なぜベイズ統計はリスク分析に向いているのか？その哲学上および実用上の理由

なぜベイズ統計はリスク分析に向いているのか？

～その哲学上および実用上の理由～(ver 2.0)

林岳彦国立環境研究所環境リスク研究センター

[email protected]

2010.6.17国環研生物系若手セミナー＊ブログup用改変版＊

mailto:[email protected]

mailto:[email protected]

今日の話

ベイズ

ベイズにまつわるエトセトラ

確率概念

仮説検定

リスク分析

本日のメニューI.「確率」の哲学的諸概念と　リスク解釈にとっての意味

II. 仮説検定の｢筋違い｣さ　とベイズの本質的な利点

III.デフォルトあるいは糊代　としての事前分布の利用

35min

30min

概念的

統計的

実務的25min




35min

30min

概念的

統計的

実務的25min

I.「確率」の哲学的諸概念と　リスク解釈にとっての意味

1-1「確率」とは何か？

1-2 確率概念とリスクの解釈




I-1.「確率」とは何か？

・クロロホルムによってガンになる確率・2050年までに地球の気温が2度以上上昇　する確率・人為的な温室効果ガスが温暖化の原因で　ある確率 (IPCC曰く90%)

・今年広島カープが優勝する確率・国環研任期付がパーマネントになる確率

・コインを投げてオモテが出る確率


・クロロホルムによってガンになる確率・2050年までに地球の気温が2度以上上昇　する確率・人為的な温室効果ガスが温暖化の原因で　ある確率 (IPCC曰く90%)

・今年広島カープが優勝する確率・国環研任期付がパーマネントになる確率


Risk = f(Effect,Probability)

発がんリスクが10-6


頻度型確率確信度型確率

伝統的統計学ベイズ統計学

確率

確信度型確率

頻度型確率

用語法はHacking (2001)に基づく

Probability

Frequency-type probability

個人説

間個人説

論理説

頻度説

傾向説

Logical theory

Personal theory

Inter-personal theory

Frequency theory

Propensity theory

Belief-type probability

確率概念の分類・古典確率・確率の公理

AN Kolmogolov (1903-1987)

コルモゴルフの確率測度の定義（公理）確率の公理

http://en.wikipedia.org/wiki/Image:Kolmogorov-m.jpg 第一公理

第二公理

第三公理

全ての事象の起こる確率は０と１の間である

全事象Sの起きる確率は１である

可算個の排反事象に対する和の法則が成り立つ

http://ja.wikipedia.org/wiki/確率空間より引用

http://en.wikipedia.org/wiki/Image:Kolmogorov-m.jpg

http://en.wikipedia.org/wiki/Image:Kolmogorov-m.jpg

http://ja.wikipedia.org/wiki/

http://ja.wikipedia.org/wiki/

確率

確信度型確率

頻度型確率


Probability


個人説

間個人説

論理説

頻度説

傾向説

Logical theory

Personal theory


Frequency theory

Propensity theory



PS Laplace (1749-1827)

「場合の数の比」としての確率

その事柄の起こりうる場合の数

同程度に起こりうる全体の場合の数

ある事柄の起こる確率 =

古典的確率概念

http://en.wikipedia.org/wiki/File:Pierre-Simon_Laplace.jpg





1/52



理由不十分の原理

http://bsoza.com/money_02.php



確率

確信度型確率

頻度型確率


Probability


個人説

間個人説

論理説

頻度説

傾向説

Logical theory

Personal theory


Frequency theory

Propensity theory



頻度説「ある事柄が起こる頻度」としての確率

RA Fisher(1890-1962)

例：コインを投げてオモテがでる確率

http://en.wikipedia.org/wiki/File:Karl_Pearson_2.jpg http://en.wikipedia.org/wiki/File:R._A._Fischer.jpg

K Pearson(1857-1936)

http://en.wikipedia.org/wiki/File:Karl_Pearson_2.jpg

http://en.wikipedia.org/wiki/File:Karl_Pearson_2.jpg

http://en.wikipedia.org/wiki/File:R._A._Fischer.jpg

http://en.wikipedia.org/wiki/File:R._A._Fischer.jpg


2/5=0.4? N→∞形而上学的跳躍

コイン投げの試行数

オモテが出た割合

p→0.5


2/5=0.4?

p→0.5

N→∞形而上学的跳躍





http://en.wikipedia.org/wiki/File:John_Maynard_Keynes.jpg

In a long run,we are all dead





・容疑者Xが犯人である確率

頻度説は繰り返し事象にのみ適用可


・2050年までに地球の気温が2度以上　上昇する確率・人為的な温室効果ガスが温暖化の原因で　ある確率 (IPCC曰く90%)

・今年広島カープが優勝する確率

頻度説

確率

確信度型確率

頻度型確率


Probability


個人説

間個人説

論理説

頻度説

傾向説

Logical theory

Personal theory


Frequency theory

Propensity theory



傾向説「対象に内在する傾向」としての確率

あくまでも我々の認識作用に関わらないものとして確率を定義

コインを投げて表がでる確率コインの物理的性質

ウランの同位体の分裂確率ウランの物理的性質K Popper

(1902-1994)

http://en.wikipedia.org/wiki/File:Karl_Popper.jpg



確率

確信度型確率

頻度型確率


Probability


個人説

間個人説

論理説

頻度説

傾向説

Logical theory

Personal theory


Frequency theory

Propensity theory



論理説「合理的信念の度合い」としての確率

条件E→Xである確率が50%


JM Kaynes(1983-1946)

「確率論」1921

E→Xの確からしさの定量的記述






演繹的推論

帰納的推論

前提E→Xである確率が100%

条件E→Xである確率が中間的な%



「確率論」1921





主観的なものでは全くない！

理由不十分の原理あるいは優れた知性による直感

(条件E→Xという)論理的関係に内在するもの



「確率論」1921







歪んだコインの問題


オモテ・ウラ・ヨコ？



確率

確信度型確率

頻度型確率


Probability


個人説

間個人説

論理説

頻度説

傾向説

Logical theory

Personal theory


Frequency theory

Propensity theory



個人説「個人が持つ信念の度合い」としての確率

異なる個人は異なる信念の度合いを

持ちうる

(条件E→Xという)個人的な信念の度合いの記述としての確率FP Ramsey

(1903-1930)

http://sms.cam.ac.uk/institution/PHIL

B de Finetty(1906-1985)

http://it.wikipedia.org/wiki/Bruno_de_Finetti





画像画像画像

個人説「個人が持つ信念の度合い」としての確率

うろこ雲→次の日雨

10% 70%30%画像画像

画像画像

個人説

心理学者A Bさん

＊各種認知バイアスにも注意

次の日雨→ p×1000円not次の日雨→ (1-p)×1000円

pを選んで

個人確率

数値化の問題は「賭けの枠組み」で解決

p=0.3

・容疑者Xが犯人である確率・コインを投げてオモテが出る確率

・2050年までに地球の気温が2度以上　上昇する確率・人為的な温室効果ガスが温暖化の原因で　ある確率・今年広島カープが優勝する確率

個人説個人確率の適用範囲は広い

できますとも！

確率論数学OK！

ラムジー＝デ・フィネッティの定理 (Dutch book argument)

「必敗の賭け」にはならない合理的な賭け比率の選び方をする限りその個人確率はコルモゴルフの確率の公理を満たす

個人説個人確率に数学は適用できるの？







確率

確信度型確率

頻度型確率


Probability


個人説

間個人説

論理説

頻度説

傾向説

Logical theory

Personal theory


Frequency theory

Propensity theory



間個人確率

間個人説

個人確率

個人確率

個人確率

個人確率

間個人確率

間個人確率

個人集団「ある集団が持つ信念の度合い」としての確率

間個人確率

間個人説

個人確率

個人確率

個人確率

個人確率

間個人確率

間個人確率

個人集団「ある集団が持つ信念の度合い」としての確率

人為的な温室効果ガスが温暖化の原因で　ある確率は90% (IPCC)

確率

確信度型確率

頻度型確率


Probability


個人説

間個人説

論理説

頻度説

傾向説

Logical theory

Personal theory


Frequency theory

Propensity theory



例：ボールが青である”確率”BOX

Bag

一個色を見ずに取り出す

2/10

例：ボールが青である”確率”BOX

Bag

一個色を見ずに取り出す

もう一個取り出したら青だった

2/10

1/9

確率

確信度型確率

頻度型確率


Probability


個人説

間個人説

論理説

頻度説

傾向説

Logical theory

Personal theory


Frequency theory

Propensity theory


I-Iのまとめ・確率の公理

Frequency

Degree of belief

・古典確率

・古典確率

確率

確信度型確率

頻度型確率


Probability


個人説

間個人説

論理説

頻度説

傾向説

Logical theory

Personal theory


Frequency theory

Propensity theory


I-Iのまとめ

・古典確率・確率の公理

Frequency

リスクの解釈に与える影響は？

Degree of belief




I-2.確率概念とリスクの解釈

リスク認知

単一事象

情報量

交換可能性4つの例X

個人説間個人説頻度説傾向説

サイコロを1000回振ります。1の目がでる一回あたりの確率は？

if rational

1/6

例をもとに考えてみる１

画像

実弾は6弾中１弾のみ

Aさんが死ぬ確率は？頻度説個人説

シリンダーは回す0or11/6

A

roomロシアンルーレットをやってください

例をもとに考えてみる２

A B CD E F K

リスク評価者

実弾は6弾中１弾のみシリンダーは固定する

A→B→C→D→E→Fの順

リスクは一人あたり1/6

room

頻度説個人説

0or11/6

例をもとに考えてみる３

実弾は6弾中１弾のみシリンダーは固定する

A→B→C→D→E→Fの順A B CD E F K

0,1/6,1/2

リスクは一人あたり1/6

メモメモ

1/20

1/6

room

頻度説個人説

0or1

例をもとに考えてみる４

頻度説

サイコロ的問題ロシアンルーレット的問題

単一事象

頻度説

化学物質のリスクってどっち？

ふりかえってみる１

ふりかえってみる２リスク認知

A B CD E F K

リスク評価者

外から見ることが「科学的」なの？

交換可能性

ふりかえってみる３情報量

どれが”正しい”リスク評価なの？

A B CD E F K

メモメモ

1/20

1/61/6

一回まとめてみる頻度的確率個人確率

一義性

情報に

困る点

良い点確率的計算

リスク

一義的人それぞれも可

依存しない依存する繰り返し事象にしか

適用できない非論理的でありうる

“科学的”だと思われているリスク認知の問題まで

扱える

適用可適用可=どっちが向いてるかな？

事実上個人確率としかいいようがない1

10-6暴露の分布感受性の分布

試験動物の毒性試験PRTRデータなど

外挿外挿

外挿

外挿

外挿

外挿

外挿

外挿

事実上個人確率としかいいようがない1

10-6暴露の分布感受性の分布

試験動物の毒性試験PRTRデータなど

外挿外挿

外挿

外挿

外挿

外挿

外挿

外挿

計算結果としての「確率論的リスク」は専門家の合意に基づく一連の推定手順

により構成された（間）個人確率に基づくリスクの表現だと思う

いろいろ利点があるから2

人はfrequencyではなくdegree of beliefで動く

・繰り返し事象も非繰り返し事象もOK　（特定個人のリスク評価も可能）

・リスク認知の問題も扱える

個人確率間個人確率

個人

集団個人確率

個人確率個人確率

間個人確率個人

個人確率

・情報量の違いに対応可

画像

リスク評価が“主観”確率でいいの？

私はあなたと違って客観的に物事を見れるんです

頻度的確率（a.k.a客観確率）

リスク評価が“主観”確率でいいの？

Evidence Transparency Logic

客観性の高い(間)個人確率

確率概念は大きく分けて確信度型と頻度型の２つある

頻度型確率は繰り返し事象における比率　→厳密だが、適用範囲は狭い

個人確率は信念の度合い　→柔軟であり、適用範囲は広い

化学物質のリスク評価における確率概念はどっち？→個人確率だと私は思う

I全体のまとめ

IIのpreview

頻度型確率確信度型確率

伝統的統計学ベイズ統計学




概念的

統計的

実務的

35min

30min

25min

II.仮説検定の｢筋違い｣さとベイズの本質的な利点

II-1 仮説検定とは

II-2 仮説検定はなぜ｢筋違い｣か

II-3 ベイズの本質的な利点





そもそも統計とは“数え上げる”ことにより現象の法則性を発見する

記述統計学

統計的推測

全数調査国勢調査

部分全体

帰納的推論

State→statistics

経験科学の発展の礎

標本

母集団未知ではあるが固定されたパラメータ値

標本抽出

モデル（母集団は対数正規分布

するとか）

推測難しい数学

演繹

頻度論的な統計的推測の枠組み

推測

仮説検定の論理構成「２群間に差があるか？」差がないと仮定する（帰無仮説）

データから統計量Xを求める

「データから求めたX」以上に極端となるXの値が帰無仮説が正しいという仮定のもとで得られる確率pを計算

帰無仮説は棄却（差があると”判断”）

帰無仮説は棄却不可（差があるとはいえないと判断）

p<有意水準p>有意水準

t検定, U検定, F検定,カイ二乗検定, etc...

発病率θ=0.01

暴露群もθ=0.01だと仮定する（帰無仮説）

発病者数 (r=16) が統計量

(n=1000)

帰無仮説（θ=0.01）が正しいときn=1000で発病者数rが16以上となる確率を計算

(既知)

仮想例：発病率に差があるか？

1000人中16人発病

暴露群θ=0.016^

0 5 10 15 20 25 30

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Index

r.series

発病者数 r(n=1000）

帰無仮説の基での確率

p<0.05

θ=0.016^r=16


0 5 10 15 20 25 30

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Index

r.series

発病者数 r(n=1000）


p<0.05

θ=0.016^暴露群は有意に発病率が高い

r=16

仮想例：発病率に差があるか？帰無仮説のもとでは5%以下の確率でしか起こらない稀な事象が起こった

ちなみに：p値の意味は？p<0.05で

帰無仮説が棄却

帰無仮説が正しい確率が5%以下

対立仮説が正しい確率が95%以上

帰無仮説が正しいときに（全く同じ調査方法で）

今回のデータが得られる確率が5%以下





デミングの批判WE Deming(1900-1993)品質管理の神

日本復興の立役者

http://ja.wikipedia.org/wiki/ファイル:W._Edwards_Deming.jpg

http://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB:W._Edwards_Deming.jpg


デミングの批判WE Deming

実際の問題はAとB、二つの処理の違いが有意かどうかなどではない。（両者に）差異があるとすると・・その差異がどんなにわずかなものであっても実験をかなりの回数くり返せば有意となる。

(1900-1993)品質管理の神

日本復興の立役者

サルツブルグ「統計学を拓いた異才たち」より引用

http://ja.wikipedia.org/wiki/ファイル:W._Edwards_Deming.jpg



0 5 10 15 20 25 30

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Index

r.series

発病者数 (サンプル数1000人中）


p<0.05


θ=0.016^r=16

0 5 10 15 20 25 30

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Index

r.series



p<0.05

仮想例：発病率に差があるか？暴露群は有意に発病率が高い

θ=0.016^r=16


0 5 10 15

0.00

0.05

0.10

0.15

Index

r.series


p<0.05


θ=0.016^r=8


0 5 10 15

0.00

0.05

0.10

0.15

Index

r.series


p<0.05

仮想例：発病率に差があるか？暴露群の発病率は有意差なし

θ=0.016^r=8


800 1000 1200 1400 1600 1800 2000

0.000

0.002

0.004

0.006

0.008

0.010

0.012

Index

r.series


r=1050p<0.05θ=0.016^r=1600



800 1000 1200 1400 1600 1800 2000

0.000

0.002

0.004

0.006

0.008

0.010

0.012

Index

r.series


r=1050p<0.05θ=0.016^r=1600

暴露群は有意に発病率が高い


有意差の意味って？

n=1000, θ=0.16

n=500, θ=0.16

n=100000, θ=0.16

ありあり？

なしなし？？

ありあり？

ありあり？？

有意差リスク

｢有意差｣はリスクの指標とはならない！

^

^

^

^n=100000, θ=0.106

そもそも：目的が違う仮説検定リスク分析

帰無仮説対立仮説vs

真理あるいはその近似としての

最も尤もらしい仮説

“架空の敵”

実利効用を最大化する意思決定の支援

データ

予測・制御

データ

“科学の文法”K.Pearson

そもそも：目的が違う

データマイニング探索的データ解析

情報量規準仮説検定

バラメータ値の定量的推定

定量的推定統計的推論の３つのフェーズリスク解析のゴール

そもそも：目的が違う

データマイニング探索的データ解析

情報量規準仮説検定

バラメータ値の定量的推定

定量的推定統計的推論の３つのフェーズリスク解析のゴール

リスク分析の専門書には仮説検定の話は殆んど全く出てこない

仮説検定の実害（１）不毛かつ非本質的な議論の元凶の一つ

｢有意差なし｣と｢リスクなし｣の混同

0/1的リスク認識の一つの源

2007年5月20日（日）13時から17時，東京大

学医学部教育研究棟鉄門記念講堂において，日本

薬剤疫学会主催，日本計量生物学会共催による標

記特別シンポジウムが開催された．参加者は277

人であった．座長は大橋靖雄，津谷喜一郎が務め，

司会進行にあたった．プログラムは次のとおりで

ある．なお，本シンポジウムは録画され，UMIN

VOD：蓄積型インターネット放送（http://www.

umin.ac.jp/vod/）に公開されている．

プログラム

１．景山茂（日本薬剤疫学会理事長)

「はじめに」

２．佐藤俊哉（京都大学)

「薬剤疫学研究を理解するためのキーワード

解説」

３．横田俊平（横浜市立大学)

「インフルエンザに伴う随伴症状の発現状況

に関する調査研究(1)―研究の背景，インフ

ルエンザ罹患後の臨床症状と治療薬剤の概

要―」

４．藤田利治（統計数理研究所)

「インフルエンザに伴う随伴症状の発現状況

に関する調査研究(2)―臨床症状と治療薬剤

の関連についての統計解析―」

５．浜六郎（NPO法人医薬ビジランスセンター)

「タミフルは中枢抑制作用により異常行動死

や突然死を起こす」

６．総合討論

上記4人に，水口雅（東京大学)，別府宏圀（医

薬品・治療研究会）を含め，必要に応じてス

ライドなどを用い議論．

７．丹後俊郎（日本計量生物学会会長)

「おわりに」

背景

インフルエンザ罹患時における異常行動とタミ

フル (一般名：リン酸オセルタミビル）の関係に

ついては，2006年頃から服用後の転落死などがメ

ディアで報道されるようになり，社会的にも注目

を集めるようになった．2007年2月にはタミフル

を服用したとみられる中学生の転落死が2例報告

されたのを受け，厚生労働省は，インフルエンザ

治療に関わる医療関係者に向けての注意喚起とし

て「異常行動発現のおそれがあること，保護者は

少なくとも2日間患者がひとりにならないように

配慮すること，を説明するように」とした．しか

し，その後もタミフル服用後の転落・骨折事例が

報告されたことから，厚生労働省は中外製薬に対

し，タミフルに関する「緊急安全性情報」発出を

報告

特別シンポジウム「インフルエンザ罹患後の異常行動と薬剤疫学」

開催報告

特別シンポジウム組織委員会：

八重ゆかり（東京大学大学院疫学・予防保健学博士後期課程)

津谷喜一郎（東京大学大学院薬学系研究科医薬政策学)

大橋靖雄（東京大学大学院医学系研究科公共健康医学専攻生物統計学)

〒113-0033東京都文京区本郷7-3-1

薬剤疫学 Jpn J Pharmacoepidemiol,12(2)Dec 2007：25

「ある特定区間における有意差のあるなし」

誰得？

仮説検定の実害（２）筋違いな適用が多すぎ

予測が目的ならモデル選択等を使って！

データが正規分布に従うか？適合度検定しよう

有意差なし正規分布でOK!

正規分布を仮定したモデルでリスクの予測

さよなら仮説検定：生態リスク

Unclassified ENV/MC/CHEM(98)18

Organisation de Coopération et de Développement Economiques OLIS : 27-Jan-1998Organisation for Economic Co-operation and Development Dist. : 28-Jan-1998__________________________________________________________________________________________

Or. Eng.ENVIRONMENT DIRECTORATECHEMICALS GROUP AND MANAGEMENT COMMITTEE

OECD SERIES ON TESTING AND ASSESSMENTNumber 10

Report of the OECD Workshop on Statistical Analysis of Aquatic Toxicity Data

61088

Document complet disponible sur OLIS dans son format d'origineComplete document available on OLIS in its original format

Unclassified

ENV

/MC

/CH

EM(98)18

Or. Eng.

ワークショプの結論：1. 無影響濃度は毒性試験のサマリー

としては段階的に廃止していくべきである

Report of the OECD Workshop on Statistical Analysis of Aquatic Toxicity

1998

さよなら仮説検定：生態リスク

死亡率

化学物質濃度(mg/L)

対照区

０ 4 16 64 256 1024

*

*

*有意差あり

無影響濃度

毒性の強さの指標：無影響濃度

EPA/630/R-94/007February 1995

THE USE OF THE BENCHMARK DOSE APPROACH IN HEALTH RISK ASSESSMENT

Risk Assessment Forum U.S. Environmental Protection Agency

Washington, DC 20460

EPA/630/R-94/007February 1995

THE USE OF THE BENCHMARK DOSE APPROACH IN HEALTH RISK ASSESSMENT

Risk Assessment Forum U.S. Environmental Protection Agency

Washington, DC 20460

無毒性量よりもベンチマーク容量を毒性指標として使っていくべきである

さよなら仮説検定：ヒト健康

THE USE OF THE BENCHMARK DOSE APPROACH IN HEALTH RISK

1995

さよなら仮説検定：保全生態学

2003

信頼区間を考えよう

発病率θ0.000 0.005 0.010 0.015 0.020 0.025 0.030

●

●

●n=500

n=1000

n=100000

type-IItype-I暴露群における発病率θの90%信頼区間

ちなみに：区間推定の解釈

0.013 < θ < 0.02990%信頼区間が

θの真の値が0.013～0.029の間にある確率が90%

全く同じ方法で調査および信頼区間の算出を繰り返したときに

100回中90回はθの真の値がそれらの区間に含まれる

わかりにくい！

標本


標本抽出


するとか）


演繹


仮説検定とリスク分析はそもそもの目的が違うので相性が悪い

結論：仮説検定は使わないのが吉

有意性は誤解の元になりやすい指標

頻度主義は区間的推定に向かないと思う

II-2のまとめ

区間推定的／モデル選択的な方向で！





標本


標本抽出


するとか）



ベイズにとってパラメータとは未知ではあるが固定されたパラメータ値

未知パラメータは確率的に分布する

？

パラメータの値パラメータの値

確率

確率

頻度主義ベイズ主義（個人確率）



？分からなさ


確率

確率




？


確率


確率

全く分からない確率

？



？


確率

確率


とてもよく分かってます？

パラメータの値

ベイズによる統計的推測の枠組み

パラメータの値

確率＋データ

事前分布事後分布

ベイズの定理

ベイズの定理事後分布　尤度＊事前分布

!

p(param | data) =f (data | param)p(param)

f (data | param)p(param)dparam"

データを得る前の確信の度合い

データを得た後の確信の度合い

パラメータとデータの適合ぐあい（モデル）

∝

仮想例：発病率の推定暴露群

θ=0.016^ 1000人中16人発病

事後分布＝尤度＊事前分布

€

p(θ | r =16) =f (r =16 |θ)p(θ)f (r =16 |θ )p(θ )θ∫

事後分布！€

p(θ | r =16)∝ Be(16 +1,1000 −16 +1)

仮想例：発病率の推定


0.00 0.01 0.02 0.03 0.04

020

4060

80100

x

y

x <- seq(0, 0.04, length=100)y <- dbeta(x,1+16,1000-16+1)plot(x, y,type="h")

発病率θ

事後分布

暴露群θ=0.016^

仮想例：発病率の推定


0.00 0.01 0.02 0.03 0.04

020

4060

80100

x

y

x <- seq(0, 0.04, length=100)y <- dbeta(x,1+16,1000-16+1)plot(x, y,type="h")

90%信用区間

発病率θ

事後分布

暴露群θ=0.016^

ベイズ的な区間推定の解釈

0.011 < θ < 0.02490%信用区間が

わかりやすい！

θが0.011～0.024の間にある確率が90%

ベイズとリスク分析の相性の良さ

常にEffect sizeとProbabilityの情報の全体を取り扱う

0.00 0.01 0.02 0.03 0.04

0.6

0.8

1.0

1.2

1.4

x

y

0.00 0.01 0.02 0.03 0.04

020

4060

80100

x

y+データ→ Effect Size

Probability事前分布事後分布ProbabilityEffect size

発病率θ発病率θ


0.00 0.01 0.02 0.03 0.04

0.6

0.8

1.0

1.2

1.4

x

y

0.00 0.01 0.02 0.03 0.04

020

4060

80100

x




Risk = f(Effect size,Probability)


0.00 0.01 0.02 0.03 0.04

0.6

0.8

1.0

1.2

1.4

x

y

0.00 0.01 0.02 0.03 0.04

020

4060

80100

x




Risk = f(Effect size,Probability)

トミーマツ

画像

ベイズとリスクの相性はばっちり

それぞれの手法の見ているもの

0.00 0.01 0.02 0.03 0.04

020

4060

80100

x

y

発病率θ

ベイズ


0.00 0.01 0.02 0.03 0.04

020

4060

80100

x

y

発病率θ

ベイズ

区間推定


0.00 0.01 0.02 0.03 0.04

020

4060

80100

x

y

発病率θ

仮説検定ベイズ

区間推定

＊本質論じゃなくて実用的な話

Bootstrap×最尤法でも良くね？

発病率θ0.00 0.01 0.02 0.03 0.04

020

4060

80100

x

y

発病率θ

ｄｄｄ

乱暴に言うと実はbootstrapとMCMCってユーザー視点から見ると実は似てるかも

bootstrap

ｄ

データセット

データセット

データセット

尤度と事前情報に応じてパラメータを

乱数的に生成データセットを乱数的に生成

0.00 0.01 0.02 0.03 0.04

020

4060

80100

x

yEffectProbability

パラメータの事後分布

ベイズ（MCMC）

0.00 0.01 0.02 0.03 0.04

020

4060

80100

x

y

EffectProbability

パラメータの推定分布

最尤推定

＊本質論じゃなくて実用的な話

Bootstrap×最尤法でも良いかも

発病率θ0.00 0.01 0.02 0.03 0.04

020

4060

80100

x

y

発病率θ

＊ただし事前分布を積極的に利用しない場合に限る

II全体のまとめ仮説検定の枠組みはリスク分析には向かない→区間推定的／モデル選択的方向で

ベイズ推定は常にprobabilityとeffect size全体の情報を取り扱う→リスク分析に向く！

実用上はbootstrapとベイズは大差ないかもしれない　→事前分布の利用がキモ

IIIへ




概念的

統計的

実務的

35min

30min

25min

III.デフォルトあるいは糊代としての事前分布の利用

III-2 助け合いvia事前分布：階層ベイズ

III-3 “糊代”としての事前分布の利用

III-1 リスク分析と事前分布





事前分布とは


確率＋データ

事前分布事後分布

データを得る前のパラメータの値に関する確信の度合いを示す

事前分布とはデータを得る前のパラメータの値に

関する確信の度合いを示す

データがない場合の推定値

デフォルト値！

デフォルト値をベイズ的に眺める

データがないデフォルト値

事前分布

よくあるリスク評価手法

ベイズ解析

0.00 0.01 0.02 0.03 0.04

0.6

0.8

1.0

1.2

1.4

x

y

リスク分析

0.00 0.01 0.02 0.03 0.04

0.0

0.2

0.4

0.6

0.8

1.0

x

y

デフォルト値をベイズ的に眺める

データがないデフォルト値

事前分布

よくあるリスク評価手法

ベイズ解析

0.00 0.01 0.02 0.03 0.04

0.6

0.8

1.0

1.2

1.4

x

y

リスク分析

0.00 0.01 0.02 0.03 0.04

0.0

0.2

0.4

0.6

0.8

1.0

x

y

事前分布はリスク評価においてより好ましい性質をもつ”デフォルト”である

ちなみに：事前分布とデータの関係

事前分布事後分布データ

0.00 0.01 0.02 0.03 0.04

020

4060

80

x

y

0.00 0.01 0.02 0.03 0.04

010

2030

40

x

y

0.00 0.01 0.02 0.03 0.04

010

2030

4050

6070

x

y

n=500

0.00 0.01 0.02 0.03 0.04

050

100

150

200

250

300

x

y

0.00 0.01 0.02 0.03 0.04

050

100

150

200

250

300

350

xy

n=10000

ちなみに：事前分布とデータの関係

0.00 0.01 0.02 0.03 0.04

020

4060

80

x

y

0.00 0.01 0.02 0.03 0.04

010

2030

40

x

y

0.00 0.01 0.02 0.03 0.04

010

2030

4050

6070

x

y

n=500

0.00 0.01 0.02 0.03 0.04

050

100

150

200

250

300

x

y

0.00 0.01 0.02 0.03 0.04

050

100

150

200

250

300

350

xy

n=10000

データが多い場合も少ない場合も一貫したやり方で対応できるリスク分析の枠組みが構築可能

事前分布事後分布データ

事前分布はどう決める？1

2

3

無情報分布（平らな分布）

データそのものから決める

他のものから決める・過去の研究・知見など

・専門家へのインタビュー・歴史的コンセンサス

最尤法とほぼ同等の結果が得られる

階層ベイズモデル

猫の手も借りたいときの奥の手

III-1のまとめ

事前分布は柔軟な”デフォルト”である　→リスク分析の枠組みと親和性が高い

事前分布の決め方はいろいろある　→データそのものから決める III-2へ





発病率一定θ=0.0001

2000地域人口は100～10000

人の一様分布

2000地域の発病率の仮想データを乱数的に作成

例：地域別発病率仮想データの解析仮想データの作成

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0 2000 4000 6000 8000 10000

0.0000

0.0010

0.0020

0.0030

y.new

z1.new

人口

発病率

例：地域別発病率仮想データの解析

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0 2000 4000 6000 8000 10000

0.0000

0.0010

0.0020

0.0030

y.new

z1.new

人口

発病率

人口が小さいほど高リスク？


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0 2000 4000 6000 8000 10000

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0 2000 4000 6000 8000 10000

0.0000

0.0010

0.0020

0.0030

y.new

z1.newバイアスを避けたい！


人口

発病率

疾病地図における“小地域問題”

経験ベイズ法を使ってみよう

地域ごとの発病率 θは連続的に分布すると仮定

頻度

事前分布として利用

発病率 θ

地域ごとのデータそのものから最尤推定

経験ベイズ法を使ってみよう

地域ごとの発病率 θは連続的に分布すると仮定

Gamma(0.1,1115)0.000 0.005 0.010 0.015 0.020 0.025 0.030

050

100

150

200

250

x

y.temp

2000地点のデータそのものから最尤推定頻度

発病率 θ

事前分布を使って解析してみる

＋地点ごとのデータ

事後分布

頻度

Gamma(0.1,1115)

0.000 0.005 0.010 0.015 0.020 0.025 0.030

050

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250

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y.temp

0.000 0.005 0.010 0.015 0.020 0.025 0.030

0100

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x

y.temp

事前分布

発病率 θベイズの定理

地域別発病率の推定結果

人口

発病率

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0 2000 4000 6000 8000 100000.0000

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z1.new

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0 2000 4000 6000 8000 10000

0.0000

0.0010

0.0020

0.0030

y.new

z1.new

0.01

0.000 0.005 0.010 0.015 0.020 0.025 0.030

050

100

150

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x

y.temp

事前分布

地域別発病率の推定結果

人口

発病率

経験ベイズ推定値通常の方法

発病率 θの領域レベルでの分布

地域レベルでの発病率 θ



事前分布を介したデータの”助け合い”！

階層ベイズ法のご利益

実例：環境中濃度分布推定

地点データ地点データ地点データ

領域レベル

個々のモニタリングデータ



目的：地点ごとの環境中濃度分布の推定

解析のためのベイズモデル（ｅ．９．，ＩＥ３２１ｎｅｒｊｅｅｅｔａｌ．，２００４；Ｃｏｃｃｈｉｅｔａｌ．，２００７）と比較した場合の特徴としては，本研究における解析法は地点名と濃度データのみからなる一般的なモニタリングデータに対しても適Ｊｌｊ可能であることが挙げられる。本稿では解析の例示として，１１２１９８-２００５年度に東京都が行ったクロロホルムの河川濃度モニタリングデータに対する解析を示した。

２。モデルと方法近年，計算機能力の発達に伴い，確率的推論に

関わる科学分野においてベイズ法の利用が急速に広まってきている。ベイズ法に特徴的な利点としてはご事前悄報を解析に取り入れることができること，意志決定理論との親和性が非常に高いこと，不確実性を明示的な形で取り扱うことができること，新しい情報を順次取り入れながら推測を更新できる（順応的管理に応用できる）ことなどが挙げられる。これらの利点は特にリスク分析に関する分野において非常に本質的なものであり，リスク分析の分野においてもベイズ統計が標準的な手法として定着してきている（中西等２００８）。ベイズ法についての一般的な解説については中妻（２００７），伊庭等（２００５），渡部（１９９９），Ｓｐｉ（・１１ｅｌｈａｌｔ；（ｎ`ｅｔａｌ．（２０００），丹後（２０００）に詳しい。

階層ベイズモデル本研究では，地点内分散と地点問分散について

明示的に区別した解析を行うため，階層ベイズモデルを用いた解析を行った。階層ベイズモデルとは，「統計モデルにおける確率分布のパラメータ自体がさらに超パラメータにより規定される事前分布を持つ」という階層構造を持つベイズモデルを指す（石黒等２（ＸＭ）。階層ベイズモデルは複数の要素が様々な段階で関与する複雑な現象を，比較的単純なサプモデルの階層的組み合わせを用いて柔軟にモデル化できる有用な手法として広く用いられてきている。トｉｙｐｅｒ・ｐｌｌｌ’ａ「１１ｅｔｅｒｓ

Ｐａｒａｍｅｔｅｒｓ

Ｄａｔａ

Ｆｉ９．１Ｔｈｅｄｉａ９「ａｍｏｆｏｕｌ・ｈｉｅｒａｒｃｈｉｃａｌｍｏｄｅｌ，０ｖａｌｓ：ｒａｎｄｏｍｌｙ１１（ｊｒｌｅｒａｔｅｄｖａｌ・ｉａｂｌｅｓ．Ｒｅｃｔａｎ１１１ｅｓ：ｎｘｅｄｐａｒａｍｅｔｅｒｌ；ｔｈａｔａｒｅｅｓｔｉｍａｔｅｄｗｉｔｈｕｎｃｅｒｔａｉｎｔｋｓ．Ｓｅｅｔｅｘｔｆｏｒｆｕｒｔｈｅ【‘（ｌｅｔａｉｌｓ．

本研究で用いた階層モデルの要約図をＦｉｇ．１に示した。まず，地点内での濃度分布の常川対数他は正規分布に従うと仮定し，地点Ｈこおける月目の測定濃度の常用対数値ろを，次の統計モデルにより記述した。

ｘドｙ（ ∂ ，球）（１）ここでりまｊｔ！１点Ｈこおける常用対数濃度の平均値，叶はＪｔ！！点ｆ内における常用対数濃皮の分散，ｊゝｒ（１？。叶）は平均∂ｊ，分散吋で規定される正規分布を表している。一般に化学物質の地点毎の平均濃度の分布は対数正規分布によく適合することが知られており（ＭＣＢｅａｎａｎｄＲｏｖｅｒｓ１９９８），各地点における常用対数濃度の平均値θ，はより高次のレベル（本研究では便宜的に「領域レベル」と呼ぶ）で見た場合にはまた正規分布に従うと一般に考えられる。そのとき，各地点におけるθ，の事前分布を

４～ｙ｛Ｓβ。。。。（ｙな。。）（２）により表すことができる。ここで・∂４。と ″‰。は領域レベルで見たときの∂，の分布の平均と分散を表す超パラメータである。一方，領域レベルで見たときには，各ｊｔｋ点に；ｌｚ：μする常川丿吋１２Ｓ（濃皮の分散吋は逆ガンマ分布に従うと仮定する。この仮定はベイズモデルにおいては標準的な仮定であり，必ずしも経験的な裏付けを伴うものではないが，従来の環境中濃度推定モデルにおいても同様の仮定が用いられている（ｅ．９．，Ｃｏｃｃｈｉｅｔａｌ．２００７）。この場合，精度（分散の逆数１／叶として定義される）の事前分布を以下のガンマ分布で表すことができる。

１／イ～Ｇαｓｓα（α１．，βここで・ら。・ｊｌ？吻。は精度の分布の形状とスケールを表す超パラメータである。これらの超パラ

メータに対しては事前の情報は殆ど存在しないため超事前分布として％。については平均Ｏ・分散１０６の正規分布を・Ｉ／″ｔ。・“ｗ，・・？４-については形状パラメータ，スケールパラメータがともに０．００１のガンマ分布を与えた。これらの超事前分布は平坦な分布型を持ち，実質上・７：）無悄報分布であると言える。実際の解析においては，不検出他と報告されて

いるデータも取り扱うために，式（１）の代わりに次の式を用いた。

鳶″ｙ（θ。回μぴｇ’回ら，ｇか吸）（４）ここで・扨゜“り・砂Ｚ’ら）は鳶。が不検出値であ゜た場合に，そのろのとりうる範囲がｊゝｒ（θ。吋）の

- ４９ -

環境中濃度推定の階層ベイズモデル

領域レベル分布パラメータ

地点レベル分布パラメータ


林・柏木（2009）


Ｔａｂｌｅｌ，ＳｕｍｍａｒｙｏＨｈｅｅｒｌｖｉｒｏｎｍｅｎ抽ｌｃｏｎ（冶ｎｔｒａｔｉｏｎｄａｔａｆｏｒｃｈｌｏｒｏｆｏｒｍｉｎｐｕｂｌｉｃｓｕｒｆａｃｅｗａｔｅｒｓｉｎ‘「ｏｋｙｏ

ては，データ数が多い方が予測される平均値が小さくなる傾向がみられた。例えばレ地点１では１４報告値，地点７では１報告値の全てが不検出であるが，中央値を比べると，地点１の方がより小さい推定値が得られている。

標準偏差の事後分布の推定結果地点毎の標準偏差・７，およびｊｌｋ点問での平均値の

標準偏差ｒ‰。の事後分布の要約をＩ；｀ｉｇ．２ｂに示す。ここで，それぞれの標準偏差の値は地点内あるいは地点間での濃度の変１助性Ｖの大きさに対応する。一方，エラーバーは９０％ベイジアン信頼区間を示

しており，変！Ｒり性Ｖの大きさの推定における不確実性Ｕの大きさを示している．Ｊｔｋ点毎のらの事後分布の中央値が最も高い；１１！！点は地点４であり，事後分布の中央ｆｉとは０．２９，９０％ベイジアン信頼区間は（０．１？３，０．３９）であった。地点毎のらの事後分布の中央値が最も低い地点は地点２３であり，事後分布の中央値は０．２６，９０％ベイジアン信頼区間は（０．１９・０．３２）μｇ／Ｌであ゜た・゛ゆ，の事後分布の中央値は０．４３，９０％ベイジアン信頼区間は（０．２９，０・６９）であ゜た・‘７，とヾ‰。の事後分布の中央値を比較すると，東京都のクロロホルムの河川中濃度においては，地点内での変動性Ｖよりも地点間での変動性Ｖの方が大きいことが示唆される。

（ａ）ｊｊ　Ｒ　ｊＯＩ　Ｗ

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Ｓｉｔｅｎｕｍｂｅｒ（ｊ）

１

０５１０　ｊＷ

Ｆｉｇ．２Ｐｏｓｔｅｒｉｏｒａｎｄｐｒｅｄｉｃｔｅｄｄほｒｉｂｕｔｉｏｎｓ．Ｃｏｒｌ｛；ｅｎｔｒａｔｉｏｎｓａｒｅｓｈｏｗｎｉｎｌｏ１１１ｏ，ｓｃａｌｅ．Ｅｒｒｏｒｂａｒ


ｖａｌｕｅＳｉｎｌ）ｚｌｒｅｌｌｔｌｌｅｓｉｓａｒｅｔｈｃｐｅｒ｛ごｔｌｎｔａ８ｅｏｆｎｏｌｌ（ｊｅｔｅｃｔｅｄ（）ｂｓｅｒｖａｔｉｏｎｓｉｎｔｈｅｌ７ｃ（）ｏｒｔｅｄｄａｌａｌｉ）「ｅａｃｈｓｉｔｅ．Ｍｅｄ：Ｍｅｄｉａｎｖａｌｕｅ，Ｍａｘ：Ｍａｘｉｍｕｎｌｖａｌｕｅ，

３。結果平均値の事後分布の推定結果地点毎の対数濃度の平均他θ，および：地点毎の

平均値の平均値％。。の事後分布の要約をＦｉｇ．２ａに示す・：１也点毎のθ，の事後分布の中央値が最も高い地点は地点４であり，事後分布の中央値は０．ｆ５３，ｚｇ／Ｌ，９０％ベイジアン信頼区間は（０．３４，０．７９）μｇ／Ｌであった。地点毎の∂，の事後分布の中央値が最も低い地点は地点１４であり，事後分布の中央値は４Ｓ）。０４３μｇ／Ｌ，９０％ベイジアン信頼区間は（０．００９９Ｊ．０９４）μｇ／Ｌであった。一方，θ，の平均値１‰。の事後分布の中央イ１１｛は０．１１μｇ／Ｌ，９０％ベイジアン信頼区間は（０．０６３，０．１６）μｇ几であっ

た。概して，不検出データの数が多いＪ也削こおいてθ，の事後分布の幅が大きくなる傾向が見られた。例えば，地点１と地点３ではどちらも１４回の測定が行われているが，地点１では不検出データが１４個であり，地点３では不検出データは３個である。平均値の事後分布の幅は地点１の方が地点３よりもかなり大きく，これはデータ数が同じであっても不検出データが多いほどパラメータ推定における不確実性Ｕが大きいことを反映している。一方，全ての報告値が不検出であった場合におい

- ５１ -

林・柏木（2009）地点番号

環境中濃度の推定結果

推定濃度分布











１１１　０１１　　　０１１１１１　　０１１１４１　　　０１１１１１　０１１　　　０８１１　　　ｏｌ　　０１１９１　ｌ　０１０１１１　９１１１１　　　ｌ　ｉ　　　０１１１１　　１９１　　　１　　１１１　　　１　１１１１　　１１１１１１９１２２　　ｌ　

く


ｌ　

ｉＷＮＩ

く



１

０５１０　ｊＷ







- ５１ -


14data (ND100%)

4data (ND100%)

8data (ND13%)

1data (ND100%)


推定濃度分布











１１１　０１１　　　０１１１１１　　０１１１４１　　　０１１１１１　０１１　　　０８１１　　　ｏｌ　　０１１９１　ｌ　０１０１１１　９１１１１　　　ｌ　ｉ　　　０１１１１　　１９１　　　１　　１１１　　　１　１１１１　　１１１１１１９１２２　　ｌ　

く


ｌ　

ｉＷＮＩ

く



１

０５１０　ｊＷ







- ５１ -


14data (ND100%)

4data (ND100%)

8data (ND13%)

1data (ND100%)


推定濃度分布

200datain total


地点パラメータ

事前分布を介したデータの”助け合い”

領域パラメータ





地点パラメータ地点パラメータ

III-2のまとめ階層ベイズモデルでは

データの背後に存在する構造を組み込むことにより

事前分布を介した”助け合い”が可能になる





事前分布

リスク分析とは





個別と一般をつなぐ

地点パラメータ

領域パラメータ

地点パラメータ地点パラメータ



領域データ






領域データp(地点パラメータ,領域パラメータ|観測データ）p(観測データ|地点パラメータ）p(地点パラメータ|領域パラメータ）p(領域パラメータ）

∝ × ×

個別論個別論個別論

個々のデータ

一般論

個々のデータ個々のデータ

階層ベイズモデルの一般化



個々のデータ

一般論

個々のデータ個々のデータ

階層ベイズモデルの一般化



一般論

地点データ

p(個別パラメータ,一般パラメータ|データ）p(データ|個別パラメータ）p(個別パラメータ|一般パラメータ）p(一般パラメータ）

∝ × ×

観察とプロセスをつなぐベイジアンPBPKモデル

観測データ

階層ベイズ

Chu et al. (2009) in TAP

プロセス


観測データ

階層ベイズプロセスp(プロセス, パラメータ|データ）

p(プロセス|パラメータ）p(パラメータ）

∝ × ×

p(データ|プロセス,パラメータ）



観測データ

階層ベイズプロセスp(プロセス, パラメータ|データ）

p(プロセス|パラメータ）p(パラメータ）

∝ × ×

p(データ|プロセス,パラメータ）

分布データ

移動分散モデル


体内濃度データ

食物網モデル

過去と未来をつなぐデータの追加による推定の逐次更新

確率＋新しい

データ

事前分布

確率

事後分布

確率＋新しい

データ確率

確率＋新しい

データ確率

異なるソースを繋ぐ

海外における新型感染症データ

異なるソースを事前分布として取り込む

国内における新型感染症データ

国内における新型感染症の毒性の推定値

＋事後分布事前分布

ある新型感染症の日本における毒性を推定したい

少ない！

推定の数珠つなぎベイジアンネットワーク

風が吹く土ぼこり盲人

三味線

猫が減る鼠が増える

桶が齧られる桶屋が儲かる

インフルエンザ

タミフル処方

異常行動


インフルエンザ

タミフル処方

異常行動重篤化


Towards optimization of chemical testing under REACH: A Bayesian networkapproach to Integrated Testing Strategies

Joanna Jaworska a,*, Silke Gabbert b, Tom Aldenberg c

aModeling & Simulation Biological Systems, Procter and Gamble, Temselaan 100, 1853 Strombeek-Bever, Brussels, BelgiumbDepartment of Social Sciences, Environmental Economics and Natural Resources Group, Wageningen University, P.O. Box 8130, 6700 EW Wageningen, The NetherlandscRIVM, Antonie van Leeuwenhoeklaan 9, P.O. Box 1, NL-3720BA Bilthoven, The Netherlands

a r t i c l e i n f o

Article history:Received 15 May 2009Available online xxxx

Keywords:Integrated Testing StrategiesConceptual requirements for ITSdevelopmentBayesian inferenceBayesian networksQuantitative Weight-of-Evidence

a b s t r a c t

Integrated Testing Strategies (ITSs) are considered tools for guiding resource efficient decision-making onchemical hazard and risk management. Originating in the mid-nineties from research initiatives on min-imizing animal use in toxicity testing, ITS development still lacks a methodologically consistent frame-work for incorporating all relevant information, for updating and reducing uncertainty across testingstages, and for handling conditionally dependent evidence. This paper presents a conceptual andmethodological proposal for improving ITS development. We discuss methodological shortcomings ofcurrent ITS approaches, and we identify conceptual requirements for ITS development and optimization.First, ITS development should be based on probabilistic methods in order to quantify and update variousuncertainties across testing stages. Second, reasoning should reflect a set of logic rules for consistentlycombining probabilities of related events. Third, inference should be hypothesis-driven and should reflectcausal relationships in order to coherently guide decision-making across testing stages. To meet theserequirements, we propose an information-theoretic approach to ITS development, the ‘‘ITS inferenceframework”, which can be made operational by using Bayesian networks. As an illustration, we examinea simple two-test battery for assessing rodent carcinogenicity. Finally, we demonstrate how running theBayesian network reveals a quantitative measure of Weight-of-Evidence.

! 2010 Elsevier Inc. All rights reserved.

1. Introduction

The new European chemicals regulation REACH (CEC, 2006) isconsidered a paradigm shift regarding risk assessment and riskmanagement of chemicals (Hansen and Blainey, 2006; Führ andBizer, 2007; Van Leeuwen et al., 2007). The REACH regulation hasintroduced a single regulatory framework with common standardsfor all substances produced, or marketed, in amounts above onemetric ton per year (Schoerling, 2003; Petry et al., 2006; Hengstleret al., 2006), and has turned the burden of proof for chemical haz-ard and risk assessment from regulatory agencies to chemicalindustry (Führ and Bizer, 2007). In addition, REACH explicitly sup-ports developing and using testing schemes providing an alterna-tive to gold standard in vivo testing (CEC, 2006, Title II, Article13). These conceptual changes, the challenging information needsdefined in REACH, and the large number of substances that is ex-pected to be tested have stimulated the discussion on how to eval-uate existing data and generate new information in order to meetinformational requirements in an optimal way.

Integrated Testing Strategies (ITSs) are assumed to speed uphazard and risk assessment of chemicals, while reducing testingcosts and animal use (Grindon et al., 2006; Van Leeuwen et al.,2007; Ahlers et al., 2008; Lilienblum et al., 2008). Hence, ITSs areconsidered tools for guiding resource efficient decision-makingon chemicals’ hazard and risk management. To meet this goal,Blaauboer and Andersen (2007) have pointed to the need for anew toxicity testing and risk analysis paradigm. Looking at thecurrent practice of ITS development, as proposed in the REACHguidance documents of the European Chemicals Agency (ECHA),the selection of tests at a particular stage of an ITS is largely deter-mined by the annual tonnage volume of chemicals (see, for exam-ple, the ITSs proposed in ECHA, 2008a,b,c).

Furthermore, inference in ITSs is linked to a descriptive Weight-of-Evidence (WoE) approach, which is a step-wise procedure forintegration of data and for assessing the equivalence and adequacyof different types of information. This approach aims at ensuringoptimal integration of information from different sources and vari-ous aspects of uncertainty (Ahlers et al., 2008). Though both currentITS andWoE approaches are undoubtedly useful tools for systemiz-ing chemical hazard and risk assessment, they lack a consistentmethodological basis for making inferences based on existing infor-mation, for coupling existing informationwith newdata fromdiffer-

0273-2300/$ - see front matter ! 2010 Elsevier Inc. All rights reserved.doi:10.1016/j.yrtph.2010.02.003

* Corresponding author. Fax: +32 2 5683098.E-mail address: [email protected] (J. Jaworska).

Regulatory Toxicology and Pharmacology xxx (2010) xxx–xxx

Contents lists available at ScienceDirect

Regulatory Toxicology and Pharmacology

journal homepage: www.elsevier .com/locate /yr tph

ARTICLE IN PRESS

Please cite this article in press as: Jaworska, J., et al. Towards optimization of chemical testing under REACH: A Bayesian network approach to IntegratedTesting Strategies. Regul. Toxicol. Pharmacol. (2010), doi:10.1016/j.yrtph.2010.02.003

Towards optimization of chemical testing under REACH: A Bayesian network approach to

Integrated Testing Strategies

3.3. Bayesian networks as an operational tool of the ITS inferenceframework

The conceptual ITS inference framework requires to be imple-mented through a computational tool in order to become opera-tional. We propose to use Bayesian networks, which are defined asgraphical models of probabilistic relationships between variablesof interest in a decision-making context (Pearl, 1988; Heckermanet al., 1995; Jensen and Nielsen, 2007). A Bayesian network is a for-mal tool that is based on the axioms of probability theory (Doyle,1992). Its objective is to combine complex, and possibly conflicting,information by probabilistic reasoning and to generate the final re-sult as a posterior probability distribution, which, in our case, isthe probability that a chemical compound has a particular property.

Bayesian networks can be regarded as decision-support frame-works because of their ability to explain causal relationships and toserve as prediction models (Castillo et al., 1997; Pearl, 2000;Kjaerulff and Madsen, 2008). Bayesian networks have emerged inthe mid 1980s, and have been shown to be remarkably effectivefor encoding uncertain knowledge and using available and newdata to support decisions (Charniak, 1991; Holmes and Jain,2008). As a result, they have become widely used for more thantwo decades in many different fields such as, for example, causallearning (Steyvers et al., 2003; Gopnik et al., 2004; see Pourretet al., 2008 for an overview of applications).

Bayesian networks have been frequently applied in medicaldiagnosis and clinical decision-making (see, for example, Spiegel-halter et al., 1989; Heckerman et al., 1995; Wang et al., 1999;Dendukuri and Joseph, 2001; Georgiadis et al., 2003; Branscumet al., 2005; Engel et al., 2006; Onisko, 2008; Bhattacharjee et al.,2008; Gevaert et al., 2006). As discussed in Guzelian et al. (2005)and in Hoffmann and Hartung (2006), medical diagnosis and chem-ical testing show several conceptual similarities. This suggests theapplicability of Bayesian network approaches to support thedevelopment of chemical testing strategies.

Bayesian networks offer several merits for structuring safety-related decisions. First, they graphically illustrate causal relation-ships between the target variable, and the observed, as well asthe unobserved, variables. The graph representation, showing ex-plicit dependencies between variables, is a well-developed tool inknowledge acquisition, verification, and explanation processes(Pearl and Paz, 1987; Pearl, 1988, 2000).

Second, chemical safety evaluation frequently relies on incom-plete data sets that will differ from chemical to chemical. Bayesiannetworks can handle incomplete and different data sets in order toreach consistent conclusions.

Third, an important feature that can be addressed by Bayesiannetworks is quantification of conditional dependence betweentests. As discussed in Section 2.3, current ITS schemes cannot ad-dress this issue. Ignoring conditional dependence between tests,however, often leads to an erroneous reduction of apparent uncer-tainty when presented with information from multiple tests(Gardner et al., 2000). Conditional dependence may also arisewhen both tests essentially measure the same effect of a com-pound. In extreme cases of conditional dependence, the situationis much like doing the same test twice: no additional informationis obtained beyond what the first result already revealed. Last, theknowledge representation in a Bayesian network is fully isotropicand allows to reason based on the entire evidence.

4. An illustrative example: Bayesian network of 2 in vitro geno-toxicity tests for rodent carcinogenicity potential assessment

As stated in the introduction, the main objective of our paper isto make a conceptual proposal for improving ITS development. To

better illustrate the useful features of Bayesian networks forchemical safety evaluation and ITS development, we provide in thissection a simple example of a Bayesian network for two in vitrotests, used for supporting decision-making on the in vivo activityof a chemical. We use a stylized example which is not meant torepresent a complete ITS. Rather, we attempt to exemplify the ba-sic characteristics of Bayesian inference, and we illustrate how theinference framework can support decision-making on chemicalhazard and risk in situations as presented in Fig. 1.

Let us assume that two in vitro genotoxicity tests, the Ames testand Mouse Lymphoma Assay (MLA), are potentially available to as-sess the carcinogenic potential of a chemical (Fig. 2). The selectionof tests, to be used as information inputs to a Bayesian network, ac-counts for the possible mode(s), or mechanism(s), of action. Thisillustrates the important role of expert knowledge for constructinga Bayesian network.

In Fig. 2, the direction of the arcs symbolizes probabilistic causal-ity. Anarrow fromtheyes/no (binary) variable ‘Carcinogenic’ to eachtest indicates that carcinogenicity increases the probability that thetest result is positive. Probabilistic causality is expressed as a condi-tional probability: Pr!TjC", denoting the probability of a test result,given (‘|’) carcinogenicity status. However, we are interested in theprobability of carcinogenicity given the results obtained from thein vitro test(s). Using Bayes’ Theorem as shown in Eq. (1) in Section3.2, this can be expressed as the inverse conditional probabilityPr!CjT", which can be calculated by transforming the prior belief(probability) of a chemical being carcinogenic and the conditionalprobability Pr!TjC" into the posterior probability Pr!CjT". Hence,applying Bayes’ Theorem allows inferential reasoning to go intothe opposite direction of the probabilistic causality arrows. Thisillustrates the isotropic nature of Bayesian networks.

The arc connecting theAmes testwith theMLA test (Fig. 2) showsthatweconsider conditional dependencebetween the tests,which isreasonable, since none of the tests is a perfectly accurate gold stan-dard test (Dendukuri and Joseph, 2001) and given the evidence pro-vided by Tennant et al. (1987) and by Kim and Margolin (1994).When running the Bayesian network, it is important to account forconditional dependencebetween tests even ifwe expect it to be onlyweak (for example when underlying biological mechanisms ob-served by the tests are different). Ignoring conditional dependencemay lead to an overestimation of the posterior probability of the tar-get variable (in our case ‘‘Carcinogenicity potential”).

When considering conditional dependence between the twotests, there are two options: to either draw an arrow from theAmes test to the MLA test, or the reverse. Note that, when the con-ditional probabilities are specified in accordance with eitherchoice, this is entirely equivalent. In the present configuration ofthe network (Fig. 2), we have chosen the result of the Ames testto depend on Carcinogenicity only, while the result of the MLA testboth depends on Carcinogenicity and the result of the Ames test.This is just a matter of convenience, since more often the Ames testwill be available rather than the MLA test and the Ames test willusually be conducted first. As a result, the tests will become condi-tionally correlated and inferential reasoning can go either way.

Carcinogenic

T1Ames T2MLA

Fig. 2. Bayesian network of two in vitro genotoxicity tests to reason about theunobservable state of in vivo rodent carcinogenicity. The arrows (arcs) modelprobabilistic causality quantified by conditional probabilities.

J. Jaworska et al. / Regulatory Toxicology and Pharmacology xxx (2010) xxx–xxx 5

ARTICLE IN PRESS

Please cite this article in press as: Jaworska, J., et al. Towards optimization of chemical testing under REACH: A Bayesian network approach to IntegratedTesting Strategies. Regul. Toxicol. Pharmacol. (2010), doi:10.1016/j.yrtph.2010.02.003

Ames試験 MLA試験


III-3のまとめ

III-3のまとめ事前分布＆ベイズの定理個別と一般観測とプロセス

過去と未来

異なるソース

推論の数珠つなぎ

事前分布はデフォルトである事前分布は糊代である

事前分布＆ベイズ解析は21世紀のリスク解析において非常に有効なツールキットを提供する

III全体のまとめ

ふう

1

II

III

リスクは個人確率で捉えたほうがよさそうだ

ベイズはEffect sizeとProbabilityの全体を常に考慮

事前分布はデフォルトあるいは糊代である

ベイズ統計は個人確率に基づく

ベイズはリスク解析と相性がとても良い

ベイズはリスク解析の実務においても非常に有効なツールキットを提供

全体をまとめます

Risk Bayes

ご清聴いただきありがとうございました

CONCLUSION

+ =

Education

なぜベイズ統計はリスク分析に向いているのか？ その哲学上および実用上の理由

なぜベイズ統計はリスク分析に向いているのか？その哲学上および実用上の理由