多目標(biāo)lec2014fall決策分析基本概念與方法

主講:丁晶晶11234 1

DeGroot:Optimalstatistical Berger:Statisticaldecision ysisandBayesian 22 第

第

33 決策問題 決策問題 決策問題 ） TheFoundationsof ysisRevisited(R.A. Whatyoucan

Alternatives:options(finiteorinfinite)tobeWhatyouWhatyouRiskanduncertaintyrandomfactorsoutsidethe 態(tài)decisionmaker.Eachofthesepossiblesituations態(tài)bytherandomfactorsisreferredtoasapossiblestate tativemeasureofthevaluetothedecision oftheconsequencesof (6)

DecisionWhatyouWhatyoucando?

Whatyou 決策樹（DecisionDecisiontreesprovidesausefulofwayofvisually yingthedecisionproblemandthenorganizingthecomputationalworktoachieveadecision.Decisiontreesconsistofnodesandarcs(alsoknownasAdecisionnode,representedbyasquare,indicatesthatadecisionneedstobemadeatthatpointintheAchancenode,representedbyacircle,indicatesthatarandomeventoccursatthatpoint. ysisysisvs.GameYouraretryingtodeterminewhether willenteramarketornot.choosesitsentrydecisionfrom{Enter,Stayout}choosesitspricefrom{Low,High}Naturepicksdemand,D,tobeRecessionwith0.3orNormalwithprobability PayoffLowHighStayPayoffsto )iftheeconomyisIfthereisarecession,thepayoffofeachyerwhooperatesinthemarketis60lowerasshownin ysis(DecisionNotethatalthoughwealreadyspecifiedtheprobabilitiesofNature’sysis(Decision0.7forNormal,wealsoneedtospecifyaprobabilityfor ’smove,whichissetatprobability0.5ofLowpriceandprobabilityof0.5ofHighprice.High

Stay

TheexpectedpayoffofNewcoStayout: GameHigh

LowHigh

Stay

Low

Thedecision SupposeNewcohasHighprice:Lowprice:0.7*(-50)+0.3*(-110)=-Conclusion:highpriceispreferredifNewconentersSupposeNewcohasstayedout:Highprice:Lowprice:0.7*50+0.3(-Conclusion:highpriceispreferredifNewcon ’sEnter:0.7*100+0.3*40=Stayout:

Conclusion: 44 St.Petersburg達(dá)到。玩家可以獲得的是2（k-1）。如果這個(gè)賭局可以無限期進(jìn)行下去，的資源是無個(gè)賭局。但是實(shí)際上有人，幾乎沒有人愿意支付DanielCommentariesoftheImperialAcademyofScienceofSaintPetersburg(1738)Econometrica,(1954):ExpositionofaNewTheoryontheMeasurementofRisk Logarithmicutilityfunction:y=U(x)=blog(x/a)(a為初 ,假設(shè)初始效用為0。utility,expected“Rationalpeoplethinksatdecreasingmarginalutility：效用的增量與已擁有 annandMorgensternTheoryofgamesandeconomicbehaviorExpectedutilityJohnVon Ifpeopledonotbelievethatmathematicsissimple,itisonlybecausetheydonotrealizehowcomplicatedlifeisQ:Ameremachinecan’treallythink,canA:Youinsistthatthereissomethingamachinecannotdo.Ifyouwill lmepreciselywhatitisamachinecannotdo,andthenIcanalwaysmakeamachinewhichwilldojustthat! 布布 Riskvs.Theriskversusuncertaintydistinctionisoftenusedtodistinguishbetweentheorieswhichdonotmakeprobabilityassignmentsusethe(subjectiveorobjective)assignmentofmathematicalprobabilities)(risk) probabilityistherelativefrequencyoftheoccurrenceofaneventinalargesetofrepetitionsoftheexperiment(orinalargeensembleofidenticalsystems)andis,assuch,apropertyofaso-calledrandomvariableprobabilityisnotdefinedasafrequencyofoccurrencebutasthe usibilitythatapropositionistrue,giventheavailableinformation.Probabilitiesarethen—intheBayesianview—notpropertiesofrandomvariablesbuta tativeencodingofourstateofknowledgeaboutthesevariables. SubjectiveProbabilityandProbability“Whetherornotthedifference(betweensubjectiveprobabilityandobjectiveprobability)isimportantphilosophically,wedon’tfeelitshouldmakeanydifferenceoperationally…”Atriple(?,F,P)iscalledaprobabilityspaceprovided?isanyset,Fisaσ-algebraofsubsetsof?,andPisaprobabilitymeasureonF. Anexampleofill-definedBertrand’sparadox.Takeacircleofradius2inchesin neandchooseachordofthiscircleatrandom.Whatistheprobabilitythischordintersectstheconcentriccircleofradius1inch? TheprobabilityofeventAisI’mabouttotossacoin.WhatistheprobabilityofitcomingupHead?Iconsiderparkingmycaronthestreettonight.Whatistheprobabilitythatitwillbestolenovernight?Iamabouttoundergoamedicaloperation.WhatistheprobabilitythatIwillsurviveit?WhatistheprobabilityofwarintheMiddleEastthis 根據(jù)決策人對(duì)的了解去設(shè)定，這樣的概率反映了決策人對(duì)掌握的知識(shí)所建立起來的信念，稱為主 ImportanceofpriorinformationAlady,whoaddsmilktohertea,claimstobeableto whethertheteaorthemilkwaspouredintothecupfirst.Inalloftentrialsconductedtotestthis,shecorrectlydetermineswhichwaspouredfirst.Adrunkenfriendsayshecanpredictthe eofaflipofafaircoin.Intentrialsconducedtotestthis,heiscorrecteachConclusion:theclaimsareWhatabout usibilityofthe ThomasBayes:(1701–AprilEnglishAnessaytowardssolvingaprobleminthedoctrineofchances,Bayes(1763) ysis:Theapproachtostatisticswhichformallyseekstoutilizepriorinformation. Wald(1950)：Statisticaldecisionfunctions理論觀點(diǎn)： Raiffa,RobertSchlaiffer和JohnPratt等，開始應(yīng)用決在這樣的背景下，決策分析一詞首先由Stanford的RonaldHoward（1966）在Decision ysis:applieddecisiontheory一文中提出。 55 ReviewofLectureAdecisionmaker,frame,whatyouwant,whatyouknow,whatyoucando，

St.Petersburg Riskvs.Theriskversusuncertaintydistinctionisoftenusedtodistinguishbetweentheorieswhichdonotmakeprobabilityassignmentsusethe(subjectiveorobjective)assignmentofmathematicalprobabilities)(risk) TheprobabilityofeventAisI’mabouttotossacoin.WhatistheprobabilityofitcomingupHead?Iconsiderparkingmycaronthestreettonight.Whatistheprobabilitythatitwillbestolenovernight?Iamabouttoundergoamedicaloperation.WhatistheprobabilitythatIwillsurviveit?WhatistheprobabilityofwarintheMiddleEastthis probabilityistherelativefrequencyoftheoccurrenceofaneventinalargesetofrepetitionsoftheexperiment(orinalargeensembleofidenticalsystems)andis,assuch,apropertyofaso-calledrandomvariableprobabilityisnotdefinedasafrequencyofoccurrencebutasthe usibilitythatapropositionistrue,giventheavailableinformation.Probabilitiesarethen—intheBayesianview—notpropertiesofrandomvariablesbuta tativeencodingofourstateofknowledgeaboutthesevariables. SubjectiveProbabilityandProbability“Whetherornotthedifference(betweensubjectiveprobabilityandobjectiveprobability)isimportantphilosophically,wedon’tfeelitshouldmakeanydifferenceoperationally…”Atriple(?,F,P)iscalledaprobabilityspaceprovided?isanyset,Fisaσ-algebraofsubsetsof?,andPisaprobabilitymeasureonF. Anexampleofill-definedBertrand’sparadox.Takeacircleofradius2inchesin neandchooseachordofthiscircleatrandom.Whatistheprobabilitythischordintersectstheconcentriccircleofradius1inch? 根據(jù)決策人對(duì)的了解去設(shè)定，這樣的概率反映了決策人對(duì)掌握的知識(shí)所建立起來的信念，稱為主 ImportanceofpriorinformationAlady,whoaddsmilktohertea,claimstobeableto whethertheteaorthemilkwaspouredintothecupfirst.Inalloftentrialsconductedtotestthis,shecorrectlydetermineswhichwaspouredfirst.Adrunkenfriendsayshecanpredictthe eofaflipofafaircoin.Intentrialsconducedtotestthis,heiscorrecteachConclusion:theclaimsareWhatabout usibilityofthe ThomasBayes:(1701–AprilEnglishAnessaytowardssolvingaprobleminthedoctrineofchances,Bayes(1763) ysis:Theapproachtostatisticswhichformallyseekstoutilizepriorinformation. Wald(1950)：Statisticaldecisionfunctions理論觀點(diǎn)： Raiffa,RobertSchlaiffer和JohnPratt等，開始應(yīng)用決的RonaldHoward（1966）在Decision applieddecisiontheory一文中提出。 主講:丁晶晶 經(jīng)驗(yàn) 層次 分 分 計(jì) 第 分 經(jīng)驗(yàn) 層次 分 分 計(jì) 第 分 HowdowedeterminethesubjectiveAretheremoreorlessrationalwaystoassignsuchArewesurethattheprobabilityapparatusistherightonetocaptureourintuition? Axiom 1Subjectivedeterminationofprior2umumEntropy34UsingMarginalDistributiontoDeterminethe5Hierarchical 1 概率盤（TheProbability 了獲得這一概率，決策人詢問一個(gè)石油經(jīng)銷商。 TheHistogram TheRelativeLikelihood例子：假定θ∈[0,1].先選擇“最可能點(diǎn)”和“最不可以θ=0為基準(zhǔn)來確定θ=1/4,?,?的可能性。假設(shè)θ=1/4的可能性為θ=0可能性的1.5θ=1/2和1為θ=0可能 TheRelativeLikelihood 布為正態(tài)。假設(shè)估計(jì)其分布的中位數(shù)，1/4,?分位點(diǎn)分別為0，-1和1.N(0,2.19) 小結(jié)小結(jié)對(duì)似然法得到的圖形差距不大,會(huì)起到簡(jiǎn)化作用影響決策結(jié)果：( , , ,( 22 Theprincipleofindifference:iftherearenpossiblees,andthereisnoreasontoviewoneasmorelikelythananother,theneachshouldbeassignedaprobabilityof1/n 位置參數(shù)無信息先驗(yàn)—（location隨位置參數(shù)無信息先驗(yàn)—（location隨 f(x f(

對(duì)任意,有 標(biāo)度參數(shù)無信息先驗(yàn)—（scale

1f(

，正態(tài)分布N 2 1f(x ) P* A)c

c))c1d(c

c

ThePrincipleofindifferenceorTheprincipleofinsufficientreasonTransformationinvariantpriors(locationinvariantpriorandscaleinvariantprior) 33 極大熵先驗(yàn)分布 umEntropyTheAmericanClaudeShannon(1916–2001)wroteAmathematicaltheoryofcommunicationin1948,anarticlethatcreatedinformationtheory熵的概念：表示的平均信息量，為無序程度（不確定程度或者不可預(yù)測(cè)程度）的測(cè)度，即熵越無序（越不確定、越不可Question1:概率大 極大熵先驗(yàn)分布umEntropy極大熵先驗(yàn)分布umEntropy i 00

i) : . 44 =? (|? 例題：假設(shè)X表示被測(cè)試者的得分，~(,，其中θ()(,，其中參數(shù)都為已知。求解X的邊際分? 2i:? 21 =

222

– Example 例題：假設(shè)=1,根據(jù)經(jīng)驗(yàn)樣本的均值為1，預(yù)測(cè)的方差為3.利用矩方法確定()。i:n 有X~ ))。根據(jù)矩方法推論=1， =3=1，=212,), Example)Example)i:n ,( )),其中為常數(shù) )∝

exp[? ]= ? 55Hierarchical 層次先驗(yàn)分布（Hierarchicalprior）也稱作多階段先 參數(shù)θ=(θ1,…,θp).參數(shù)θi,i.i.d.分布為N( 為確定值100， .請(qǐng)確 |,)i:1st |

]2nd , = ?Hierarchical =∫ 1 ?

?10010

內(nèi)部因素外部因素

公司管 公司操 客戶償債意愿 P(X7|X5,X6)P(X5, 1ststage P(X5|X1,X2)P(X1,X) P(X6|X3,X4)P(X3,4)4P(X1,X2 P(X1)P(X2P(X3,X4

2ndstage3rdstage ReviewofLectureexistence

OutlineofLecture1Subjectivedeterminationofprior2umumEntropy34UsingMarginalDistributiontoDeterminethe5Hierarchical

.

.

?

.

? . ≥ 44 =? (|? 例題：假設(shè)X表示被測(cè)試者的得分，~(,，其中θ()(,，其中參數(shù)都為已知。求解X的邊際分? 2i:? 21 =

222

– Example 例題：假設(shè)=1,根據(jù)經(jīng)驗(yàn)樣本的均值為1，預(yù)測(cè)的方差為3.利用矩方法確定()。i:n 有X~ ))。根據(jù)矩方法推論=1， =3=1，=212,), Example)Example)i:n ,( )),其中為常數(shù) )∝

exp[? ]= ? EmpiricalEmpirical EmpiricalEmpirical

EmpiricalEmpirical Empirical Empirical 55Hierarchical 層次先驗(yàn)分布（Hierarchicalprior）也稱作多階段先 參數(shù)θ=(θ1,…,θp).參數(shù)θi,i.i.d.分布為N( 為確定值100， .請(qǐng)確 |,)i:1st |

]2nd , = ?Hierarchical =∫ 1 ?

?10010

內(nèi)部因素外部因素

公司管 公司操 客戶償債意愿 P(X7|X5,X6)P(X5, 1ststage P(X5|X1,X2)P(X1,X) P(X6|X3,X4)P(X3,4)4P(X1,X2 P(X1)P(X2P(X3,X4

2ndstage3rdstage 主講:丁晶晶 經(jīng)驗(yàn) 層次 分 分 計(jì) 第 分 經(jīng)驗(yàn) 層次 分 分 計(jì) 第 分 HowdowedeterminetheutilityAretheremoreorlessrationalwaystoassignsuchArewesurethattheutilityapparatusistherightonetocaptureourintuition? 12Definitionofutilityandaxiom3Existenceofutility 11 定義在非空集X上的關(guān)系，,forallx,y,z

together ,forall ,forallx, 稱的(anti-symmetric):if implythatx=y,forallx,y∈ 完全的(連通的)(complete):ifeither ,forallx,y∈ 偏序（partialorder）ifarelationistransitive,reflexiveandanti-symmetric,完全序（completeorderorlinearorder）：ifarelationisapartialorderandcomplete,等價(jià)關(guān)系（equivalentrelation）：ifitisreflexive,symmetricandtransitive.

，P1優(yōu)于P2，或者 理性偏好關(guān)系（rationalpreference ? xyyz那么xx~yy~z那么x~xyz那么x 22 優(yōu)先關(guān)系的單調(diào)性 33 優(yōu)勢(shì)集（uppercontourset）I(x)={yyx}，所有劣勢(shì)集（lowercontourset）：S(xyxy}都是閉 公理3:在X上的優(yōu)先關(guān)系公理3:在X上的優(yōu)先關(guān)系是連續(xù)的,即對(duì)于任何確定的x，它的劣勢(shì)集Ix)公理3(等價(jià)形式):在X上的優(yōu)先關(guān)系是連續(xù)的,即這種優(yōu)先關(guān)系在極限條件可以保持:對(duì)任意的數(shù)列{(xn,yn)},有xnyn( n);如果x limxn,y , 那么有xy從下往上證從上往下證（seethenext 優(yōu)（劣）勢(shì)集為閉集?limxn=x limyn=y?y? 優(yōu)（劣）勢(shì)集為閉集?limxn=x limyn= 1,y?xn(n>N1) 2,yn?x(n>N2)y? 優(yōu)（劣）勢(shì)集為閉集?limxn=x limyn= 1,y?xn(n>N1) 2,yn?x(n>N2)y?limlimyn= 優(yōu)（劣）勢(shì)集為閉集?limxn=x limyn= 1,y?xn(n>N1) 2,yn?x(n>N2)y? 3,yn?y(n>N3) ,?( yk(n)?ylimyn= 優(yōu)（劣）勢(shì)集為閉集?limxn=x limyn= 1,y?xn(n>N1) 2,yn?x(n>N2) 3,yn?y(n>N3) ,?( yk(n)?yyn?x(n>N=limyn=y? 優(yōu)（劣）勢(shì)集為閉集?limxn=x limyn= 1,y?xn(n>N1) 2,yn?x(n>N2) 3,yn?y(n>N3) ,?( yk(n)?yyn?x(n>N=limyn=y? 優(yōu)（劣）勢(shì)集為閉集?limxn=x limyn= 1,y?xn(n>N1) 2,yn?x(n>N2) 3,yn?y(n>N3) ,?( yk(n)?yyn?yk(m)(n>N4)yn?x(n>N=limyn=y? 優(yōu)（劣）勢(shì)集為閉集?limxn=x limyn=y?y? 1,y?xn(n>N1)

2,yn?x(n>N2)yn?yn?x(n>N=limyn= 3,yn?y(n>N3)

,?( yk(n)?yyn?yn?yk(m)(n>N4)xn?yk(m)(n>N4) 優(yōu)（劣）勢(shì)集為閉集?limxn=x limyn=y?y? 1,y?xn(n>N1)

2,yn?x(n>N2)yn?xyn?x(n>N=limyn=x? 3,yn?y(n>N3)

,?( yk(n)?yyn?yn?yk(m)(n>N4)xn?yk(m)(n>N4) 優(yōu)（劣）勢(shì)集為閉集?limxn=x limyn=yk(m)?yk(m)?(k(m)y? 1,y?xn(n>N1)

2,yn?x(n>N2)yn?xyn?x(n>N=limyn=x? 3,yn?y(n>N3)

,?( yk(n)?yyn?yn?yk(m)(n>N4)xn?yk(m)(n>N4) 優(yōu)（劣）勢(shì)集為閉集?limxn=x limyn=yk(m)?yk(m)?(k(m)y? 1,y?xn(n>N1)

2,yn?x(n>N2)yn?xyn?x(n>N=limyn=x? 3,yn?y(n>N3)

,?( yk(n)?yyn?yn?yk(m)(n>N4)xn?yk(m)(n>N4) 證明策略：（構(gòu)造法 任意(0,0,…,0)?x=(x1,…,xK?證明存在a(x)(1,1,…,1)~ 任意(0,0,…,0)?x=(x1,…,xK?證明存在a(x)(1,1,…,1)~連續(xù)性S(x)={a為正實(shí)數(shù)x?a(1,1,…,1)} 任意(0,0,…,0)?x=(x1,…,xK?證明存在a(x)(1,1,…,1)~S(xS(x)={a為正實(shí)數(shù)x?a(1,1,…,1)}I(x)∪S(x)I(x)∩S(x) 任意(0,0,…,0)?x=(x1,…,xK?證明存在a(x)(1,1,…,1)~S(S(x)={a為正實(shí)數(shù)x?a(1,1,…,1)}?∈,I(x)∪S(x)I(x)∩S(x) 任意(0,0,…,0)?x=(x1,…,xK?證明存在a(x)(1,1,…,1)~S(x)={S(x)={a為正實(shí)數(shù)x?a(1,1,…,1)}?∈,I(x)∪S(x)I(x)∩S(x) ReviewofLecture

公司管

內(nèi)部因素

公司操

外部因素

證明策略：（構(gòu)造法 任意(0,0,…,0)?x=(x1,…,xK?證明存在a(x)(1,1,…,1)~ 任意(0,0,…,0)?x=(x1,…,xK?證明存在a(x)(1,1,…,1)~連續(xù)性S(x)={a為正實(shí)數(shù)x?a(1,1,…,1)} 任意(0,0,…,0)?x=(x1,…,xK?證明存在a(x)(1,1,…,1)~S(xS(x)={a為正實(shí)數(shù)x?a(1,1,…,1)}I(x)∪S(x)I(x)∩S(x) 任意(0,0,…,0)?x=(x1,…,xK?證明存在a(x)(1,1,…,1)~S(S(x)={a為正實(shí)數(shù)x?a(1,1,…,1)}?∈,I(x)∪S(x)I(x)∩S(x) 任意(0,0,…,0)?x=(x1,…,xK?證明存在a(x)(1,1,…,1)~S(x)={S(x)={a為正實(shí)數(shù)x?a(1,1,…,1)}?∈,I(x)∪S(x)I(x)∩S(x) （傳遞和完全性 經(jīng)驗(yàn) 層次 分 分 計(jì) 第 分 經(jīng)驗(yàn) 層次 分 分 計(jì) 第 分 11 1;...;Simple compound 公理體系（independence0 2,當(dāng)且僅當(dāng)1+(1 ) 2+(1 )0 1，1~2,當(dāng)且僅當(dāng)1+(1 )3~ 2+(1 ) (1 (1 ∈(,)LL

(p1,c1;p2,c2)

(p1,c1;p2,c0)vs. (3)若p1<p2,則P1P2. p2,則P2 設(shè)抽獎(jiǎng)T(xc1;1xc2且c1c2.若對(duì)于滿足優(yōu)劣關(guān)系c1cc2的任意結(jié)果值c,則必存在xp(0p1使得T=(p,c1;1-p,c2)~c.其中抽獎(jiǎng)c稱為抽獎(jiǎng)的確定當(dāng)p稱為c的無差異概率. 一個(gè)簡(jiǎn)單抽獎(jiǎng)P'=(p',c*,1-p',co)使得P'~P.其中??max , ? n pjqj,這里qj( 無差異概率滿足效用函數(shù)定義序關(guān)系；（性質(zhì)） )f ) 22 Def1:apreferencerelationisriskaverseifforanyprospectp,[Ep]p.Def2: beapreferencerepresentedbytheutilityfunctionU.Thepreferencerelationisriskaverseifandonlyifuisconcave.Def3:Apreferencerelationisriskaverseifandonlyifforallp,Ep CE(p). Def1Def?P= Def1?Def]?)=])≥U(P)=∑ Def2?DefJensen’sInequality, ) )=[E(p)]? Def1Def ~[ Risk Fair抽獎(jiǎng)

Arrow-PrattMeasuresofRiskAbsoluteriskr u Relativeriskr w0u Arrow-PrattmeasuresofriskArrow’smeasureofriskabsoluteriskaversion(pu(wo+ )+(1-p)U(wo–)=relativeriskaversion(pu(wo )+(1-p)U(wo = Arrow-PrattmeasuresofriskPratt’smeasureofriskabsoluteriskaversion(pu(wo+ )+(1-p)U(wo–)=U(wo-π)?)relativeriskaversion(pu(wo )+(1-p)U(wo =U(wo)- 風(fēng)險(xiǎn)厭惡的比較(1)Thepreferencerelation1ismoreriskaversethanifforanyprospectpand[c],p1[c]impliesthatp2[c](2)Thepreferencerelationismoreriskaversethan2ifCE1(p) CE2(p)forallp. 風(fēng)險(xiǎn)厭惡的比較(3)Letu1andu2bevNMutilityfunctionsrepresent-moreriskaversethan2iftheby is

(4)Letu1andu2bedifferentiablevNMutilityfunctionsrepresenting1and2respectively.Ther2(x)≤r1(x),whereri(x)isabsoluteriskaversion 風(fēng)險(xiǎn)厭惡的比較 ismoreriskaversethan(2) ) (2) ) (1)? 2(=con ()) ()))=con )) ) ()

= =? ())=? ((4)r2(x)≤(4)r2(x)≤′1() ( ′ 1 (1)=>(1)?(1)? =con?,

?,,,0 < ()+(1 ( ()) ())+(1 ()),~2( 1 ,z,[

1 =2((=2((1 ,z~2( 1 ,z<1((,;(1),[ (, 1 假設(shè)存在2(( 1 1((,;(1 ),( 1 ?( 1 ConstantAbsoluteRiskAversion– eAw,U

UConstantRelativeRiskAversion– U

1),

U ReviewofLecture

) ? )= ）CARAand )

)

經(jīng)驗(yàn) 層次 分 分 計(jì) 第 分 經(jīng)驗(yàn) 層次 分 分 計(jì) 第 分 Mean-VarianceCriterion(Sec.StochasticDominance(Sec. 11(Sec. 評(píng)價(jià)函數(shù)：(a 2 假設(shè)有n種有價(jià)。資產(chǎn)組合用(X1,X2,…,Xn)表示，益率為用Ri表示，其均值為，方差為，那么資產(chǎn)nXjX X

,Rj i

cov)。分析N X 2 X

2XXcov(R,R 1 [homogeneousexpectations] R斜率： )M

)M i，標(biāo)準(zhǔn)差為i.其與市場(chǎng)組合按 a2a22i 2M M

d/d/a M

R

iM iM R 22 （Sec. ki jkr)r) ,ki jkr)r) 量, Sec5.1 第一等隨機(jī)優(yōu)勢(shì)（First-orderstochastic滿足以下條件稱F第一等隨機(jī)占優(yōu)G，記為 I,有F(x) G(x)xl,有F(xl G(xl允許（probabilisticallyadmissible） Absolutedominancevs.First-order性質(zhì)：Absolutedominancefirst-order

Pr(X Y* aHint:X X,Y G1(F(X)),X~a 如果不做處理，病情可能，患者因此可能或中可能會(huì)在手術(shù)中或者導(dǎo)致殘疾。醫(yī)生應(yīng)該如何

P()=P殘疾0.25P治愈)=0.70P()=0.01P(殘疾)= P(治愈)=

00 例當(dāng)當(dāng)時(shí) 不是必要條件 u(x)dF(x)

u(x)dG(x) ? = 0 F0

F 第二等隨機(jī)優(yōu)勢(shì)（second-orderx R,有 x, F u(x)dF u(x)dG(x) ，則E(X E(Y ? = E(X F F0 定義與定理（Sec5.6, 如果F G，則F?G是不可能的如果I=[0,1],F=G,則對(duì)于所有嚴(yán)格凹的、嚴(yán)格遞增的u有,>(,)，當(dāng)且僅當(dāng)對(duì)于所有嚴(yán)格凹的、嚴(yán)格遞減的u有,>(,)于所有嚴(yán)格凹的u有,>(,)。如果F G,則F一般情況下，F(xiàn) G和F 設(shè)X和Y是集Q中的有限個(gè)點(diǎn)，={ ,…, }? 有<<<.==≥,==≥=∑=,? ≤=≥=≤=∈)? 分布函數(shù)F和G只交叉一次，那么 如果F G,則FF 如果F G，則條件F ，對(duì)于F?G或者F? ReviewofLecturen種有價(jià)

1)∈1)∈, )(2)∈, )(? ∈ 有 ) ( 有 ) ,有 () ( ) ( 經(jīng)驗(yàn) 層次 分 分 計(jì) 第 分 經(jīng)驗(yàn) 層次 分 分 計(jì) 第 分 OutlineofLecture Claxton,K., ann,PJ.,etal.Bayesianvalueofinformation AnApplicationtoa ModelofAlzheimer’sDisease.InternationalJournalofTechnologyAssessmentinHealthCare,17:1(2001),38–55. 11 , =

]? ,

0 ： (： () EX| (X ] X (XX [ 若θ取n個(gè)值θj（j=l,2,n），H取m個(gè)值Xi（i＝1,p(X1 p(X1 p(X1 p(X2

1 p(X2 2

p(X2 n p(Xm|1 |

p(Xm

p(Xm n 風(fēng)險(xiǎn)函數(shù): (X (x))f(xS風(fēng)險(xiǎn): (X (X

opt(X (X 例2 準(zhǔn)則(Bayesp(X1 p(X1 p(X1 p(X2p(Xm

1 p(X21 p(Xm

2 2

p(X2 np(Xm n 決策分析正規(guī)型(Normalform 決策準(zhǔn)則：如果行動(dòng)規(guī)則1的風(fēng)險(xiǎn)小于行動(dòng)規(guī)則2在同樣先驗(yàn)分布()下的風(fēng)險(xiǎn)值，r(,1)r(,則定義行動(dòng)規(guī)則1優(yōu)于行動(dòng)規(guī)則2 r * min[r )] ,X

(x,A))f(x

,

(x, f(x X

, *(x, f(x 令*(x, ,X

A(x , *(x,amin f(x a

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(x,A))f(x

,

(x, f(x X

, *(x, f(x 令*(x, ,X

A(x , *(x,amin f(x ≤≤ a

,a) *(x,A) f(x| (X (X))f(x ) (X))f(x 對(duì)任意x如下定義** a

,a)f(x|

,a)f(x

|

,a)f | (|x) , 例例 050 1| /4Pr(X 1|W 1/3,Pr(X 0|W 2/3 。 AtanygivenHe,decisionmaker,hasaprobabilitydistributionfortheparameterΘ.AstimeHegainsinformationaboutΘfromvarioussourcesandusesthisinformationaboutΘtorevisehisdistributionforFromtimetoWhenhemustchooseadecisionwhoseconsequencesarerelatedtoΘ,hewillselectadecisionthatisoptimalagainsthiscurrentdistributionforΘ.

(Sec = = = = (

= (

( = ( ( ()|=

( |

( |== Improper 例3 x/e4 e

2(X2

1/

4e4

1/e4 1/

2e 3 ，則完全信息值xi對(duì)狀態(tài)θ的期望收益值稱為完全信息價(jià)值的期望值（expectedvalueofperfectinformation E maxQ E

Q

( , minL 狀態(tài)暢銷滯銷概率生產(chǎn)不生產(chǎn)00

maxQa

–EQaopt

5 maxQ a

補(bǔ)充信息價(jià)值：全部補(bǔ)充信息值xi價(jià)值的期望值，稱EVAI（ExpectedValueofAdditionalInformation）。 1：EVAI ){E ( (x), )]} ,(((EVAI (

aopt –E

(E)E

(x), 050 1| w 050 4 1| 2/ EVAI與EVPI E(Q(*

Qt

*(x))f(x

Qt) *(x))f(x ,a*( ))f(x 例

也可以整批部件都查就用于設(shè)備裝配，這樣在設(shè)備最終 θ1次品率θ2次品率 θ1次品率θ2次品率0 p(θ1|h1)=0.84，p(θ2p(θ1|h2)=0.44，p(θ2h1h2 p(θ1|h1)=0.87，p(θ2p(θ1|h2)=0.5033，p(θ2p(θ1|h3)=0.14，p(θ2 p(θ1|h1)=0.8905，p(θ2p(θ1|h2)=0.5621，p(θ2p(θ1|h3)=0.1685，p(θ2p(θ1|h4)=0.031，p(θ2決策結(jié)果：如果h1或h2發(fā)生，則 )，如果h3或h4發(fā)生，則a1(全檢 通常有CS(N)=Cf+CvN，（N非零 收益(ExpectedNetGainfromSampling)，記作ENGS(N)是抽樣決策的重要指標(biāo)，以此確定抽樣工作的必要性。當(dāng)ENGS(N)為正時(shí)，抽樣分布抽樣方案 選擇不抽檢，期望損失 為“合格”，期望損失2400，概率 為“全合格”，期望損失1642，概率為“兩合格，一不合格”，期望損失6568，概率為“一合格，兩不合格”，期望損失2527，概率 例 ReviewofLecture?,

| ?(

?) ,∈ | ∈ ) , |

)∈argmin |∈ OutlineofLectureClaxton,K., ann,PJ.,etal.Bayesianvalueofinformation AnApplicationtoa ModelofAlzheimer’sDisease.InternationalJournalofTechnologyAssessmentinHealthCare,17:1(2001),38–55. 3 ，則完全信息值xi對(duì)狀態(tài)θ的期望收益值稱為完全信息價(jià)值的期望值（expectedvalueofperfectinformation E maxQ E

Q

( , minL 狀態(tài)暢銷滯銷概率生產(chǎn)不生產(chǎn)00

maxQa

–EQaopt

5 maxQ a

態(tài)分布( )。不生產(chǎn)利潤(rùn)為0.計(jì)算EVPI. maxQa

–EQaopt 1 2Unitnormalloss2()

(′)

? ?

? ( 補(bǔ)充信息價(jià)值：全部補(bǔ)充信息值xi價(jià)值的期望值，稱EVAI（ExpectedValueofAdditionalInformation）。 1：EVAI ){E ( (x), )]} ,(((EVAI (

aopt –E

(E)E

(x), 050 1| w 050 4 1| 2/ EVAI與EVPI

t

*(x))f(x

Qt) *(x))f(x ,a*( ))f(x ,a*( 例

也可以整批部件都查就用于設(shè)備裝配，這樣在設(shè)備最終 θ1次品率θ2次品率 θ1次品率θ2次品率0 p(θ1|h1)=0.84，p(θ2p(θ1|h2)=0.44，p(θ2h1h2 p(θ1|h1)=0.87，p(θ2p(θ1|h2)=0.5033，p(θ2p(θ1|h3)=0.14，p(θ2 p(θ1|h1)=0.8905，p(θ2p(θ1|h2)=0.5621，p(θ2p(θ1|h3)=0.1685，p(θ2p(θ1|h4)=0.031，p(θ2決策結(jié)果：如果h1或h2發(fā)生，則 )，如果h3或h4發(fā)生，則a1(全檢 通常有CS(N)=Cf+CvN，（N非零 收益(ExpectedNetGainfromSampling)，記作ENGS(N)是抽樣決策的重要指標(biāo)，以此確定抽樣工作的必要性。當(dāng)ENGS(N)為正時(shí)，抽樣分布抽樣方案 選擇不抽檢，期望損失 為“合格”，期望損失2400，概率 為“全合格”，期望損失1642，概率為“兩合格，一不合格”，期望損失6568，概率為“一合格，兩不合格”，期望損失2527，概率 例 44Case A1A1 A2A2 A3A3 A其他A其他 IntroductionIntroduction&ture IntroductionIntroduction&ture IntroductionIntroduction&ture Introduction&tureAlthoughpreviousinvestigationsprovidesomeinsightintothesequestions,researchinthisareagenerally…focuseson…ratherIntroduction&ture Separate turepositioningthestudyinrelationtotheexistinglitureintheIntroduction,andtherebysettingthesceneandexiningthemotivationforthepaperfocusedonthepaperswhicharedirectlyrelevanttotheEmphasizethefindingsofpreviousresearch-notjusttheresearchMethodologiesandnamesofvariablesstudied…Itshouldbefocusedonwhatisneededforthespecificstudy.…thedevelopmentofthetheoreticalunderpinningsforthepaper.Theli turereviewshould,…buildatheoreticalframework…orthetheoreticalbasis.Iwouldnormallyexpecttheli turereviewtoleaddirectlytotheresearchquestions MainResearchmethods(orthemodelResults yses(orProportionsandDiscussion Lossfunction

SamplingSampling

modelofAD ResearchResearchmethods:aProbabilistic ResearchResearchmethods:aProbabilistic Results:TheResultsof ExpectedNet$In$1064 Inferenceisirrelevanttotreatment Results:EVPIfortheChoiceBetween Results:EVPIfortheChoiceBetween Results:implicationof Results:EVPIforModel Discussion& Discussion& Discussion& Discussion&Inconsistentresults,unexpected Whattheauthorreally WhatWhattheauthorreallymeant) ReviewofLecture ( (

Introduction&Li Separate tureResearchRM2.1,RM2.2,Results Discussionandconclusion(DC1,DC2,DC3,DC4) OutlineOutlineofLectureMarkovMultipleCriteriaDecisionMaking,MultipleattributeUtilityTheory:Thenexttenyears(ManagementScience,1991,38(5):645-654)MultipleCriteriaDecisionMaking,Multi-attributeUtilityTheory:RecentplishmentsandWhatLiesAhead(ManagementScience,2008,54(7):1336–1349) 11 多目標(biāo)、多準(zhǔn)則和多屬性 多目標(biāo)、多準(zhǔn)則和多屬性（MultipleCriteria多目標(biāo)、多準(zhǔn)則和多屬性之間的比較從而做出權(quán)衡，叫做多屬性決策（Multipleattributedecisionmaking,MADM）順序的叫做多目標(biāo)決策（Multipleobjectivedecisionmaking,MODM）Hwang,C.L.Yoon,K.L.,1981.MultipleAttributeDecisionMaking:MethodsandApplications.Springer-Verlag,NewYork. 多目標(biāo)、多準(zhǔn)則和多屬性陳珽1987. BasicsofMCDM(Chapters7,MultipleAttributeDecisionMaking(Chapters8,FiniteMAUT,AHP,DEA,TOPSIS,MultipleObjectiveDecisionMaking(ChapterInfiniteKuhn-Tuckercondition,GoalProgramming,11 性水利工程建設(shè)問題中的發(fā)電這一目標(biāo)可以用年發(fā)電量防洪效益只能用下游免遭洪 非劣解（non-inferiorsolution）：沒有其它的方意大利維多改進(jìn)(ParetoImprovement)的定義是:一種變化，在 j

j R

T T Trisk(x) T

Treward)T T1T T

xj

1T

jx 1T T

x Rjj

1T

Rj1 T 1 Markowitz-Type

1T

xjRj

1T

x Rj j xxj j

1Tt 1T x

jx j 如果越大，則更重 如果

TT xRT 1

1xRjT 1 1

1T

j

tTtt–yt

j(t) j

j

,22n個(gè)決策指標(biāo)（決策屬性）fj(1jn)，m個(gè)可行方案ai(1im)，構(gòu) 最大速度)最大速度)最大負(fù)載千克 低低 高高高

mx 在決策矩陣 (xmx

則矩陣 (yij) n稱為向量歸一標(biāo)準(zhǔn)化陣y2經(jīng)過歸一化處理后，指標(biāo)值 |yij y2 正向仍是正向，負(fù)向仍是負(fù)向)正、逆向指標(biāo)沒有發(fā)生變化(正向仍是正向，負(fù)向仍是負(fù)向)在決策矩陣 (x 中，對(duì)于正向指標(biāo)f，取

maxx x

ij，對(duì)于逆向指標(biāo)f，取x* minx，： x xj

則矩陣 n稱為線性比例標(biāo)準(zhǔn)化矩陣1最優(yōu)值為（1）經(jīng)過線性比例處理后，指標(biāo)值0 |yij| (2)正、逆向指標(biāo)均變?yōu)檎?最優(yōu)值為 (x maxx，x。minx

ijm

ij minx，x。maxx x。

j (y

ijm 10低高1357997531例例 低高高2.01502.015005903 00772.2180055X 低高高

例例 n個(gè)決策指標(biāo)f1,f2,...,fn按三級(jí)比例標(biāo)度兩兩相對(duì)比較評(píng)分，分值

當(dāng)fi和fj同等重要時(shí) 顯然 a當(dāng)f不如f 例計(jì)111040000000001110411111依據(jù)系統(tǒng)的程序，采用意見的方式，即專家之間不得互相討論，不發(fā)生橫向聯(lián)系，只能與人員，通過多輪次專家對(duì)問卷所提問題的看法，經(jīng)過反復(fù)征詢、歸納、修改 設(shè)有n個(gè)決策指標(biāo)，組織m個(gè)專家咨詢，得到權(quán)重估計(jì)值i1, i2, in m從而權(quán)重的平均估計(jì)值：m1jmjmi1

如果方差較大，就需要請(qǐng)專家重新估計(jì) 法1)用適當(dāng)?shù)姆椒ù_定個(gè)指標(biāo)的權(quán)重，設(shè)權(quán)重向量為：n (w,w,..., )T,其 (2)對(duì)決策矩陣作標(biāo)準(zhǔn)化處理，標(biāo)準(zhǔn)化矩陣Y (yij)m 并且標(biāo)準(zhǔn)化后所有指標(biāo)都為正向指標(biāo)3)求出各方案的線性加權(quán)指標(biāo)值：nu wjyij m 4)以線性加權(quán)值為判據(jù)，選擇最大者為最滿意方案，即：n max m

低高高20.800.560.950.820.711.000.860.690.430.560.721.001.000.800.560.950.820.711.000.860.690.430.560.721.001.001.000.880.950.900.710.56(2)用線性比例法法得到標(biāo)準(zhǔn)化矩陣:

因此，最滿意方案為a3a3a1a4a例例)

理想解法TOPSIS(TechniqueforOrderPreferenceSimilaritytoIdealSolution) 方案ai到理想點(diǎn)A和負(fù)理想點(diǎn)A的距離分別為：n(x–xn(x–x j n(x–j) SS, S相對(duì)貼近度：C ，顯然0 S S 相對(duì)貼近度越大，方案越理想！注意量綱和屬性重要性的影響!例1）用適當(dāng)?shù)姆椒ù_定權(quán)重向量為： (0.2,0.1,0.1,0.1,0.2,0.3)0.460.4670.3660.5050.50530.4810.6700.5830.6590.4550.5980.2880.3120.4200.4880.5300.4140.6730.5210.5130.4390.5050.46030.4810.372Y0.090.09340.0360.05060.0500.0960.2010.1160.0650.0450.0590.05770.1110.0840.0480.0530.0410.1340.1560.1020.0430.0500.0460.09620.111確定理想解和負(fù)理想解：V [0.1168,0.0659,0.0531,0.0414,0.1347,0.2012V [0.0841,0.0366,0.0455,0.0598,0.0577,0.1118計(jì)算各方案的相對(duì)貼近度：C 0.643,C 0.268,C 0.613,C .312 因此，最滿意方案為a1a1a3a4aijm ij (x) (y) (v) (wy) ijm ij ij

ij ijm

1 jj

n v ( j y)y)n

d 1jwj

* 0 作拉格郎日函數(shù)： (ij– 1 w令 0,得 (ywj w

yj w得：

0

1][

(y–

*)2jj (jj 1

–y*)

ij m 1 (y–y*) nnv(– n ij(–y*2w2jj y 例 0.5828

11Y(yij)56=6.96091，000 (0.21020.24640.17390.08010.1123 22OurdecisionmakingOurfieldsofconcernnormallypresumeasingledecisionmakerwhoistochooseamonganumberofalternativesthatheorsheevaluatesonthebasisoftwoormorecriteriaorattributes.Thealternativescaninvolverisksanduncertainties;theymayrequiresequentialactionsatdifferent andthesetofalternativesmightbeeitherfiniteorinfinite.Inthesimplestcases,thedecisionmakeractstoizeautilityorvaluefunctionthatdependsonthecriteriaorKeyDecisionmaker:responsibleformakingadecision(orsequenceofdecisions)Alternatives:options(finiteorinfinite)tobemadeRiskanduncertainty:randomfactorsoutsidethecontrolofdecisionmaker.Eachofthesepossiblesituationsdeterminedbytherandomfactorsisreferredtoasapossiblestateofnature tativemeasureofthevaluetothedecisionmakeroftheconsequencesof MCDMassumesthatrelevantobjectivefunctionvaluesorattributesareknownwithcertainty.ManyproblemsinMCDMareformulatedasmultipleobjectivelinear,integerornonlinearmathematicalprogrammingproblemsMorecomputationalresourcesareThereisusuallynoattempttocapturethedecisionmaker’sutilityorvaluefunctionmathematically.MAUTissometimessubsumedunderMCDM,butisusuallytreatedsepara ywhenrisksoruncertaintieshaveasignificantrolethevaluefunctionisInformationobtainedfromassessmentusuallyfeedsintotheparentproblemtorankalternatives,makeachoice,etc.(discretealternativeproblems)Manyapproachestomultiplecriteriadiscreteproblemsattempttorepresentaspectsofadecisionmaker’sutilityorvaluefunctionmathematicallyandt