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Introduction to Bayesian Statistics (Modern epidemiology Chap. 18)

Shuntaro Sato
December 10, 2019

Introduction to Bayesian Statistics (Modern epidemiology Chap. 18)

Modern epidemiology Chap. 18の「Introduction to Bayesian Statistics」をもとに,「疫学へのベイズ統計の導入」を焦点に話を進めています.

最近話題のベイズ統計モデリングとは異なります.

Shuntaro Sato

December 10, 2019
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  1. Chapter 18
    Introduction to Bayesian Statistics
    Chuntaro (@Shuntarooo3)
    1
    Modern Epidemiology

    View full-size slide

  2. 2
    Overview
    w ϕΠζͷྺ࢙
    w Ӹֶσʔλ΁ස౓࿦తํ๏Λ༻͍Δ͜ͱͷ໰୊఺
    w ϕΠζਪఆ
    w ࣄલ෼෍
    w ໬౓
    w ࣄޙ෼෍
    w ۩ମྫ
    w ࣄલ෼෍Λ3$5Ͱද͢ͱʁ
    w ֊૚Ϟσϧ
    w Ϛϧίϑ࿈࠯ϞϯςΧϧϩ๏
    w චऀͷࢥ͍

    View full-size slide

  3. Introduction: History
    3
    w $IBQʙ͸ɼӸֶͰ༻͍Δස౓࿦తํ๏Λઆ໌
    w $IBQ͸ɼϕΠζతํ๏Λ঺հ




    5IPNBT#BZFT
    https://ja.wikipedia.org/wiki/τʔϚεɾϕΠζ, https://ja.wikipedia.org/wiki/ϐΤʔϧʹγϞϯɾϥϓϥε, https://en.wikipedia.org/wiki/Ronald_Fisher, https://en.wikipedia.org/wiki/
    Jerzy_Neyman, https://errorstatistics.com/2018/08/11/egon-pearsons-heresy-3/
    1JFSSF4JNPO
    -BQMBDF
    3"'JTIFS
    +/FZNBO
    &1FBSTPO
    ϕΠζ౷ܭͷ։ൃ Ұൠతͳ
    ౷ܭֶ ϕΠζ౷ܭͷ
    ࠶ڵ

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  4. Introduction: Motivation of Bayesian
    4
    3BOEPNJ[FE
    USJBMT
    3BOEPNTBNQMF
    TVSWFZT
    'SFRVFOUJTU
    NFUIPET
    ʢස౓࿦తํ๏ʣ
    0CTFSWBUJPOBM
    EBUB
    w %BUBͷੜ੒ϝΧχζϜ
    w മ࿐ͷׂ෇͚
    w)FBWJMZOPOSBOEPN
    w1PPSMZVOEFSTUPPE

    'SFRVFOUJTU
    NFUIPET
    ʢස౓࿦తํ๏ʣ
    ˡେৎ෉ʁ
    Ӹֶݚڀͷେ෦෼
    ࢖༻
    #BZFTJBOBQQSPBDI
    ͜ΕΒΛ͏·͘
    ଊ͑ΒΕΔ͔΋

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  5. FREQUENTISM VERSUS SUBJECTIVE BAYESIANISM (p. 329)
    5
    w ղੳʹ࠾༻ͨ͠Ծఆ΍Ϟσϧ͕ओ؍తͰ͋Δ
    w ዞҙੑ͕ੜ͡Δ
    ස౓࿦೿͔ΒϕΠζ೿΁ͷ൷൑
    w ͢΂ͯͷ౷ܭతਪ࿦ʹڞ௨͍ͯ͠Δ
    w ϕΠζΞϓϩʔν͸ɼ೚ҙͷཁૉΛ໌Β͔ʹ͍ͯ͠Δ͚ͩ

    View full-size slide

  6. Subjective probabilities should not be arbitrary [1/4] (p. 330)
    6
    ࣄલʢ֬཰ʣ෼෍ɿQSJPS QSPCBCJMJUZ
    EJTUSJCVUJPO
    w ୯ʹɼQSJPSͱ΋ݴ͏
    w ɹɹɹɹɹɹɹͱ.PEFSO&QJ͸ද͢ɽҰൠతʹ͸ɼ
    w σʔλΛ؍࡯͢Δલʹ෼ੳऀ͕ʢओ؍తʹʣܾఆͨ͠฼਺ͷ෼෍
    QSJPS TVCKFDUJWF QBSBNFUFS

    w །Ұͷύϥϝʔλ͕33 3JTL3BUJP
    ͱ͢Δ
    RR > RRmedian
    RR < RRmedian
    ಉ͡
    ֬৴౓
    Pr(RR > RRmedian
    ) = Pr(RR < RRmedian
    )
    RRlower
    RRupper
    ͜͜ʹ33͕͋Δ
    ֬৴౓͸ Pr(RRlower
    < RR < RRupper
    ) = 0.95
    Pr(parameters) f(θ)

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  7. Subjective probabilities should not be arbitrary [2/4] (p. 330)
    7
    TVCKFDUJWFQSJPS͸ओ؍త͗͢ͳ͍͔ʁ
    w ͔֬ʹݸਓؒͰҟͳΔ͔΋͠Εͳ͍
    w ͜Ε͸ዞҙతͰ͋Δ͜ͱΛҙຯ͢Δ΋ͷͰ͸ͳ͍
    w ଛࣦΛ࠷খݶʹ཈͑Δ͜ͱΛ໨ඪʹϨʔεʹṌ͚Δͱ͖ʹɼ୭΋
    ແ࡞ҝʹબ͹ΕͨڝٕऀʹṌ͚Δͷ͕߹ཧతͩͱ͸ࢥΘͳ͍
    w աڈͷσʔλʹج͍ͮͯɼṌ͚Δର৅ऀΛબͿ͸ͣ
    w ϕΠζ౷ܭͰ͸ɼઌߦݚڀͷ݁ՌΛࣄલ֬཰ʹ൓ө͢Δ

    View full-size slide

  8. 8
    ໬౓ɿMJLFMJIPPE
    w ϕΠζ౷ܭʹ͓͚Δೖྗ͸ɼࣄલ֬཰ͱ໬౓
    w ɹɹɹɹɹɹɹɹɹͱ.PEFSO&QJ͸ද͢ɽҰൠతʹ͸ɼ
    w ύϥϝʔλ͕༩͑ΒΕͨͱ͖ʹɼσʔλ͕ಘΒΕΔ֬཰
    ֬཰
    Pr(data ∣ parameters) f(X ∣ θ)
    Pr(data ∣ parameters)
    ύϥϝʔλΛఆ਺ɼ
    σʔλΛม਺
    ύϥϝʔλΛม਺ɼ
    σʔλΛఆ਺
    ໬౓
    Subjective probabilities should not be arbitrary [3/4] (p. 330)

    View full-size slide

  9. 9
    ໬౓͸ద੾͔ʁ
    w ద੾ͳ֬཰෼෍Ͱ໬౓Λද͍ͯ͠Δ͔͸ɼ
    ස౓࿦೿΋ϕΠζ೿΋ಉ͡Α͏ʹٙ೦ʹ࣋ͭ
    w ͋ΔళฮɹͷΞΠεͷചΓ্͛ɹΛ༧ଌ͢Δ
    w ചΓ্͛ͷฏۉΛɹͰද͢
    w ചΓ্͛ͷฏۉɹ͸ɼؾԹɹͱ஍Ҭɹʢ੢೔ຊ͔౦೔ຊ͔ʣͷ
    ӨڹΛड͚ΔͱԾఆ͢Δ
    w ؾԹ͕౓্͕Δ͝ͱʹɹສԁചΓ্্͕͕͛Δ
    w ੢೔ຊ͸౦೔ຊʹൺ΂ͯɼɹສԁചΓ্্͕͕͛Δ
    w ചΓ্͛ͷ෼ࢄ͸ɹͰද͢

    i Yi
    μi Ti
    Li
    βT
    βL
    σ2
    μi
    = β0
    + βT
    Ti
    + βL
    Li
    μi
    Yi
    ∼ Normal(μi
    , σ2) (i = 1,...,n)
    NPEFMJOH
    ˡద੾͔ͳʁ
    Subjective probabilities should not be arbitrary [4/4] (p. 330)

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  10. The posterior distribution (p. 330)
    10
    ࣄޙʢ֬཰ʣ෼෍ɿQPTUFSJPS QSPCBCJMJUZ
    EJTUSJCVUJPO
    w ୯ʹɼQPTUFSJPSͱ΋ݴ͏
    w σʔλ͕༩͑ΒΕͨޙͷ฼਺ͷ৚݅෇͖෼෍
    Pr(parameters ∣ data) =
    Pr(data ∣ parameters) Pr(parameters)
    Pr(data)
    Pr(parameters ∣ data) ∝ Pr(data ∣ parameters) Pr(parameters)
    ࣄޙ֬཰
    ໬౓ ࣄલ֬཰
    पล໬౓ɹˡύϥϝʔλͷؔ਺Ͱͳ͍
    Χʔωϧɿ,BSOFM
    ࣄલ֬཰
    ໬౓
    ࣄޙ֬཰

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  11. Frequentist-Bayesian parallels (p. 331)
    11
    w ʮύϥϝʔλ͸ස౓࿦೿ʹ͸ݻఆ͞Εͨ΋ͷͱͯ͠ѻΘΕΔ͕ɼ
    ϕΠζ೿͸ϥϯμϜʹѻ͏ʯͱ͍͏ͷ͸ؒҧ͍
    w ͲͪΒ΋ڵຯͷ͋Δύϥϝʔλ͸ݻఆ͍ͯ͠Δ
    ਅ஋͸ݻఆ
    ϕΠζ೿
    ස౓࿦೿
    ύϥϝʔλͷෆ࣮֬ੑΛ
    ࣄޙ෼෍Ͱද͢
    ύϥϝʔλ͸఺ਪఆ͠ɼ
    ͦͷਫ਼౓͸۠ؒਪఆͰද͢
    w ස౓࿦తํ๏΋ϕΠζతํ๏΋໬౓ʹ݁Ռ͕ґଘ͢Δ
    w ద੾ͳϞσϦϯάΛߦ͏ඞཁ͕͋Δ

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  12. Empirical Priors (p. 331)
    12
    w ࣄલ෼෍͕ओ؍తͳ΋ͷͰ͋ͬͯ͸ͳΒͳ͍ͱ͢ΔͳΒ͹ɼ
    ԿΛ࢖͑͹ྑ͍ͷ͔ʁ
    w 4UFJOFTUJNBUJPOɿॖখਪఆ
    w &NQJSJDBM#BZFTɿϋΠύʔύϥϝʔλ΋σʔλ͔Βਪఆ
    w 1FOBMJ[FEFTUJNBUJPOɿਖ਼ଇԽʢॖখਪఆʣ
    w SBOEPNDPF⒏DJFOUɿϋΠύʔύϥϝʔλ΋σʔλ͔Βਪఆ
    w SJEHFSFHSFTTJPOɿਖ਼ଇԽʢॖখਪఆʣ
    w ڞ໾ࣄલ෼෍ɿޮ཰ͷྑ͍ܭࢉ
    w ແ৘ใࣄલ෼෍ɿ੄ͷ޿͍෼෍

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  13. Frequentist-Bayesian divergences [1/2] (p. 332)
    13
    ;ʔΜ

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  14. Frequentist-Bayesian divergences [2/2] (p. 332)
    14
    ֬৴۠ؒʢDSFEJCMFJOUFSWBM$SF*ʣ
    w ϕΠζ౷ܭͰਪఆ͢Δ৴པ۠ؒ
    w ৴༻۠ؒͱ΋͍͏
    w ͋Δύϥϝʔλͷ֬৴۠ؒͰ͸ɼ
    ύϥϝʔλࣗ਎͕෼෍ʢࣄޙ෼෍ʣ͢Δ
    w $SF*͸ɼͦͷ۠ؒ಺ʹύϥϝʔλ͕ଘࡏ͢Δ֬཰͕
    ৴པ۠ؒʢDPOpEFODFJOUFSWBM$*ʣ
    w ස౓࿦Ͱਪఆ͢Δ৴པ۠ؒ
    w $*ʹਅͷύϥϝʔλ͕ଘࡏ͢
    Δ֬཰͸ɼ͔
    IUUQTSQTZDIPMPHJTUDPNE$*
    $*ΛΞχϝʔγϣϯʹͨ͠αΠτ

    View full-size slide

  15. Frequentist fantasy versus observational reality (p. 332)
    15
    3BOEPNJ[FE
    USJBMT
    3BOEPNTBNQMF
    TVSWFZT
    'SFRVFOUJTU
    NFUIPET
    ʢස౓࿦తํ๏ʣ
    0CTFSWBUJPOBM
    EBUB
    w %BUBͷੜ੒ϝΧχζϜ
    w മ࿐ͷׂ෇͚
    w)FBWJMZOPOSBOEPN
    w1PPSMZVOEFSTUPPE

    'SFRVFOUJTU
    NFUIPET
    ʢස౓࿦తํ๏ʣ
    ࢖༻
    Ӹֶʹ͓͍ͯස౓࿦తํ๏Λ༻͍Δ͜ͱ͸ɼ
    ؍࡯σʔλΛ͔͋ͨ΋ݫີʹઃܭ͞Εɼ
    ؅ཧ͞ΕͨϥϯμϜԽ࣮ݧ͔ΒಘΒΕͨͱߟ͑ΔΑ͏ͳ΋ͷ
    'BOUBTZ

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  16. Summary (p. 333)
    16
    w ϕΠζతํ๏΁ͷ൷൑͸ɼ
    w ࣄલ෼෍͕ዞҙత
    w ࣄલ෼෍͕ɼ༗֐Ͱಛผͳํ๏Ͱओ؍తʢʁʣ
    ɹͰͳ͚Ε͹ͳΒͳ͍
    w ؍࡯ݚڀͰɼ೔ৗతʹ࢖ΘΕ͍ͯΔղੳϞσϧ΋ዞҙతͰ͋Δ
    w ϕΠζతํ๏͸ɼස౓࿦తํ๏ͱಉ౳͔ͦΕҎ্ͷՊֶతࠜڌΛ
    ༩͑Δ
    w લఏ͕ʮݟ͑ΔԽʯ͞Ε͍ͯΔ͔Β
    w ൷൑తʹਫ਼ࠪ͞ΕΔඞཁ͸͋Δ

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  17. SIMPLE APPROXIMATE BAYESIAN METHODS [1/2] (p. 333)
    17
    Pr(parameters ∣ data) =
    Pr(data ∣ parameters) Pr(parameters)
    Pr(data)
    ࣄޙ֬཰
    ໬౓ ࣄલ֬཰
    पล໬౓
    ܭࢉେม f(x) =


    −∞
    f(x ∣ θ)f(θ)dx
    ϕΠζ౷ܭͷ࠶ڵ
    w 1$ͷൃୡ
    w ϞϯςΧϧϩΞϧΰϦζϜ
    ΧςΰϦσʔλʹదͨ͠ϕΠζۙࣅ͸ɼස౓࿦ͷۙࣅͱಉఔ౓ʹਖ਼֬
    w YදΛྫʹͱΓɼϕΠζతํ๏ͱස౓࿦తํ๏Λൺֱ

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  18. SIMPLE APPROXIMATE BAYESIAN METHODS [2/2] (p. 333)
    18
    w YදΛྫʹͱΓɼϕΠζతํ๏ͱස౓࿦తํ๏Λൺֱ
    w PVUDPNF͸ඇৗʹك
    w SJTLSBUJP SBUFSBUJP PEETSBUJPΛ۠ผ͠ͳ͍ˠ33Ͱදݱ
    w ɹɹɹ͸ਖ਼ن෼෍ʹै͏
    w ਖ਼ن෼෍Ͱ͸࠷ස஋ɼதԝ஋ɼฏۉ͸۠ผ͠ͳ͍
    w ɹɹɹɹɹ͸ର਺ਖ਼ن෼෍ʹै͏
    w ɹɹɹɹɹɹɹɹɹɹɹ͕੒ΓཱͭͨΊɼҎԼٞ࿦ʹ͸
    Λ༻͍Δ
    ln(RR)
    RR = eln(RR)
    medianRR = emedian ln(RR) medianRR

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  19. INFORMATION-WEIGHTED AVERAGING (p. 334)
    19
    w ৘ใʢ·ͨ͸ਫ਼౓ʣ͸ɼ෼ࢄͷٯ਺

    w ϕΠζతํ๏ʹ͓͚Δ৘ใʹΑΔॏΈ෇͚͸ɼ
    ස౓࿦తํ๏ʹ͓͚Δٯ෼ࢄॏΈ෇͚ʹجͮ͘ਪఆʹରԠ
    w ࣄલ෼෍ͱ໬౓͸ਖ਼ن෼෍ʹै͏ͱԾఆ͢Δ
    w ͜ͷԾఆ͸े෼ͳେ͖͞ͷαϯϓϧαΠζΛඞཁͱ͢Δ
    w DFMMTJ[F͸·ͨ͸͋Ε͹ྑ͍ͩΖ͏
    information = precision =
    1
    variance

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  20. A single two-way table [1/5] (p. 334)
    20
    $BTFDPOUSPMTUVEZGSPN4VBWJUZFUBM

    DAVID A. SAVITZ, HOWARD WACHTEL, FRANK A. BARNES, ESTHER M. JOHN, JIRI G. TVRDIK, CASE-CONTROL STUDY OF CHILDHOOD CANCER AND EXPOSURE TO 60-HZ
    MAGNETIC FIELDS, American Journal of Epidemiology, Volume 128, Issue 1, July 1988, Pages 21–38, https://doi.org/10.1093/oxfordjournals.aje.a114943
    w ډॅ஍ͷ࣓քͱখࣇന݂පͱͷਖ਼ͷؔ࿈Λࣔͨ͠ݚڀ
    w ઌߦݚڀͰ͸Ոఉ಺഑ઢͱന݂පͱͷਖ਼ͷؔ࿈͸ใࠂ͞Ε͍ͯͨ
    w ډॅ஍ͷ࣓քʢڧ͍ిքޮՌʣͱന݂පͱͷؔ࿈͸௿͍ͱߟ͑
    ΒΕ͍ͯͨ
    தࠃిྗHPΑΓʢhttp://www.energia.co.jp/energy/emf/emfa1.htmlʣ
    w ࣓ք͸ɼిྲྀ͕ྲྀΕΔ෺ͷपғʹൃੜ
    w ిք͸ɼిѹ͕͔͔ͬͨ෺ͷपғʹൃੜ

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  21. A single two-way table [2/5] (p. 334)
    21
    w ࣄલ33ͷ֬཰Λ ͱԾఆ
    w $POUSPMʹൺ΂ͨ$BTFͷ࣓քമ࿐Φοζͷൺ
    exp(prior mean − 1.96 prior SD) = 1/4
    exp(prior mean + 1.96 prior SD) = 4
    Prior mean of ln(RR) =
    ln(1/4) + ln(4)
    2
    = 0
    Prior SD of ln(RR) =
    ln(4) − ln(1/4)
    2 ⋅ 1.96
    = 0.707


    Prior variance of ln(RR) = 0.7072 = 1/2
    Prior ln(RR) ∼ N(0, 1/2)
    ࣄલ෼෍ͷܭࢉ
    $POUSPMʹൺ΂ͨ$BTFͷ࣓քമ࿐Φοζൺ͸/VMM
    33ͷ৴༻۠ؒ

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  22. A single two-way table [3/5] (p. 334)
    22
    DAVID A. SAVITZ, HOWARD WACHTEL, FRANK A. BARNES, ESTHER M. JOHN, JIRI G. TVRDIK, CASE-CONTROL STUDY OF CHILDHOOD CANCER AND EXPOSURE TO 60-HZ
    MAGNETIC FIELDS, American Journal of Epidemiology, Volume 128, Issue 1, July 1988, Pages 21–38, https://doi.org/10.1093/oxfordjournals.aje.a114943
    X = 1
    (>= 3 mG)
    X = 0
    (< 3 mG)
    Case 3 33
    Table odds ratio = RR
    estimate = 3.51
    Controls 5 193 95% CI = 0.80, 15.4
    Estimated RR = sample OR = (3 ⋅ 193)/(5 ⋅ 33) = 3.51
    ؍࡯σʔλͰͷܭࢉ
    Estimated variance ln(OR) = 1/3 + 1/33 + 1/5 + 1/193 = 0.569
    95 % CI of OR = exp[ln(3.51) ± 1.96 ⋅ 0.5691/2] = 0.80, 15.4

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  23. A single two-way table [4/5] (p. 334)
    23
    DAVID A. SAVITZ, HOWARD WACHTEL, FRANK A. BARNES, ESTHER M. JOHN, JIRI G. TVRDIK, CASE-CONTROL STUDY OF CHILDHOOD CANCER AND EXPOSURE TO 60-HZ
    MAGNETIC FIELDS, American Journal of Epidemiology, Volume 128, Issue 1, July 1988, Pages 21–38, https://doi.org/10.1093/oxfordjournals.aje.a114943
    Posteriror variance for ln(RR) ≃
    1
    1
    1/2
    + 1
    0.569
    = 0.266
    ࣄޙ෼෍ͷܭࢉ
    ࣄલ෼෍ͷ৘ใ ؍࡯σʔλͷ৘ใ
    UPUBMJOGPSNBUJPO
    Posteriror mean for ln(RR) = expected ln(RR) given data

    0
    1/2
    + ln(3.51)
    0.569
    total information
    =
    0
    1/2
    + ln(3.51)
    0.569
    1/0.266
    = 0.587
    ࣄલ෼෍ͷॏΈ෇͚ฏۉ ؍࡯σʔλͷॏΈ෇͚ฏۉ
    Posteriror median for RR ≃ exp(0.587) = 1.8
    95 % posterior limits for RR ≃ exp(0.587 ± 1.96 ⋅ 0.2661/2) = 0.65, 4.94

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  24. A single two-way table [5/5] (p. 334)
    24






    ࣄલ෼෍
    σʔλ
    º
    ࣄޙ෼෍

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  25. Bayesian Interpretation of Frequentist results (p. 335)
    25
    Posterior mean for ln(RR) ≃
    0
    1/2
    + ln(3.51)
    0.569
    1
    1/2
    + 1
    0.569
    = 0.587
    Posterior mean for ln(RR) without prior ≃
    0 + ln(3.51)
    0.569
    0 + 1
    0.569
    = ln(3.51)
    Estimated RR = sample OR = (3 ⋅ 193)/(5 ⋅ 33) = 3.51
    Posterior mean for RR without prior ≃ exp{ln(3.51)} = 3.51
    ࣄલ෼෍ͷ৘ใ͸ͳ͍
    ස౓࿦తํ๏ Ұக
    w ස౓࿦తํ๏͔ΒಘΒΕΔ݁Ռ͸ɼࣄલ৘ใΛશ͘࢖Θͳ͍ϕΠζ࿦తํ๏͔Β
    ಘΒΕΔ݁Ռͱಉ͡
    w ࣄલ৘ใΛ࢖Θͳ͍ͱ͍͏͜ͱ͸ɼ
    ͱ͍͏33Λ΍ͱಉ͘͡Β͍͋Γ͑Δͱߟ͑Δͱಉ͜͡ͱ
    w ແ৘ใࣄલ෼෍͸໌ࣔతʹࣄલ෼෍Λ༩͑͸͍ͯ͠Δ఺Ͱස౓࿦ͱҟͳΔ
    w ͔͠͠ײ֮తʹ͋Γಘͳ͍ࣄલ෼෍΋औΓ͑Δ

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  26. Adjustment [1/3] (p. 336)
    26
    ࣓ք খࣇന݂ප
    ݚڀʢ$POGPVOEFSʣ ࣄલ৘ใͳ͠
    ݚڀ 1SJPS 33 1PTUFSJPS
    ˠ
    ˠ
    ˠ
    ݚڀ 1SJPS 33 1PTUFSJPS
    /" ˠ
    /" ˠ
    /" ˠ
    ݚڀ 1SJPS 33 1PTUFSJPS
    ˠ
    ˠ
    ˠ
    ͜͏͍͏ঢ়ଶ͔ͳʁ ݚڀ͝ͱͷQSJPS͸ͳ͍ ڞ௨ͷQSJPSΛઃఆ͢Δ
    ૚ผղੳͰަབྷΛௐ੔
    Estimated common RR = sample OR = 1.69 ln(OR) = ln(1.69) = 0.525
    95 % CI of common OR = 1.28, 2.23
    variance of ln(OR) = {
    ln(2.23) − ln(1.28)
    2 ⋅ 1.96
    }2 = 0.0201
    ؍࡯σʔλͰͷܭࢉ
    N(0, 1/2)

    View full-size slide

  27. Adjustment [2/3] (p. 336)
    27
    Posteriror variance for ln(RR) ≃
    1
    1
    1/2
    + 1
    0.0201
    = 0.0193
    ࣄޙ෼෍ͷܭࢉ
    ڞ௨ࣄલ෼෍ͷ৘ใ ؍࡯σʔλͷ৘ใ
    UPUBMJOGPSNBUJPO
    Posteriror mean for ln(RR) = expected ln(RR) given data

    0
    1/2
    + ln(1.69)
    0.0201
    total information
    =
    0
    1/2
    + ln(1.69)
    0.0201
    1/0.0193
    = 0.504
    ࣄલ෼෍ͷॏΈ෇͚ฏۉ ؍࡯σʔλͷॏΈ෇͚ฏۉ
    Posteriror median for RR ≃ exp(0.504) = 1.66
    95 % posterior limits for RR ≃ exp(0.504 ± 1.96 ⋅ 0.01931/2) = 1.26, 2.17

    View full-size slide

  28. Adjustment [3/3] (p. 336)
    28






    ڞ௨ͷࣄલ෼෍
    σʔλ
    º
    ࣄޙ෼෍

    View full-size slide

  29. Varying the prior (p. 336)
    29






    ࣄલ෼෍
    σʔλ
    ࣄޙ෼෍






    σʔλ
    ࣄޙ෼෍
    ࣄલ෼෍








    ࣄલ෼෍
    σʔλ
    ࣄޙ෼෍




    *OGPSNBUJPO
    ಉ͙͡Β͍ͩͱ
    ࣄલ෼෍ʹۙ͘ͳΔ
    σʔλͷ৘ใ͕େ͖͍ͱ
    σʔλͷ෼෍ʹۙ͘ͳΔ
    w ࣄલ෼෍ͷ৘ใ͕େ͖͍ͱ
    ਫ਼౓ͷߴ͍ࣄલ෼෍ʹۙ͘ͳΔ
    w /VMMΛӽ͢ʁ

    View full-size slide

  30. Bayes versus semi-bayes (p. 337)
    30
    logit(Pr(Y = 1 ∣ X, C)) = β0
    + β1
    X + β2
    C
    ࣓ք
    ʢ9ʣ
    খࣇന݂ප
    ʢ:ʣ
    $$POGPVOEFS
    β1
    ∼ N(0, 1/2)
    ৘ใͳ͠
    ηϛ΂Πζ

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  31. PRIOR DATA: FREQUENTIST INTERPRETATION OF PRIORS [1/3] (p. 337)
    31


    ࣄલ෼෍ Prior ln(RR) ∼ N(0, 1/2)
    ͜ͷࣄલ෼෍ͱಉ͡৘ใΛ༩͑Δස౓࿦తͳ৘ใ͸Կ͔ʁ
    3$5ΛԾ૝తʹߟ͑Δ
    R
    //
    / 9
    ˠ"
    / 9
    ˠ"
    X = 1 X = 0
    Case A1 A0
    Controls N1-A1 N0-A0
    Total N1 N0
    &RVBMBMMPDBUJPO///
    3BSFEJTFBTF/"
    Estimated RR = (A1
    /N)/(A0
    /N) = A1
    /A0
    Estimated variance for ln(RR) =
    1
    A1
    +
    1
    A0
    +
    1
    N − A1
    +
    1
    N − A0
    =
    1
    A1
    +
    1
    A0
    N − A1
    ≃ N, 1/N ≃ 0
    XIFSF

    View full-size slide

  32. PRIOR DATA: FREQUENTIST INTERPRETATION OF PRIORS [2/3] (p. 337)
    32
    Estimated variance for ln(RR) =
    1
    A1
    +
    1
    A0
    =
    1
    A
    +
    1
    A
    =
    2
    A
    =
    1
    2
    Prior ln(RR) ∼ N(0, 1/2) ʹ߹ΘͤΔ
    Estimated RR = A1
    /A0
    = 1 → A1
    = A0
    = A
    A1
    = A0
    = A = 4
    X = 1 X = 0
    Case 4 4
    Controls 99,996 99,996
    Total 100,000 100,000


    ࣄલ෼෍
    Prior ln(RR) ∼ N(0, 1/2)

    ࣄલ෼෍ͱಉ౳ͷ৘ใΛ༗͢Δঢ়گ͸ۃ୺Ͱ͸ͳ͍͔ʁ
    ࣄલ෼෍ͷݟ௚͠

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  33. PRIOR DATA: FREQUENTIST INTERPRETATION OF PRIORS [3/3] (p. 337)
    33
    X = 1 X = 0
    Case 4 4
    Controls 99,996 99,996
    Total 100,000 100,000


    ࣄલ෼෍
    Prior ln(RR) ∼ N(0, 1/2)
    ࣄલ෼෍͔Βɼಉ౳ͷσʔλΛٯࢉ
    ্هσʔλΛҰͭͷ૚ɼ؍࡯σʔλΛҰͭͷ૚ͱ͠ɼ૚ผղੳ
    ස౓࿦తํ๏ͷιϑτ΢ΣΞͰϕΠζਪఆΛ͓͜ͳ͏ʹ͸ʁ
    X = 1 X = 0
    Case 3 33
    Controls 5 193

    ૚ผղੳ
    ؍࡯σʔλ

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  34. Reverse-bayes analysis (p. 338)
    34


    ࣄલ෼෍
    σʔλ
    ࣄޙ෼෍


    σʔλ͕໌ࣔ͞Ε͍ͯͳ͍ͨΊΘ͔Βͳ͍
    ࣄޙ෼෍ͷ֬৴۠ؒԼݶΛͱ͢Δͱʜ
    RRprior
    = 1
    95 % posterior limits for RRprior
    = 0.85, 1.18
    ٯࢉ
    ඞཁͳࣄલ෼෍ͱͦΕʹରԠ͢Δσʔλ
    ࣄޙ෼෍ΛԾઆͱͯ͠ઃఆͨ͠৔߹ɼࣄલ෼෍͸Ͳ͏ͳΔ͔ʁ
    X = 1 X = 0
    Case 250 250
    Total 7,500,000 7,500,000
    ͋·Γʹ΋ۃ୺

    View full-size slide

  35. Priors with non-null center (p. 339)
    35


    ࣄલ෼෍ Prior ln(RR) ∼ N(ln(2), 1/2)
    Posteriror variance for ln(RR) ≃
    1
    1
    1/2
    + 1
    0.569
    = 0.266
    Posteriror mean for ln(RR) ≃
    ln(2)
    1/2
    + ln(3.51)
    0.569
    total information
    =
    0
    1/2
    + ln(1.69)
    0.0201
    1/0.266
    = 0.956
    Posteriror median for RR ≃ exp(0.956) = 2.60
    95 % posterior limits for RR ≃ exp(0.956 ± 1.96 ⋅ 0.2661/2) = 0.95, 7.15




    σʔλ
    ࣄޙ෼෍

    X = 1 X = 0
    Case 4 4
    Total 100,000 200,000

    View full-size slide

  36. Choosing the sizes of the prior denominators (p. 339)
    36
    w ɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹͰ͋Ε͹ɼ
    رগ࣬ױੑ͸ҡ࣋Ͱ͖ͯ໰୊ͳ͍
    w SJTLSBUJPSBUFSBUJPPEETSBUJPͷۙࣅ͕ҡ࣋Ͱ͖Δ
    w ͜Ε·Ͱͷܭࢉʹ࢖͍ͬͯͨɹɹɹɹͷۙࣅʹ͸Өڹ͢Δ͸ͣ
    N1
    > 100 ⋅ A1
    and N0
    > 100 ⋅ A0
    N ≫ A

    View full-size slide

  37. Non-normal priors (p. 340)
    37
    Prior ln(RR) ∼ N(μ, σ2) → Prior RR ∼ logNormal(μ, σ2)
    ͜Ε·Ͱͷࣄલ෼෍
    ɹɹɹɹɹɹɹɹɹͳΒ͹ɼɹɹɹɹɹɹɹɹɹɹͷํ͕ਖ਼֬
    N1
    ≫ A1
    and N0
    ≫ A1
    Prior RR ∼ F(2A1
    , 2A0
    )
    A1
    ≠ A0
    , or A1
    , A0
    < 3 ͷͱ͖ɼ͜ͷ෼෍ͷ࢖༻͸ෆద
    log Normal F F
    RR 95% CreI RR CreI Distribution
    A1 = A0 = A = 4 (1/4, 4) 93.3% F(8, 8)
    A1 = A0 = A = 3 (1/5, 5) 92.8% F(6, 6)

    View full-size slide

  38. Further extensions (p. 340)
    38
    w߱ѹༀͷޮՌΛݕূ͢Δྟচࢼݧ
    wಉҰର৅ऀʹରͯ͠౤༩લ͔Β౤༩ޙʹ͔͚ͯෳ਺ճ݂ѹଌఆ
    w ಉҰର৅ऀͷ݂ѹͷਪҠ͸૬ؔ͋Δ
    w ର৅ऀ͝ͱʹϕʔεͷ݂ѹҧ͏
    w ର৅ऀ͝ͱʹԼ͕Γํ΋ҧ͏
    .VMUJMFWFM )JFSBSDIJDBM
    NPEFMJOH
    μij
    = β0
    + βX
    Xi
    + βT
    Tij
    + βXT
    Xi
    Tij
    + r0i
    Yij
    ∼ Normal(μij
    , σ2) (i = 1,...,n; j = 1,...,k)
    β0
    , βX
    , βT
    , βXT
    , ri
    ∼ N(0, 1002)
    ूஂͷฏۉͱͯ͠ճؼ܎਺ ݸਓ͝ͱͷҧ͍
    ˞ର৅ऀ͝ͱʹԼ͕
    Γํ͕ҧ͏͜ͱ͸ߟ
    ྀͨ͠ϞσϧͰ͸͋Γ
    ·ͤΜ

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  39. CHECKING THE PRIOR (p. 340)
    39
    ࣄલ෼෍ΛՃ͑ͯϕΠζਪఆ͢Δલʹɼ
    ؍࡯σʔλͱͷۉҰੑΛ֬ೝ͢Δ΂͖
    Frequentist estimate − prior estimate
    ( frequentist variance + prior variance)2
    = χscore
    Ծఆɿࣄલ෼෍͕ਖ਼ن෼෍ʹै͏ɼ؍࡯঱ྫ਺͕े෼େ͖͍
    1஋͕খ͍͞৔߹ɼ
    ࣄલ෼෍ͱ؍࡯σʔλͰॏΈ෇͚ਪఆ͢Δͷ͸ෆదͰ͋Δ͜ͱΛࣔ͢
    ݕఆͯ͠ͳ͍͚Ͳɼ݁ہ1஋ग़͢Μ͔ʔʔʔ͍

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  40. DISCUSSION Data alone say nothing at all (p. 341)
    40
    wස౓࿦తํ๏͸ϕΠζతํ๏͔ΒಘΒΕΔ݁ՌΑΓ΋؍࡯σʔλΛ
    Α͘൓ө͢Δ
    wස౓࿦తํ๏͸ϥϯμϜԽൺֱࢼݧɼϥϯμϜαϯϓϦϯάʹجͮ
    ͍͍ͯΔ
    w͍ͭ΋ͷ࿩͠
    wਪ࿦Ͱ͸ͳ͘σʔλΛهड़͢Δ͚ͩͳΒ͹ɼ
    ද΍άϥϑͰཁ໿͢΂͖

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  41. Data priors as a general diagnostic device (p. 341)
    41
    wേଇ෇͖໬౓๏͸ɼϕΠζతͳݟํ͔Β΋ղऍͰ͖Δ
    wଞͷ͍Ζ͍Ζͳ౷ܭख๏΋ϕΠζతͳݟํ͕Ͱ͖Δ͔΋

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  42. The role of Markov-Chain Monte Carlo [1/3] (p. 342)
    42
    Pr(parameters ∣ data) =
    Pr(data ∣ parameters) Pr(parameters)
    Pr(data)
    ࣄޙ෼෍
    ฏۉɼதԝ஋ɼύʔηϯλΠϧ఺Λऔಘ͍ͨ͠ˠܭࢉෳࡶ
    .$.$ʢ.BSLPW$IBJO.POUF$BSMPʣ๏
    ʢϚϧίϑ࿈࠯ϞϯςΧϧϩ๏ʣΛ༻͍ͯ౷ܭྔΛऔಘ͠Α͏
    w Ϛϧίϑ࿈࠯Λར༻ͯ͠ɼࣄޙ෼෍ʹै͏ཚ਺Λੜ੒͢Δख๏
    w ཚ਺ੜ੒ΞϧΰϦζϜͱͯ͠ɼϝτϩϙϦεɾϔΠεςΟϯά
    ε๏΍ϋϛϧτχΞϯɾϞϯςΧϧϩ๏͕͋Δ

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  43. The role of Markov-Chain Monte Carlo [2/3] (p. 342)
    43
    w .$.$๏͸ຊ࣭తʹɼ෼ੳత౷ܭख๏ΑΓ΋ϩόετੑ͕௿͍
    ʢ೥࣌఺ͷ8JO#6(4ͷओுʣ
    w .$.$͔Βಘͨղੳ݁ՌΛνΣοΫ͢Δʹ΋෼ੳత౷ܭख๏͕༗ӹ
    w ෳࡶͳϞσϦϯάΛߦ͏৔߹Ͳ͏͢Δͷʁ
    w ׬શʹਖ਼֬ͳϞσϧͰɼे෼௕࣮͘ߦʢੜ੒͢Δཚ਺ͷݸ਺ʣ
    Ͱ͖Ε͹ɼ෼ੳత౷ܭख๏ΑΓ΋ਖ਼֬ͳ݁ՌΛੜ੒Ͱ͖Δ
    w Ϟσϧ͸ਖ਼ղ͔෼͔Βͳ͍͔Βɼ
    .$.$͸ޡΓʹޡΓΛੵΈॏͶΔ͚͔ͩ΋
    w શͯͷํ๏͕ͦ͏Ͱ͸ͳ͍͔ʁ
    w ߟ͑͏Δଥ౰ͳϞσϦϯά͢Ε͹Α͍ͷͰ͸ʁ

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  44. The role of Markov-Chain Monte Carlo [3/3] (p. 342)
    44
    w ϕΠζ౷ܭϞσϦϯά͸ɼैདྷͷ౷ܭϞσϦϯάΑΓ΋ॊೈͳઃ
    ܭ͕Մೳ
    w ࠷ۙ͸ɼ8JO#6(4Ͱ͸ͳ͘4UBO͕Α͘ར༻͞ΕΔ
    w 3Ͱ΋3TUBOͱ͍͏΋ͷ͕࢖͑Δ
    w CSNTͱ͍͏QBDLBHF΋
    w NBD04$BUBMJOB͸4UBO͔Βਖ਼ࣜʹΞοϓσʔτ͢Δͳͱܯࠂ
    ग़͔ͨΒ஫ҙͯ͠ʂʂ
    w ϕΠζ౷ܭϞσϦϯά໘ന͍Α
    w ݕఆ͢Δ͜ͱͰͳ͘ɼਪఆʹڵຯ͕͋ΔͳΒੵۃతʹ࢖ͬͯྑ͍
    ͷͰ͸ʁ

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  45. Connections to sensitivity analysis (p. 342)
    45
    w ස౓࿦తํ๏ʹΑΓɼݻఆ͞ΕͨύϥϝʔλΛมԽͤͯ͞ɼ
    ͦΕ͕౷ܭྔʹ༩͑ΔӨڹΛධՁ͢Δ
    w ࣄલ෼෍ΛมԽͤͯɼͦΕ͕౷ܭྔʹ༩͑ΔӨڹΛධՁ͢Δ

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  46. Some cautions on use of priors (p. 342)
    46
    w ࣄલ෼෍ΛऔΓೖΕͯղੳΛ͓͜ͳ͏ͷ͸༰қͰ͋Δ
    w ࣄલ৘ใΛར༻͢Δ͜ͱ͸ɼ
    ਪ࿦Λվળ͢Δํ๏ͱͯ͠ड͚ೖΕΒΕ͍ͯΔ
    w ޡͬͨ৘ใΛઃఆ͢Δͱɼޡͬͨ݁Ռ͕ಘΒΕΔ
    w ස౓࿦తํ๏Ͱಘͨ݁ՌͱϕΠζతํ๏Ͱಘͨ݁ՌΛൺֱ͢Δ΂͠
    w 1஋Λ࢖ͬͯɼซ߹͕ద͍ͯ͠Δ͔νΣοΫ͢Δ

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  47. CONCLUSIONS (p. 343)
    47
    w ϕΠζతํ๏͸ɼස౓࿦తํ๏͔ΒಘΒΕͨਪఆ஋ʹɼࣄલ৘ใ
    ΛՃ͑Δ͜ͱͰ͓͜ͳ͏͜ͱ͕Ͱ͖Δ
    w ૚ผղੳͷΑ͏Ͱ͋Δ
    w ස౓࿦తํ๏ʹΑΔ݁Ռͷఏ͕ࣔඪ४Ͱ͋ͬͯ΋ɼ
    ϕΠζΛ౷ܭڭҭʹऔΓೖΕΔͷ͸༗༻Ͱ͋Ζ͏

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  48. 48
    Overview
    w ϕΠζͷྺ࢙
    w Ӹֶσʔλ΁ස౓࿦తํ๏Λ༻͍Δ͜ͱͷ໰୊఺
    w ϕΠζਪఆ
    w ࣄલ෼෍ɼ໬౓ɼࣄޙ෼෍
    w ۩ମྫ
    w ࣄલ෼෍Λ3$5Ͱද͢ͱʁ
    w ֊૚Ϟσϧ
    w Ϛϧίϑ࿈࠯ϞϯςΧϧϩ๏
    w චऀͷࢥ͍
    ࣄલ֬཰
    ໬౓
    ࣄޙ֬཰

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  49. ࢀߟจݙͱҾ༻จݙ
    അ৔ਅ࠸
    3ͱ4UBOͰ͸͡ΊΔϕΠζ౷ܭϞσϦϯάʹΑΔσʔλ෼ੳೖ໳
    ߨஊࣾ
    w ࠓ࠷΋৽͘͠෼͔Γ΍͍͢ϕΠζϞσϦϯάͷॻ੶
    w 4UBO΍CSNTͷઆ໌΋๛෋
    দӜ݈ଠ࿠
    4UBOͱ3ͰϕΠζ౷ܭϞσϦϯάڞཱग़൛
    w ΑΓ೉ղ͕ͩ෼͔Γ΍͍͢ɽ
    w 4UBOͷਂΈ͸ͪ͜Βͷํ͕ཧղ͠΍͍͢
    େؔਅ೭
    ϕΠζਪఆೖ໳ɽΦʔϜࣾ
    w ෺ޠܗࣜͰອը΋͋Γɼͱ͖ͬͭ΍͍͕͢ɼ಺༰͸͚ͬ͜͏ϔϏʔ
    w ػցֶशͱͷϦϯΫΛҙ͍ࣝͯ͠Δ͔΋
    Ԟଜ੖඙ɼ຀ࢁ޾࢙ɼӝੜਅ໵
    3Ͱָ͠ΉϕΠζ౷ܭೖ໳ɽٕज़ධ࿦ࣾ
    ਢࢁರࢤ
    ϕΠζਪ࿦ʹΑΔػցֶशೖ໳ɽߨஊࣾ
    ๛ాलथ
    ͸͡Ίͯͷ౷ܭσʔλ෼ੳɽே૔ॻళ
    ؠ೾σʔλαΠΤϯεץߦҕһձ
    ؠ೾σʔλαΠΤϯεWPM
    <ಛू>ϕΠζਪ࿦ͱ.$.$ͷϑϦʔιϑτؠ೾ॻళ
    49

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  50. Discussion!!
    50

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