D|θ2
    )

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  6. ϞσϧൺֱͷͨΊͷϞσϧ
    • ͜ͷϞσϧͷύϥϝʔλಉ࣌෼෍͸ҎԼʹͳΔ

    • Ϟσϧ਺ΛM, ύϥϝʔλ{θ_1, ..., θ_M} Λ Θ ͱͨ͠
    P(Θ, m|D) =
    P(D|Θ, m)P(Θ, m)

    m
    ∫ dθm
    P(D|Θ, m)
    =

    m
    ∫ dθm
    Pm
    (D|θm
    , m)Pm
    (θm
    )P(m)

    m

    m
    ∫ dθm
    Pm
    (D|θm
    , m)Pm
    (θm
    )P(m)
    P(D|Θ) ͕

    m
    Pm
    (D|θm
    )Pm
    (θm
    |m)P(m)ʹͳΔͷ͕ॏཁΒ͍͠

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  7. P(m|D)
    • ͜ͷϞσϧΛ࢖͏͜ͱͰɺσʔλD͕༩͑ΒΕͨͱ͖Ϟσ
    ϧm͕࢖ΘΕΔ֬཰P(m|D)ΛٻΊΔ͜ͱ͕Ͱ͖Δ
    P(m|D) =
    P(D|m)P(m)

    m
    P(D|m)P(m)
    P(D|m) =

    dθm
    Pm
    (D|θm
    )Pm
    (θm
    )

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  8. ࣄޙΦοζͱϕΠζϑΝΫλ
    • ϞσϧؒͰP(m|D)ͷൺΛऔΕ͹ɺͲͪΒͷϞσϧ͕΋ͬͱ
    ΋Β͍͔͠Θ͔Δ=ࣄޙΦοζ
    P(m = 1|D)
    P(m = 2|D)
    =
    P(D|m = 1)
    P(D|m = 2)
    P(m = 1)
    P(m = 2)
    • ໬౓P(m|D)ͷൺΛϕΠζϑΝΫλ(BF)ͱ͍͏

    • ࣄޙΦοζ=BF * ࣄલ֬཰ͷൺ

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  9. 10.2 2ͭͷίΠϯ޻৔ͷྫ
    • ίΠϯΛNճ౤͛ͨΒද͕zճग़ͨɻ2ͭͷ޻৔ͷͲͪΒ͔Βདྷ
    ͨίΠϯ͔?

    • ͦΕͧΕͷ޻৔ΛϞσϧͱΈͳͯ͠ɺϞσϧൺֱ͢Δ

    • fig 10.1ʹ͓͚Δӈͷਤ=໬౓ؔ਺͕ಉ͡Ͱ͋Δέʔεʹ૬౰
    z
    θ1
    θ1
    ∼ Beta(ω = 0.25,κ = 12)
    θ2
    m
    m ∼ Categorical(0.5,0.5)
    z ∼ Binomial(θm
    , N)
    θ2
    ∼ Beta(ω = 0.75,κ = 12)
    N

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  10. P(m|D)ͷൺΛٻΊΔ: ղੳղ
    • ࣄલ෼෍͕Beta(a, b)Ͱ༩͑ΒΕΔͱ͖ɺP(z,N)͸ҎԼͷࣜ
    ʹͳΔ(6ষͰઆ໌ࡁΈ)

    • আࢉ࣌͸Ξϯμʔϑϩʔ๷ࢭͷͨΊʹlogΛऔΔͱ͍͍

    • Rʹ͸ϕʔλ෼෍ͷlogΛٻΊΔlbetaؔ਺͕͋Δ
    P(z, N) =
    B(z, a, N − z + b)
    B(a, b)
    = exp(log B(z + a, N − z + b) − logB(a, b))

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  11. P(m|D)ͷൺΛٻΊΔ: ղੳղ
    • ·ͱΊΔͱɺP(D|m)͸ҎԼͱͳΔ
    P(D|m) = P(z, N|m)
    =
    B(z, am
    , N − z + bm
    )
    B(am
    , bm
    )
    am
    = ωm
    (κ − 2) + 1
    bm
    = (1 − ωm
    )(κ − 2) + 1
    ω1
    = 0.25
    ω2
    = 0.75

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  12. P(m|D)ͷൺΛٻΊΔ: ղੳղ
    • P(D|m=1) ≒ 0.000499

    • P(D|m=2) ≒ 0.002339

    • BF = P(D|m=1)/P(D|m=2) ≒ 0.213 ͱͳΔ

    • P(m = 1) = P(m = 2) = 0.5 ͷͱ͖ɺ
    P(m = 1|D)
    P(m = 2|D)
    =
    P(D|m = 1)
    P(D|m = 2)
    = 0.213
    P(m = 2|D) = 1 − P(m = 1|D)ΑΓ
    P(m = 1|D)
    1 − P(m = 1|D)
    = 0.213
    P(m = 1|D) = 0.176
    P(m = 2|D) = 0.824

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  13. P(m|D)ͷൺΛٻΊΔ: άϦουۙࣅ
    • m͸{0, 1}ͷ஋ΛऔΔ཭ࢄύϥϝʔλ

    • ਤ͕ॻ͖ʹ͍͘ͷͰɺmͷ͔ΘΓʹωΛಋೖ

    • ω͸0.25ͱ0.75ʹϐʔΫΛ࣋ͭ෼෍ɻ֤Ϟσϧͷ࠷ස஋
    ω_{1,2}ʹରԠɻ

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  14. • ӈ্: ωͷ෼෍ɻ2ͭͷࢁ͕ಉ͡ߴ͞Ͱ͋Δ=2ͭͷ஋͸౳֬཰

    • ࠨԼ: θͷपล෼෍ɻϞσϧʹରԠͨ͠2ͭͷࢁ͕͋Δ

    • ӈԼ: ω={0.75,0.25}ʹ͓͚Δθͷ෼෍
    άϦουۙࣅ: ࣄલ෼෍

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  15. • ӈ্ͷP(ω|D)ʹ͓͍ͯɺߴ͞ͷൺ͸໿5:1Ͱ͋Δ: P(m|D)ͷൺʹରԠ

    • ղੳղͰ͸mͷΈʹ஫໨͕ͨ͠ɺࠓճͷۙࣅͰ͸θͷ෼෍͕ՄࢹԽ͞Εͨ
    άϦουۙࣅ: ࣄޙ෼෍

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  16. 10.3 MCMCΛ༻͍ͨղ๏
    • ͦΕͧΕͷmʹ͍ͭͯP(D|m)Λܭࢉ͢Δํ๏

    • Ϟσϧൺֱ༻ͷϞσϧΛ௚઀ϞσϦϯά͠ɺmͷࣄޙ෼෍
    ΛٻΊΔํ๏

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  17. 10.3.1 MCMC: Ϟσϧ͝ͱ
    • ͦΕͧΕͷmʹ͍ͭͯɺP(D|m)Λܭࢉ͢Δ

    • JAGSͰθ_mΛαϯϓϦϯάͯ͠ɺ݁Ռʹରͯ͠Ή͔ͣ͠
    ͍͚͍͞ΜΛ͢ΔͱP(D|m)ʹͳΔ

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  18. Ϟσϧ͝ͱͷपล໬౓ܭࢉ
    • P(D)͸P(θ)͔ΒαϯϓϦϯάͨ͠θ_nΛ࢖ͬͯɺΣP(D|θ) / N
    ͰۙࣅՄೳ͕ͩɺ࣮༻తͰͳ͍

    • P(θ)͸֦ࢄ͍ͯ͠Δ

    • ΄ͱΜͲͷαϯϓϧʹஔ͍ͯɺP(D|θ)͸ඇৗʹখ͍͞

    • ࣄޙ෼෍P(θ|D)͔ΒαϯϓϦϯάͨ͠θΛ࢖ͬͯP(D)Λಋ
    ग़͍ͨ͠

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  19. Ϟσϧ͝ͱͷपล໬౓ܭࢉ
    P(θ|D) =
    P(D|θ)P(θ)
    P(D)
    1
    P(D)
    =
    P(θ|D)
    P(D|θ)P(θ)
    ೚ҙͷ֬཰෼෍h(θ)Λಋೖͯ͠
    =
    P(θ|D)
    P(D|θ)P(θ) ∫
    dθ′h(θ′)
    =

    dθ′
    P(θ|D)
    P(D|θ)P(θ)
    h(θ′)
    ೚ҙͷθʹ͍ͭͯɺ
    P(θ|D)
    P(D|θ)P(θ)
    ͷ஋͸ಉ͡ͳͷͰ
    =

    dθ′
    P(θ′|D)
    P(D|θ′)P(θ′)
    h(θ′)

    N

    θi
    ∼P(θ|D)
    h(θi
    )
    P(D|θi
    )P(θi
    )

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  20. Ϟσϧ͝ͱͷपล໬౓ܭࢉ
    • h(θ)ͱͯ͠͸೚ҙͷ֬཰෼෍͕࢖͑Δ͕ɺ਺஋ܭࢉͷ౎߹
    ্ɺ෼฼ͱࣅͨܗঢ়Ͱ͋Δ͜ͱ͕๬·͍͠

    • ෳࡶͳϞσϧʹ͓͍ͯɺͦͷΑ͏ͳhΛٻΊΔͷ͸೉͍͠

    • 10.3.1.1ʹ͓͍ͯ͸ɺαϯϓϦϯάͨ͠θΛݩʹhͷܗঢ়Λ
    ܾΊ͍ͯΔ
    N

    θi
    ∼P(θ|D)
    h(θi
    )
    P(D|θi
    )P(θi
    )

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  21. N
    10.3.2 MCMC: ֊૚Ϟσϧ
    • ࠓճͷέʔεͰ͸ɺ֤Ϟσϧͷࣄલ෼෍͓Αͼ໬౓෼෍͕ಉؔ͡
    ਺ͰදͤΔ

    • θΛαϯϓϦϯά͢ΔࡍʹɺmΛߟྀͯ͠ωͷ஋Λม͑Ε͹Α͍
    y
    θ
    m
    ω1
    = 0.25
    ω2
    = 0.75
    m ∼ Categorial(0.5,0.5)
    θ ∼ Beta(ω = ωm
    , κ = 12)
    yi
    ∼ Bern(θ)
    2
    ω

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  22. MCMCͷ݁Ռ
    • ্͕ࣄલɺԼ͕ࣄޙ

    • mͷࣄޙ෼෍͸ɺଞͷख๏Ͱͷ݁
    ՌͱҰக͍ͯ͠Δ

    • m=1ʹ͓͚Δθͷࣄޙ෼෍͸ɺα
    ϯϓϧશମͷ18%͔͠࢖ΘΕͯ
    ͍ͳ͍͜ͱʹ஫ҙ

    • m=2ʹ͓͚Δθͷࣄޙ෼෍͸ɺ࢒
    Γ82%͕࢖ΘΕ͍ͯΔ

    • ࢧ࣋͞Εͳ͔ͬͨϞσϧʹؔ͢Δ
    αϯϓϧ͸গͳ͘ͳΔ

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  23. 2
    10.3.2.1 ΋ͬͱҰൠతͳํ๏
    • ͜ͷࣄྫͰ͸ɺͨ·ͨ·ࣄલ෼෍ؔ਺͕શϞσϧͰಉ͡

    • Ұൠతʹ͸ɺҟͳΔؔ਺Λ࢖͍͍ͨ

    • ͷͰɺ෼͚ͯهड़͢Δͱ͜͏ͳΔ

    • આ໌ͷ౎߹্ɺࣄલ෼෍ͷύϥϝʔλ͸લͷྫͱҧ͍ͬͯΔ
    N
    y
    θ
    m
    ω1
    = 0.10
    ω2
    = 0.90
    m ∼ Categorial(0.5,0.5)
    θ1
    ∼ Beta(ω = ω1
    , κ = 20)
    θ2
    ∼ Beta(ω = ω2
    , κ = 20)
    yi
    ∼ Bern(θm
    )
    ω

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  24. ݁Ռ
    • ਤ10.5ࢀর

    • ҰԠαϯϓϦϯάͰ͖ͯ͸͍Δ͕……

    • ESS(༗ޮαϯϓϧαΠζ)<500

    • mͷࣗݾ૬͕ؔҟৗʹߴ͍

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  25. ࣗݾ૬ؔͷߴ͞
    • θ1(m=2)͓Αͼθ2(m=1)͸ࣄલ෼෍ͷΈʹै͏ͷʹରͯ͠ɺθ1(m=1)
    ͓Αͼθ2(m=2)͸ࣄલ෼෍ٴͼyʹӨڹΛड͚Δ

    • ͜ͷҧ͍͕mͷαϯϓϦϯάʹѱӨڹΛٴ΅͢
    θ1(m=1)
    θ1(m=2)
    θ2(m=1)
    θ2(m=2)

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  26. mʹΑΔθͷมԽ
    • JAGS͸gibbs sampling͍ͯ͠ΔͷͰɺύϥϝʔλΛҰݸ
    ͣͭαϯϓϦϯά͍ͯ͘͠
    θ(1)
    1
    ∼ P(θ1
    |θ(0)
    2
    , m(0), D)
    θ(1)
    2
    ∼ P(θ2
    |θ(1)
    1
    , m(0), D)
    m(1) ∼ P(m|θ(1)
    1
    , θ(1)
    2
    , D)
    θ(2)
    1
    ∼ P(θ1
    |θ(1)
    2
    , m(1), D)
    θ(2)
    2
    ∼ P(θ2
    |θ(2)
    1
    , m(1), D)
    m(2) ∼ P(m|θ(2)
    1
    , θ(2)
    2
    , D)

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  27. αϯϓϦϯάաఔ
    P(θ1
    , θ2
    , m|D) = {
    P1
    (D|θ1
    )P1
    (θ1
    )P2
    (θ2
    )P(m = 1) if m = 1
    P2
    (D|θ2
    )P1
    (θ1
    )P2
    (θ2
    )P(m = 2) if m = 2
    m(1) = 1
    θ(1)
    1
    ∼ P(θ1
    |θ(0)
    2
    , m = 1,D)
    =
    P(θ1
    , θ(0)
    2
    , m = 1|D)
    P(θ(0)
    2
    , m = 1|D)
    P(θ(0)
    2
    , m = 1|D) = P2
    (θ(0)
    2
    )P(m = 1)

    dθ1
    P1
    (D|θ1
    )P1
    (θ1
    )ΑΓ
    =
    P1
    (D|θ1
    )P1
    (θ1
    )
    ∫ dθ1
    P1
    (D|θ1
    )P1
    (θ1
    )

    View full-size slide

  28. αϯϓϦϯάաఔ
    θ(1)
    2
    ∼ P(θ2
    |θ(1)
    1
    , m = 1,D)
    =
    P(θ(1)
    1
    , θ2
    , m = 1|D)
    P(θ(1)
    1
    , m = 1|D)
    P(θ(0)
    1
    , m = 1|D) = P1
    (D|θ(1)
    1
    )P1
    (θ(1)
    1
    )P(m = 1)

    dθ2
    P2
    (θ2
    )ΑΓ
    = P2
    (θ2
    )
    P(θ1
    , θ2
    , m|D) = {
    P1
    (D|θ1
    )P1
    (θ1
    )P2
    (θ2
    )P(m = 1) if m = 1
    P2
    (D|θ2
    )P1
    (θ1
    )P2
    (θ2
    )P(m = 2) if m = 2

    View full-size slide

  29. αϯϓϦϯάաఔ
    m(2) ∼ P(m|θ(1)
    1
    , θ(1)
    2
    , D)
    =
    P(θ(1)
    1
    , θ(1)
    2
    , m|D)
    P(θ(1)
    1
    , θ(1)
    2
    |D)
    P(θ1
    , θ2
    , m|D) =
    P(D|θ1
    , θ2
    , m)
    P(θ1
    , θ2
    , m)
    P(D|θ1
    , θ2
    , m) = {
    P1
    (D|θ1
    )P1
    (θ1
    )P2
    (θ2
    )P(m = 1) if m = 1
    P2
    (D|θ2
    )P1
    (θ1
    )P2
    (θ2
    )P(m = 2) if m = 2
    P(θ1
    , θ2
    , m) = P1
    (θ1
    )P2
    (θ2
    )P(m)
    = {
    P(D|θ1
    ) if m = 1
    P(D|θ2
    ) if m = 2

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  30. αϯϓϦϯάաఔ
    • ࣍ͷm͕{1,2}ͷͲͪΒʹͳΔ͔͸ɺP(D|θ1)/P(D|θ2)ͷൺͰܾ·Δ

    • θ1ͷ΄͏͸P(D|θ1)P(θ1)͔Βੜ੒͞Ε͍ͯΔˠP(D|θ1)͕େʹͳΔ
    ֬཰͕ߴ͍

    • θ2͸P(θ2)͔Βੜ੒͞Ε͍ͯΔˠP(D|θ2)͸খʹͳΔͩΖ͏

    • ݁Ռͱͯ͠ɺm͸1ʹཹ·Δ֬཰͕ߴ͍
    m(1) = 1
    θ(1)
    1
    ∼ P(θ1
    |m = 1,D)
    ∝ P1
    (D|θ1
    )P1
    (θ1
    )
    θ(1)
    2
    ∼ P1
    (θ2
    )
    m(2) ∼ P(m|θ(1)
    1
    , θ(1)
    2
    , D)
    = {
    P(D|θ1
    ) if m = 1
    P(D|θ2
    ) if m = 2

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  31. ٙࣅࣄલ෼෍ʹΑΔ
    αϯϓϦϯάվળ
    • ໰୊: θͷ෼෍͸mͷ஋ʹΑΒͣҰఆͰ͋ͬͯ΄͍͠

    • ղܾ: P(θ_i|m=i)ʹ͍ۙ෼෍Λ༻ҙͯ͠ɺθ_i(i≠m)ʹ͍ͭͯ
    ͸ͦͷ෼෍͔ΒαϯϓϦϯά͢Δ

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  32. ٙࣅࣄલ෼෍ͷར༻
    • ٙࣅࣄલ෼෍Λ࢖Θͳ͍ϞσϧΛࣄલʹ࣮ߦ͓͖ͯ͠ɺٙࣅࣄલ෼෍
    ͷύϥϝʔλΛಘΔ

    • બ͹ΕͨϞσϧͷθ͸ී௨ʹαϯϓϦϯά͢Δ͕ɺબ͹Εͳ͔ͬͨํ͸
    ٙࣅࣄલ෼෍͔ΒαϯϓϦϯά͢Δ
    ωi,j
    , κi,j
    = {
    true prior if i = j
    pseudo prior if i ≠ j
    m ∼ Categorial(0.5,0.5)
    θ1
    ∼ Beta(ω = ω1,m
    , κ = κ1,m
    )
    θ2
    ∼ Beta(ω = ω2,m
    , κ = κ2,m
    )
    yi
    ∼ Bern(θm
    )
    2
    2 N
    y
    θ
    m
    ω

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  33. ݁Ռ
    • θͷ෼෍͸m͕มΘͬ
    ͯ΋͍͍ͩͨಉ͡ܗঢ়

    • mͷࣗݾ૬͕ؔେ෯ʹ
    Լ͕ΓɺESS=10000

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  34. ࢧ࣋͞Εͳ͍Ϟσϧͷαϯϓ
    ϧ͕গͳ͍໰୊
    • mͷࣄޙ෼෍ʹ͓͍ͯɺϞσϧ1͕બ͹ΕΔͷ͸8%

    • ͭ·ΓϞσϧ1ͷύϥϝʔλͰ͋Δθ1ͷαϯϓϧ਺͕શମ
    ͷ8%

    • αϯϓϧ਺Λ૿΍ͨ͢Ίʹ͸ɺνΣʔϯͷ௕͞Λ૿΍͢
    (ܭࢉ࣌ؒʹѱӨڹ)΄͔ʹɺϞσϧ͕ΑΓฏ౳ʹબ͹ΕΔ
    Α͏P(m)Λௐ੔͢Δ(m=1ʹόΠΞεΛֻ͚Δ)ํ๏͕͋Δ

    • P(m)Λ͍ͬͯ͡γϛϡϨʔγϣϯͨ͠৔߹Ͱ΋ɺฏ౳ͳ
    ࣄલ෼෍ʹ͓͚ΔࣄޙΦοζΛٻΊΒΕΔ
    BF =
    P(m = 1|D)
    P(m = 2|D)
    P(m = 2)
    P(m = 1)

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  35. 10.3.3 Ϟσϧ͝ͱʹҟͳΔ
    ໬౓ؔ਺ͷར༻
    • P(D|θ)Λnoise distributionͱ΋͍͏ͦ͏Ͱ͢

    • Ϟσϧ͝ͱʹҟͳΔP(D|θ)Λ࢖͍͍ͨͱ͖͸ɺ8.6.1Ͱ঺հ͠
    ͨςΫχοΫ͕࢖͑Δ

    • spy = if m = 1 then PDF(D|θ1) else PDF(D|θ2) / C

    • 1 ~ Bern(spy)

    • Ϟσϧͷಉ࣌֬཰ʹspyΛ৐͡Δ͜ͱʹͳΔ

    • C(େ͖Ίͷఆ਺)Ͱׂ͍ͬͯΔͷ͸spy͕1Λ௒͑ͳ͍Α͏ʹ

    • ૬ରతͳ஋͕ॏཁͳͷͰɺspyͷ۩ମతͳ஋͸ؔ܎ͳ͍

    • STANͩͱ΋ͬͱ௚ײతʹॻ͚ͨؾ͕͢Δ(increment_log_prob
    ؔ਺ͰϞσϧͷ֬཰ΛՃࢉͰ͖Δ)

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  36. 10.4: Ϟσϧฏۉ
    • P(y)Λ༧ଌ͍ͨ͠

    • Ϟσϧൺֱͷ݁ՌϞσϧb͕উ͍ͬͯͨͳΒɺͦͷϞσϧͰ༧
    ଌ͢Δ͜ͱ͕Ͱ͖Δ
    P( ̂
    y|D, m = b) =

    dθb
    Pb
    ( ̂
    y|θb
    , m = b)Pb
    (θb
    |D, m = b)
    • Ϟσϧ͝ͱʹ֬৴౓ׂ͕Γ౰ͯΒΕ͍ͯΔͷͰɺͦͷॏΈ
    Λ࢖ͬͯશϞσϧͷฏۉΛऔΔ͜ͱ͕Ͱ͖Δ
    P( ̂
    y|D) = ∑
    m

    dθm
    Pm
    ( ̂
    y|θm
    , m)Pm
    (θm
    |D, m)P(m|D)

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  37. 10.5: Ϟσϧͷෳࡶ౓
    • ࣄલ෼෍ʹ͓͍ͯɺύϥϝʔλͷऔΓ͏Δൣғ͕޿͍Ϟσ
    ϧΛʮෳࡶʯͳϞσϧͱݴ͍ͬͯΔͬΆ͍

    • ୯ʹύϥϝʔλ਺͕ଟ͍Ϟσϧͱ͍͏ҙຯͰ͸ͳ͍(ҎԼ
    ͷྫͰ΋ɺύϥϝʔλ਺͸ಉ͡)

    • ҰൠతʹɺෳࡶͳϞσϧͷ΄͏͕σʔλ΁ͷద߹͸༗ར

    • ޿͍ύϥϝʔλൣғͷϞσϧͷ΄͏͕ɺσʔλʹద߹
    ͢Δύϥϝʔλͷ૊Έ߹ΘͤΛؚΉՄೳੑ͕ߴ͍ͷͰ

    • ͔͠͠աద߹͸ආ͚͍ͨ

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  38. Ϟσϧൺֱͱෳࡶ͞
    • ෳࡶͳϞσϧ͸ɺࣄલ෼෍͕શମʹബ͘ࢄΒ͹͍ͬͯΔ

    • Մೳͳύϥϝʔλͷ૊Έ߹Θ͕ͤଟ͍=Ұݸ͋ͨΓͷ֬
    ཰͕௿͍

    • ୯७ͳϞσϧ͸ɺࣄલ෼෍͕ް͍

    • ϕΠζϞσϧൺֱʹ͓͍ͯ͸ɺࣄલ෼෍ͷް͕͞ࣄޙ֬཰
    ʹӨڹΛ༩͑Δ

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  39. Ϟσϧൺֱͱෳࡶ͞
    • ίΠϯ౤͛ͷϞσϧ: θ ~ Beta(a, b) Λߟ͑Δ

    • 1. ϑΣΞͩΖ͏Ϟσϧ: (a,b) = (500, 500)

    • 2. ͢΂ͯى͜Γ͏ΔϞσϧ: (a, b) = (1, 1)

    • 20ճத15ճද͕ग़ͨέʔεͰ͸ɺϞσϧ2͕উͭ

    • 20ճத11ճද͕ग़ͨΒϞσϧ1͕উͭ

    • ࣄલ෼෍ͷް͍෦෼Ͱσʔλʹద߹Ͱ͖͔͕ܾͨΊख

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  40. 10.5.1 Ϟσϧൺֱͷ஫ҙ
    • ͋ΔϞσϧ(full modelͱݺͿ)ʹରͯ͠ɺύϥϝʔλͷൣғ
    ʹ੍໿ΛՃ͑ͨϞσϧΛߟ͑Δ͜ͱ͕Ͱ͖Δ

    • ύϥϝʔλaͷ஋͸bͱಉ͡ɺͳͲ

    • full modelͷ΄͏͕ෳࡶͳͷͰɺ੍ݶϞσϧ͕ಉ͘͡Β͍
    Α͘σʔλΛදݱͰ͖ΔͳΒɺϕΠδΞϯϞσϧൺֱͰ͸
    ੍ݶϞσϧ͕બ͹ΕΔͩΖ͏

    • 9ষͷ໺ٿબखϞσϧʹ͓͍ͯɺ಺໺खͷೳྗ͸͢΂ͯಉ
    ͡Ͱ͋Δͱ͍͏੍ݶΛ͔͚ͨϞσϧ͕ߟ͑ΒΕΔ

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  41. Ϟσϧൺֱͷ஫ҙ఺
    • ߟ͑ΒΕΔ੍໿Λશ෦ࢼͦ͏ͱ͢Δͷ͸΍Ίͨ΄͏͕͍
    ͍

    • 9ύϥϝʔλʹಉ஋੍໿Λֻ͚Δ৔߹ɺ૊Έ߹Θͤ͸
    21147௨Γ

    • ੍໿Λ͔͚Δͱ͍͏͜ͱ͸ɺಛఆͷύϥϝʔλͷ૊Έ
    ߹Θͤʹ͍ͭͯࣄલ෼෍Λ0ʹ͢Δͱ͍͏͜ͱ

    • ͨͱ͑ϞσϧൺֱͰউͭͱͯ͠΋ɺ๬·͘͠ͳ͍͔΋
    ͠Εͳ͍

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  42. 10.6 ࣄલ෼෍ʹහײ
    • ϕΠζϑΝΫλʔ͸∫dθ P(D|θ)P(θ) Λ࢖͍ͬͯΔͷͰɺࣄ
    લ෼෍ʹහײ

    • ྫ: ෼ࢠଆͷϞσϧͷࣄલ෼෍ΛBeta(1,1)͔Β
    Beta(0.01,0.01)ʹͨ͠ΒɺBF͕0.12͔Β5.72ʹ

    • Ϟσϧͷ95% HDI͸ࣄલ෼෍ͷӨڹΛ΄΅ड͚ͳ͍

    • ॆ෼ͳྔͷσʔλ͕͋ΔͳΒɺϕΠζਪఆ͸Ϟσϧൺֱͱ
    ҧͬͯࣄલ෼෍ͷӨڹΛड͚ʹ͍͘

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  43. 10.6.1 ֤Ϟσϧͷࣄલ෼෍ʹ͸
    ฏ౳ʹ৘ใΛ༩͑Δ΂͖
    • ࣄલ෼෍ͷҧ͍͕BFʹӨڹΛ༩͑ΔɻͲ͏͢΂͖͔

    • σʔλʹج͍ͮͯࣄલ෼෍Λܾఆ͢Δ

    • ֤ϞσϧͰɺಉ͡σʔλʹج͍ܾͮͯΊΔ

    • ྫ: 100ճத65ճද͕ग़ͨίΠϯ౤͛

    • σʔλͷ10%(10ճத6ճද)Λ࢖ͬͯࣄલ෼෍Λิਖ਼

    • Beta(1, 1) → Beta(1+6, 1+4)

    • BF͕҆ఆ͢Δ

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