最大熵值法在stata中的运用docx.docx
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最大熵值法在stata中的运用docx.docx
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TheStataJournal(2010)
10.Number3,pp.315-330
Anintroductiontomaximumentropyand
minimumcross-entropyestimationusingStata
MartinWittenberg
UniversityofCapeTown
SchoolofEconomics
CapeTown,SouthAfrica
Martin.Wittenberg@uct.ac.za
Abstract.Maximumentropyandliiiiiiniuiiicross-entropyestimationareapplica・blewhenfacedwithill-posedestimationproblems・IintroduceaStataconniiaiidthatestimatesaprobabilitydistributionusingainaxiinumentropyorniiniinunicross-entropycriterion・Ishowhowtliisconiinandcanbeusedtocalibratesurveydatatovariouspopulationtotals・
Keywords:
st0196.liiaxentropy.liiaxiniuiiientropy,ininimumcross-entropy,surveycalibration,sampleweiglits
1Ill-posedproblemsandthemaximumentropycriterion
Alltoomanysituationsinvolvemoreunknownsthandatapoints・Standardformsofestimationareimpossiblewhenfacedwithsuchill-posedproblems(Mittelhammer.Judge,andMiller|2000|)・Oneapproachthatisapplicableinthesecasesisestimationbymaximizinganentropymeasure(Gokm:
Judge;and\Iillei]|1996|)・ThepurposeofthisarticleistointroducetheconceptandtoshowhowtoapplyitusingthenewStatacommandmaxentropy.Mydiscussionofthetechniquefollowsthetreat.mentill;Golan、Judge、nndblilier|(|1996|)・Furthermore,Ishowhowamaximumentropyapproachcanbeusedtocalibratesurvey(latatovnriouspopulationtotals.ThisapproachisequivalenttotheiterativerakingprocedureofDeming“ndStephnn|(jl940)orthemultiplicativemethodimplementedinthecalibrationonmargins(CALMAR)algorithm(Deviil二mdSHrndal;,1992,:
〔Deville:
SHrndnl,andSauto叮|,1993|)・
Theideaofmaximumentropyestimationwasmotivatedby|Jaynesl(1957,621ff)intermsoftheproblemoffindingtheprobabilitydistribution(pi,p2)・・・、pn)forthesetofvalues・・・,givenonlytheirexpectation.
n
i=l
st0196
Forconcreteness,weconsideradieknowntohaveE(x)=3.5,wherex=(1学2,3、4、5)6),andwewanttodeterminetheassociatedprobabilities・Clearly,thereareinfinitelymanypossiblesolutions,buttheobviousoneispi=P2=・••=P6=l/6・TheobviousnessisbasedonLaplace'sprincipleofinsufficientreason,whichstatesthattwoeventsshouldbeassignedequalprobabilityunlessthereisareasontothinkotherwise(Jaynes19571
©2010StataCorpLP
622)・Thisnegativereasonisnotmuchhelpif,instead,weknowthatE(x)=4.Jaynes5ssolutionwastotacklethisfromthepointofviewofShannon'sinformationtlieory.JayneswantedacriterionfunctionH(pi.p2y•・・:
pn)thatwouldsummarizetheuncertaintyaboutthedistribution.Thisisgivenuniquelybytheentropymeasure
n
H(Pbp2,...,pn)=-K刀i=l
wherepilnfe)isdefinedtobezeroifpi=0forsomepositiveconstantK・ThesohitiontoJaynesesproblemistopickthedistribution(p\,p2>・・・,pn)thatmaximizestheentropy5subjectonlytotheconstraints
E{门巧}=刀M(如
i
D,=1
i
As|Golan:
Judge:
andMiller)(|1996[8-10)show,ifourknowledgeofE{f(rr)}isbasedontheoutcomeofN(verylarge)trials,thenthedistributionfunctionp=(pi)P2,…,片)thatmaximizestheentropymeasureisthedistributionthatcangiverisetotheobservedoutcomesinthegieatestnumberofwavs,whichisconsistentwithwhatweknow・Anyotherdistributionrequiresmoreinformationtojustifyit.Degeneratedistributions,oneswherepi=1andpj=0forj7^i、haveentropyofzero・Thatistosay.theycorrespondtozerouncertaintyanclthereforemaximalinformation・
2Maximumentropyandminimumcross-entropyestimation
Moreformally,themaxinnimentropyproblemcanberepresentedas
maxp//(p)
n
=-刀必I】®)
1=1
suchthatyj
n
=〉]Xjip讥j=・・・,
⑴
n
f=l
1=1
=1
⑵
TheJconstraintsgivenin([IJcanbethoughtofasmomentconstraints,withyjbeingthepopulationmeanoftheXjrandomvariable・Tosolvethisproblem,wesetuptheLagrangianfunction
L=_p"ln(p)-A(X'p_y)—〃(p'l-1)
whereXisthenxJdatamatrix^]AisavectorofLagrangemultipliers,and1isacolumnvectorofones.
Thefirst-orderconditionsforaninteriorsolution一thatis,oneinwhichthevectorpisstrictlypositive—aregivenby
where
n
QP)=Eexp
f=l
andx?
:
isthezthrowvectorofthematrixX.
Themaximumentropyframeworkcanbeextendedtoincorporatepriorinformationaboutp.AssumingthatwehavethepriorpTobnbilitydistributionq=(q「q?
、・••第q)thenthecross-entropyisdefinedas(|6olan、Judge,nndMill云1996]11)
=P‘ln(p)—p‘ln(q)
Thecross-entropycanbethought,ofasameasureoftheadditionalinformationrecpiiredtogofromthedistributionqtothedistributionp.Thep
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