Shuntaro Sato
December 10, 2019
820

# 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．

## Shuntaro Sato

December 10, 2019

## Transcript

1. Chapter 18
Introduction to Bayesian Statistics
Chuntaro (@Shuntarooo3)
1
Modern Epidemiology

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

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

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΋

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

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)

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

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)

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)

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཰

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

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޿͍෼෍

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

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

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

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

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

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

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

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

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༻۠ؒ

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

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

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

ࣄલ෼෍
σʔλ
º
ࣄޙ෼෍

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

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)

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

28

ڞ௨ͷࣄલ෼෍
σʔλ
º
ࣄޙ෼෍

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

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

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

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௚͠

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

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୺

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

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

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)

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

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

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໿͢΂͖

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

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

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

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

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

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

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

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཰

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

50. Discussion!!
50