应用统计决策课件

应用统计决策StatisticalDecision

FeaturesofDecisionMaking

决策特征ListAlternativeCoursesofAction (PossibleEventsorOutcomes)DeterminePayoffs? (AssociateaPayoffwithEachEventorOutcome)AdoptDecisionCriteria (EvaluateCriteriaforSelectingtheBestCourseof Action)

ListPossibleActionsorEvents

可能行为与事件清单

PayoffTableDecisionTreeTwoMethodsofListingEvent(Ei)CoolWeather(E1) x11=$50

x12=$100WarmWeather(E2) x21=200 x22=125

PayoffTable

盈利表Considerafoodvendordeterminingwhethertosellsoftdrinksorhotdogs.CourseofAction(Aj)SellSoftDrinks(A1)SellHotDogs(A2)

xij

=payoff(profit)foreventiandactionjDecisionTree（决策树）:ExampleSoftDrinksFoodVendorProfitTreeDiagramHotDogsCoolWeatherCoolWeatherWarmWeatherWarmWeatherx11=$50

x21=200x22=125x12=100OpportunityLoss（机会损失）:Example

Highestpossibleprofitforanevent

Ei -Actualprofitobtainedforanaction

Aj

OpportunityLoss(lij

)Event:CoolWeather

Action:SoftDrinks Profit:$50AlternativeAction:HotDogs

Profit:$100 OpportunityLoss=$100-$50=$50OpportunityLoss:TableEvent OptimalProfitof

SellSoftDrinksSellHotDogs ActionOptimal Action

Cool Hot

100

100-50=50

100-100=0Weather

Dogs

Warm Soft

200

200-200=0

200-125=75WeatherDrinks

AlternativeCourseofActionDecisionCriteria

决策准则ExpectedMonetaryValue(EMV)

Theexpectedprofitfortakinganaction

AjExpectedOpportunityLoss(EOL)

Theexpectedlossfornottakingaction

AjExpectedValueofPerfectInformation(EVPI)

Theexpectedopportunitylossfromthebest decision

DecisionCriteria--EMV

期望货币价值

ExpectedMonetaryValue(EMV)

Sum

(monetarypayoffsofevents)

(probabilitiesoftheevents)Xij

Pi

Vj

NEMVj

=expectedmonetaryvalueofactionjxi,j

=payoffforaction

j

andeventiPi

=probabilityofevent

i

occurringi=1DecisionCriteria--EMVTableExample:FoodVendor

Pi

Event Soft

xijPi

Hot

xijPi

Drinks Dogs.50Cool $50$50

.5=$25$100$100

.50=$50.50Warm $200$200

.5=100$125$25

.50=62.50EMVSoftDrink=$125EMVHotDog=$112.50BetteralternativeDecisionCriteria--EOL

期望机会损失

ExpectedOpportunityLoss(EOL)Sum

(opportunitylossesofevents)

(probabilitiesofevents)

Lj

lijPiEOLj

=expectedmonetaryvalueofactionjli,j

=payoffforactionjandevent

iPi

=probabilityofeventi

occurringi=1NDecisionCriteria--EOLTableExample:FoodVendor

Pi

EventOpLoss

lijPi

OPLoss

lijPi

SoftDrinks HotDogs.50Cool $50 $50

.50=$25$0 $0

.50=$0.50Warm 0 $0

.50=$0$75$75

.50=$37.50

EOLSoftDrinks=$25EOLHotDogs=$37.50BetterChoiceDecisionCriteria--EVPI

完美信息期望价值ExpectedValueofPerfectInformation(EVPI)TheexpectedopportunitylossfromthebestdecisionRepresentsthemaximumamountyouarewilling topaytoobtainperfectinformation

ExpectedProfit

UnderCertainty

-

ExpectedMonetaryValueoftheBestAlternative

EVPI(shouldbeapositivenumber)

EVPIComputationExpectedProfitUnderCertainty

=.50($100)+.50($200) =$150ExpectedMonetaryValueoftheBestAlternative =$125EPVI=$25Themaximumyouwouldbewillingtospendtoobtainperfectinformation.TakingAccountofVariability:FoodVendor

2

forSoftDrink =(50-125)2

.5+(200-125)2

.5=5625

forSoftDrink=75CVforSoftDrinks=(75/125)

100%=60%

2forHotDogs=156.25

forHotdogs=12.5CVforHotdogs=11.11%ReturntoRiskRatio

收益风险比Expressestherelationshipbetweenthereturn(payoff)andtherisk(standarddeviation).RRR=ReturntoRiskRatio=RRRSoft

Drinks=125/75=1.67RRRHotDogs=9YoumightwishtochooseHotDogs.AlthoughSoftDrinkshavethehigherExpectedMonetaryValue,HotDogshaveamuchlargerreturntoriskratioandamuchsmallerCV.DecisionMakingwithSampleInformationPermitsRevisingOldProbabilitiesBasedonNewInformation

NewInformationRevisedProbabilityPriorProbability校正AdditionalInformation:WeatherforecastisCOOL. Whentheweatheriscool,theforecasterwascorrect80%ofthetime.Whenithasbeenwarm,theforecasterwascorrect70%ofthetime.

RevisedProbabilities（校正概率）

Example:FoodVendorPriorProbability先验概率F1=CoolforecastF2=WarmforecastE1=CoolWeather=0.50E2=WarmWeather=0.50P(F1|E1)=0.80 P(F1|E2)=0.30

RevisingProbabilitiesExample:FoodVendorP(E1|F1)

=P(cool)

P(

coolforecast|cool)P(coolforecast)==.73(.50)(.80)(.80)(.50)+(.30)(.50)RevisedProbability

P(F1|E1)=0.80P(F1|E2)=0.30E1=0.50E2=0.50P(E2|F1)

=P(warm)

P(coolforecast|warm)P(coolforecast)=.27RevisedEMVTableExample:FoodVendor

Pi

Event Soft

xijPi

Hot

xijPi

Drinks Dogs.73Cool $50$36.50 $100$73.27Warm $20054 125 33.73EMVSoftDrink=$90.50EMVHotDog=$106.75BetteralternativeRevisedEOLTable Example:FoodVendor

Pi

EventOpLoss

lijPi

OPLoss

lijPi

SoftDrink HotDogs.73Cool $50 $36.50 $0 0.27Warm 0 $0 75 20.25

EOLSoftDrinks=36.50EOLHotDogs=$20.25BetterChoiceRevisedEVPIComputationExpectedProfitUnderCertainty

=.73($100)+.27($200) =$127ExpectedMonetaryValueoftheBestAlternative =$106.75EPVI=$20.25Themaximumyouwouldbewillingtospendtoobtainperfectinformation.TakingAccountofVariability:RevisedComputation

2

forSoftDrinks =(50-90.5)2

.73+(200-90.5)2

.27=4434.75

forSoftDrinks=66.59CVforSoftDrinks=(66.59/90.5)

100%=73.6%

2forHotDogs=123.1875

forHotdogs=11.10CVforHotdogs=(11.10/106.75)

100%=10.4%RevisedReturntoRiskRatioExpressestherelationshipbetweenthereturn(payoff)andtherisk(standarddeviation