Benchmark Estimation for Markov Chain Monte Carlo Samples

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Benchmark Estimation for Markov Chain Monte Carlo Samples
Subharup Guha, Steven N. MacEachern and Mario Peruggia
[e-mail]
May 2002; revised November 2002
Abstract
While studying various features of the posterior distribution of a vector-valued parameter using
an MCMC sample, a subsample is often all that is available for analysis. The goal of benchmark
estimation is to use the best available information, i.e. the full MCMC sample, to improve future
estimates made on the basis of the subsample. We discuss a simple approach to do this and provide
a theoretical basis for the method. The methodology and beneflts of benchmark estimation are
illustrated using a well-known example from the literature. We obtain as much as an 80% reduction
in MSE with the technique based on a 1-in-10 subsample and show that greater beneflts accrue with
the thinner subsamples that are often used in practice.
1 Introduction
While using an MCMC sample to investigate the posterior distribution of a vector-valued parameter ,
many features of interest have the representation E[g( )] for some function g( ). A subsample of
the MCMC output is often all that is retained for further investigation of the posterior distribution.
Subsamplingisoftennecessaryincomputationallyintensiveorreal-time,interactiveinvestigationswhere
speed is essential. Examples include expensive plot processing and examination of changes in the prior
(sensitivity analysis), likelihood (robustness) or data (case in uence). ...

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BenchmarkEstimationforMarkovChainMonteCarloSamples SubharupGuha,StevenN.MacEachernandMarioPeruggia [e-mail] May2002;revisedNovember2002 Abstract Whilestudyingvariousfeaturesoftheposteriordistributionofavector-valuedparameterusing anMCMCsample,asubsampleisoftenallthatisavailableforanalysis.Thegoalofbenchmark estimationistousethebestavailableinformation,i.e.thefullMCMCsample,toimprovefuture estimatesmadeonthebasisofthesubsample.Wediscussasimpleapproachtodothisandprovide atheoreticalbasisforthemethod.Themethodologyandbenefitsofbenchmarkestimationare illustratedusingawell-knownexamplefromtheliterature.Weobtainasmuchasan80%reduction inMSEwiththetechniquebasedona1-in-10subsampleandshowthatgreaterbenefitsaccruewith thethinnersubsamplesthatareoftenusedinpractice. 1Introduction WhileusinganMCMCsampletoinvestigatetheposteriordistributionofavector-valuedparameterθ, manyfeaturesofinteresthavetherepresentationE[g(θ)]forsomefunctiong(θ).Asubsampleof theMCMCoutputisoftenallthatisretainedforfurtherinvestigationoftheposteriordistribution. Subsamplingisoftennecessaryincomputationallyintensiveorreal-time,interactiveinvestigationswhere speedisessential.Examplesincludeexpensiveplotprocessingandexaminationofchangesintheprior (sensitivityanalysis),likelihood(robustness)ordata(caseinfluence).Typically,suchstudieswould includehundredsorthousandsofchangestothemodel.Anotherreasonforsubsamplingisexpensive 1

storagespace.Practicalconstraints,likelimiteddiskspaceavailabletousersofsharedcomputing resources,oftenmakeitinfeasibletostoretheentiresampleofMCMCdrawswhentheparameterhas alargenumberofcomponents.Asubsampleisthenretainedforfutureinvestigationoftheposterior distribution. Geyer(1992)andMacEachernandBerliner(1994)showthatsubsamplingtheoutputoftheMarkov chaininasystematicfashioncanonlyleadtopoorerestimationofE[g(θ)].Thegoalofbenchmark estimationistoproduceanumberofestimatesbasedontheentireMCMCsample,andtothenuse thesetoimproveotherestimatesmadeonthebasisofthesubsample.Thebenchmarkestimatesmust bequickandeasytocompute.Theymustalsobecompatiblewithquickcomputationsforfurther, more(computationally)expensiveanalysesbasedontheeventualsubsample. Severalmotivatingperspectivesareusefultounderstandandinvestigatevariousaspectsofbench- markestimation.Thepointofviewofcalibrationestimation,developedinthesamplingliterature toimprovesurveyestimates(DevilleandSa¨rndal,1992;Vanderhoeft,2001),helpstobringallthese perspectivestogetherintoaunifiedframework.Incalibrationestimation,aprobabilitysamplefrom afinitepopulationisusedtocomputeestimatesofpopulationquantitiesofinterest.The(regression type)estimatorsarebuiltasweightedaveragesoftheobservationsinthesample,withtheweightsde- terminedsoastosatisfya(vector-valued)calibrationequationwhichforcestheresultingestimatorsto produceexactestimatesofknownpopulationfeatures.Usually,theconstraintsimposedbythecalibra- tionequationdonotdetermineauniquesetofweights.Thus,amongthesetsofweightssatisfyingthe calibrationequation,onechoosesthesetthatyieldsweightsthatareascloseaspossible(withrespect tosomedistancemetric)toafixedsetofprespecified(typicallyuniform)weights. TocastMCMCbenchmarkestimationintothetheframeworkofcalibrationestimation,weregard theMCMCoutputasafinitepopulationanda1-in-ksystematicsubsampleasaprobabilitysample drawnfromthefinitepopulation.Thissystematicsamplingdesigngiveseachunitinthepopulationa probability1/kofbeingselected,thoughmanyjointinclusionprobabilitiesare0.Inthissetting,the 2

(vector-valued)benchmarkE[h(θ)],forwhichthesubsampleestimateisforcedtomatchthefullsample estimate,correspondstotheauxiliaryinformationavailablethroughthecalibrationequation.Oncethe calibrationweightshavebeencalculated,theycanthenbeusedtocomputethecalibrationsubsample estimateofanyfeatureE[g(θ)].AsthefullMCMCsamplesizeincreases,theasymptoticperformance ofthesebenchmarkestimatorsmatchesthatofthecorrespondingcalibrationestimators. Thebenchmarkestimatorsthatweintroduceinthispapercanbeshowntobecalibrationestimators correspondingtoappropriatelychosencalibrationequationsandmetrics.Weinvestigatetwomethods ofcreatingweights:post-stratificationandmaximumentropy.Intheirsimplestform,post-stratification weightsarederivedbypartitioningtheparameterspaceintoafinitenumberofregionsandbyforcing theweightedsubsamplefrequenciesofeachregiontomatchthecorrespondingrawfrequenciesforthe entireMCMCsample.Theweightsaretakentobeconstantoverthevariouselementsofthepartition andtosumtoone. Animprovedversionofpost-stratification,(and,infact,theapproachthatinourexperiencehas generatedthemostsuccessfulestimators)beginswitharepresentationofanarbitraryfunctiong(θ)as ∞ acountablelinearcombinationofbasisfunctionsh j (θ):g(θ)= j =1 c j h j (θ).Theestimand,E[g(θ)],is P expressedasthesamelinearcombinationofintegralsofthebasisfunctions, j ∞ =1 c j E[h j (θ)].Splitting P theinfiniteseriesrepresentationofg(θ)intotwoparts,wehaveafiniteserieswhichmayprovideagood approximationtog(θ)andaninfiniteremainderseriesthatfillsouttherepresentationofg(θ).Focusing onthefiniteseries,wedeterminetheweightsbyforcingestimatesofE[h j (θ)]basedonthesubsample tomatchthosebasedonthefullsample.Inaddition,werequiretheweightstobeconstantoverthe elementsofasuitablychosenpartitionoftheparameterspaceandtosumtoone.Thisproducesa betterestimateofE[ j m =1 c j h j (θ)]thanonebasedonthesubsamplealone.Theimprovementcarries P overtoestimationofE[g(θ)]whenthetailoftheseriesisofminorimportance.Werefertothefinite setofbasisfunctionsasthe(vector-valued)benchmarkfunction. Inboththebasicandimprovedpost-stratificationapproaches,wespecifyenoughconditionsthat 3

(forvirtuallyallMCMCsamplesofreasonablesize)thereisauniquesetofweightsthatwouldsatisfy them.Thus,fromthepointofviewofcalibrationestimation,thechoiceofthedistancemetricbecomes immaterial,inthesensethatanymetricwouldyieldidenticalweights.Inthisrespect,ourpost- stratificationweightsarisefromadegenerateinstanceofaproblemofcalibrationestimation.Inthe caseofthemaximumentropyweights,however,wedonotspecifyenoughbenchmarkconditionsto maketheweightsunique.Rather,amongthesetsofweightssatisfyinganunder-determinednumberof benchmarkconditions,weselectthesethavingmaximumentropyandthis,fromthepointofviewof calibrationestimation,istantamounttochoosingaspecificdistancemetric. Benchmarkestimationyieldsimprovedweightedestimators(havingsmallervariances)basedon MCMCsubsamples.Weinvestigatetheperformanceofestimatorsbasedonweightsdeterminedaccord- ingtopost-stratificationandmaximumentropymethods.Substantialreductionsinthevariabilityof theestimatesoccurwhenexaminingexpectationsoffunctionsg(θ)thataresimilartoalinearcombi- nationofbenchmarkcomponents.Theoreticalresultssuggestthegainsthatweseeinpractice.The methodologyisillustratedonanexamplediscussedbyGeorge,MakovandSmith(1993).Wecompare estimationofE[g(θ)]foravarietyoffunctionsg(θ),showingthatthereareoftensubstantialbenefitsto theuseofbenchmarkestimates.TheextentoftheimprovementinestimationofE[g(θ)]forfunctions thatarenoticeablydifferentfromthebenchmarksisstriking,evenforvaluesofmassmallas3or4. 2Asimpleapproachtobenchmarkestimation 2.1Animprovedsubsampleestimator Letθ∈Θbeavector-valuedparameter.ImaginethatanMCMCsampleisdrawnfromtheposterior distributionofθ.Callthesequenceofdrawsθ (1) ,θ (2) ,...,θ ( N ) .Thedrawsareusedtoestimatesome featureoftheposteriordistribution.OftenthesefeaturesofinterestcanberepresentedasE[g(θ)]for 4

g(θ (ki) ),2() some(possiblyvector-valued)functiong(θ).ThemoststraightforwardestimatorforE[g(θ)]is N E ˆ [g(θ)] f = 1 g(θ (i) ),(1) N X 1=i wheretherighthandsubscriptdenotesthefullsampleestimator.Ifoneselectsasystematic1-in-k subsampleofthedata,thenaturalestimatoris n 1 E ˆ [g(θ)] s = n X 1=i whereN=knandsdenotesthesubsampleestimator.Asmentionedearlier,thisformofsubsampling alwaysleadstopoorerestimation;theunweightedsubsampleestimator(2)hasalargervariancethan thefullsampleestimator(1). Wewishtousetheinformationavailablefromthefullsampletoimprovefutureestimationbased onthesubsampleforanyfeatureE[g(θ)].Foranappropriatelychosen(andpossiblyvector-valued) functionh(θ),werefertothefeatureE[h(θ)]asthebenchmark.Wenowcreateaweightedversionof thesubsampleestimatorofE[g(θ)]asfollows: n E ˆ [g(θ)] w =w i g(θ (ki) ),(3) X 1= i where i n =1 w i =1.Theweightsw i arechosensothattheyforcetheweightedsubsamplebenchmark P estimatetoequalthefullsampleestimate: E ˆ [h(θ)] w =E ˆ [h(θ)] f .)4( ThusE ˆ [h(θ)] w andE ˆ [h(θ)] f havethesamedistributionsprovidedtheweightscanbeconstructed,and allfeaturesoftheirdistributionsconditionalonthiseventarethesame.Foravector-valuedbenchmark function,anylinearcombinationofitscoordinatesresultsinthesameestimateforboththesubsample andthefullsample,andtheestimatorshavethesamedistribution.Inparticular,thetwoestimators havethesamevariance,andwehavepossiblygreatlyincreasedprecisionforoursubsampleestimator ofE[g(θ)]. 5

Theconnectionbetweenaconditionallyconjugatestructureandlinearposteriorexpectationinex- ponentialfamiliesimpliesthat,formanypopularmodels,quantitiessuchastheconditionalposterior meanforacaseortheconditionalposteriorpredictivemeanwillbealinearfunctionofhyperparam- eters.Thestructureofthehierarchicalmodelenablesustousebenchmarkfunctionsbasedonthe hyperparameterstocreatemoreaccurateestimatesofthesequantities.Thereductioninvariability whenmovingfromE ˆ [h(θ)] s toE ˆ [h(θ)] w alsoappearswhenexaminingexpectationsoffunctionsg(θ) thataresimilartoh(θ).Functionssuchasapredictivevariancewhichdependonfirstandsecond momentswilltypicallybecloselyrelatedtobenchmarkfunctionsbasedonthehyperparametersandso theywillbemoreaccuratelyestimatedwithourtechnique. Theweightedsubsample,(w i ,θ ( ki ) )fori=1,2,...,n,isnowusedinplaceoftheunweighted subsample.Theweightsactexactlyastheywouldifarisingfromanimportancesample,andsowe obtainweightedsubsampleestimatesE ˆ [g(θ)] w forvariousfeaturesofinterestE[g(θ)]oftheposterior. Techniquesandsoftwaredevelopedforimportancesamplescanbeusedwithoutmodificationforthese weightedsamples. 2.2Somemethodsofobtainingweights n Theconstraintsthat i =1 w i =1andthatE ˆ [h(θ)] w =E ˆ [h(θ)] f willnottypicallydeterminethew i . P Withasinglerealbenchmarkfunction,wewouldhaveonlytwolinearconstraintsonthew i .We supplementtheconstraintswithaprinciplethatwillyieldauniquesetofweights.Thetwoprinciples weinvestigatearemotivatedbytheliteraturesonsurveysamplingandinformationtheory. Weightsbypost-stratification. Post-stratificationisastandardtechniqueinsurveysampling, designedtoensurethatasamplematchescertaincharacteristicsofapopulation.Thepopulation characteristicsarematchedbycomputingaweightforeachunitinthesample.Largesampleresults showthatapost-stratifiedsamplebehavesalmostexactlylikeaproportionallyallocatedstratified sample.Thistypeofstratificationreducesthevarianceofestimatesascomparedtoasimplerandom 6

sample. Inthissetting,thefullsampleplaystheroleofthepopulationwhilethesubsampleplaystherole ofthesample.Thus,theessenceofthetechniqueistopartitiontheparameterspaceinto(say)m strata,andtoassignthesameweighttoeachθ ( ki ) inastratum.Formally,supposethat{Θ j } j m =1 isa finitepartitionoftheparameterspaceΘ.LetI j (θ)denotetheindicatorofsetΘ j ,forj=1,...,m. Thenaturalapplicationofthepost-stratificationmethodtakesasthebenchmarkfunctionthevector ofthesemindicatorfunctions.Thatis,h(θ)=(I 1 (θ),I 2 (θ),...,I m (θ)) 0 .Weassignthesameweight toallsubsamplepointsbelongingtoagivenstratum.Specifically,forallisuchthatθ ( ki ) ∈Θ j ,weset w i =v j ,where,accordingto(4),thevaluesv j aredeterminedby Nn 1 v j I j (θ (ki) )=I j (θ (i) ),wherej=1,...,m. N XX i =1 i =1 Thepost-stratificationweightsarethenobtainedas: N − 1 i N =1 I j (θ ( i ) ) P j 1=i v j = P n I(θ (ki) ) ,(5) providedeachofthestratacontainsatleastonesubsamplepoint.Asinsurveysampling,withfairly wellchosenstrata,thechancethatanyofthestrataareemptyofsubsamplepointsisnegligible.The intuitivedescriptionofthepost-stratificationweightv j isastheratiooftheproportionoffullsample pointsinΘ j tothenumberofsubsamplepoints.Werefertothissubsampleestimatorasthebasic post-stratificationestimator,E ˆ [g(θ)] w,ps . Theperspectiveofabasisexpansionofg(θ)providesamoresophisticateduseofpost-stratification. Insteadofusingabasisformedfromindicatorfunctions(essentiallyaHaarbasis),alternativebases consistoffunctionsotherthanindicators.Anattractivebasis,duetoitssuccessthroughoutstatistics, isthepolynomialbasisthatgeneratesTaylorseries.Assigningequalweighttosubsamplepointswithin eachgivenpost-stratumyieldsn−mlinearconstraintsontheweights,andforcingtheweightstosum to1providesoneadditionalconstraint.Supplementingthesewithafurtherm−1linearconstraints 7

ontheweights(andalsowithmildconditionsontheposteriordistributionandsimulationmethodto guaranteeuniqueness)definestheweights.Theweightsarequicklyobtainedasasolutiontoasystemof mlinearequations.Thisversionofpost-stratificationhasproventobeextremelyeffectiveinpractice. Asanexampleofthistechnique,supposethatθisap-dimensionalvectorvaluedparameterandthat theparameterspaceΘispartitionedintom=3strata,namelyΘ 1 ,Θ 2 andΘ 3 .Post-stratificationassigns thesameweightv j toallsubsamplepointsbelongingtostratumΘ j ,wherej=1,2,3.Conditionalon theeventthatnostratumisemptyofsubsamplepoints,thiscorrespondston−3linearlyindependent constraintsontheweights.Anadditionalconstraintontheweightsisthattheysumto1. Thustwootherindependentlinearconstraintswillensurethattheweightsareunique.Forachoice oftworealfunctions,sayh 1 (θ)andh 2 (θ),ascomponentsofthebenchmarkfunction,thebenchmark estimatesaregivenbyE ˆ [h 1 (θ)] f =(1/N) i N =1 h 1 (θ (ki) ),andE ˆ [h 2 (θ)] f =(1/N) i N =1 h 2 (θ (ki) ).For PP j=1,2,3andl=1,2,letn j = i n =1 I j (θ (ki) )denotethenumberofsubsamplepointsfallinginstratum P Θ j ,andletb lj denotethesumofthefunctionh l (θ)evaluatedatthesubsamplepointsbelongingto stratumΘ j .Thus n b lj =h l (θ (ki) )I j (θ (ki) ). X 1=i Then,solvingthefollowingsystemoflinearequationsuniquelydeterminestheweights(v 1 ,v 2 ,v 3 ) forthethreestrata,providedthesquarematrixbelowisinvertible: n 1 n 2 n 3 v 1 1  b 11 b 12 b 13  v 2  =  E ˆ [h 1 (θ)] f  . b 21 b 22 b 23 v 3 E ˆ [h 2 (θ)] f Maximumentropyweights. Informationtheorydescribes,invariousfashions,theamountof informationindataaboutaparameterordistribution.InaBayesiancontext,itisoftenusedto describesubjectiveinformation(playingtheroleofdata)inordertoelicitapriordistribution.This isaccomplishedbyspecifyinganumberoffeaturesofthedistribution,typicallyexpectations,asthe “information”abouttheprior.Theprioristhenchosentoreflectthisinformationbutnomore.With 8

entropydefinedasthenegativeofinformation,thepriorwhichreflectsexactlythedesiredinformation isthatwhichmaximizesentropyamongthosepriorsmatchingtheconstraints. Inoursetting,weborrowthistechnique,matchingexactlytheinformationinthefullsamplebench- markestimates,butnomore.Letw=(w 1 ,w 2 ,...,w n )bethen-tupleofweightsgivenin(3). LetusdenotebyΩthe(possiblyempty)setofallweightn-tuplesthatsatisfy(4).ThusΩisthe set{w|w i ≥0∀i, i n =1 w i =1, i n =1 w i h(θ (ki) )=E ˆ [h(θ)] f }. PP Definition2.1Theentropyofann-tuplewbelongingtosetΩisdefinedas n En(w)=−w i lnw i , X 1= i subjecttotheconventionthat0ln(0)equals0. WeobservethatforallwbelongingtoΩ,En(w)≤En n , n ,..., n =ln(n).SinceΩisclosed, ¡¡ 111 ¢¢ thereexistsanelementw ∗ ofΩsuchthatEn(w ∗ )=sup w ∈ Ω En(w).Theseweightsw ∗ arecalled maximumentropyweights,andtheyexistwheneverΩisnon-empty. Findingmaximumentropyweightsw ∗ isthusequivalenttomaximizingEn(w)subjecttothecon- straintsw i ≥0fori=1,...,n, i n =1 w i =1,and i n =1 w i h(θ (ki) )=E ˆ [h(θ)] f .Forarealbenchmark PP functionh(θ),itcanbeshownthatthemaximumentropyweightsw ∗ areuniquewhenevertheyexist. Formostsubsamplesofreasonablesizetheyaregivenby w i ∗ =e λ 1 +λ 2 h(θ ( ki ) ) ,i=1,2,...n;)6( whereλ 2 ∈ R satisfiestheequation h(θ ( ki ) )−E ˆ [h(θ)] f expλ 2 h(θ ( ki ) )−E ˆ [h(θ)] f =0,(7) X n ³´³³´´ 1= i ´³ andλ 1 =−ln P i n =1 e λ 2 h ( θ ( ki ) ) .Thefewsubsampleswheretheweightsfailtoexistwillhaveeither h(θ ( ki ) )<E ˆ [h(θ)] f foralli,orh(θ ( ki ) )>E ˆ [h(θ)] f foralli.Forallothersubsamples,equation(7)has auniquerootbecausethelefthandsideoftheequationincreasesmonotonicallyfrom−∞to+∞as 9

λ 2 increases.Therootcanbeobtainedbynumericalmethods,andhencew i ∗ canbecalculatedvery quicklyusing(6). Whenh(θ)isanindicatorfunctionofasubsetoftheparameterspaceΘ,theanswerobtainedusing (6)matchesthepost-stratificationweightsgivenin(5).Thus,thetwoapproacheswillsometimesyield thesameresult. 3TheoreticalResults ThemotivationforsubsamplingtheoutputofaMarkovchaincarriesovertothestudyofasymptotic propertiesoftheestimators.Afirstasymptoticismotivatedbythecommonpracticeofusingprelim- inaryrunsoftheMarkovchaintoassessthedependenceandconvergencepropertiesofthechain,and thenselectingasubsamplingratefortherunofthechainusedforestimation.Thispracticeleadsusto considertheasymptoticwherekisheldfixedandngrows(CaseA).Thestrongestmotivationforsub- samplingisthatfurtheruseofthesubsampleinvolvesexpensive(slow)processing.Whenthisfurther processingisdoneinrealtime,aswithinvestigationofchangesinthepriordistributionorlikelihood, itisessentialtolimitthenumberofpointsthatarerepeatedlyprocessed.Pursuingthismotivationfor subsampling,anaturalasymptoticholdsthenumberofsubsampledpoints,n,fixedwhilelettingthe intervalbetweensubsampledpoints,k,tendtoinfinity(CaseB).Inthelimit,thesesubsampledpoints willlooklikearandomsamplefromπ.Wenotethatasymptoticsbetweenthesetwo,wherebothnand kgrow(CaseC),arealsonaturalcandidatesfortheoreticalexploration,andareusefulforfillingout therangeofasymptoticexpressionsusefulforassessingtheaccuracyofthebenchmarkestimators. Tierney(1994)containsacollectionofusefulresultsfortheasymptoticsofestimatorsbasedon outputfromaMarkovchain.Thetwoessentialtypesofresultareergodictheoremswhichguarantee strongconvergenceofanempiricalaverage(orfull-sampleestimator)toacorrespondingexpectation underthelimitingdistribution,andcentrallimittheoremswhichdescribeweakconvergenceofan appropriatelycenteredandscaledfull-sampleestimatortoanormaldistribution.Werelyheavilyon 01

hisresultstoshowasymptoticnormalityofoursubsampledestimators. CaseA.Wefirstconsidertheasymptoticwherekisheldfixedandntendsto∞.Theergodic theoreminTierney’spaperappliedtothesub-sampledchain(whichisitselfMarkovian)allowsusto concludethat,asntendsto∞,E ˆ [g(θ)] s tendstoE[g(θ)]almostsurely.Thenexttheoremestablishes theasymptoticnormalityofthebasicpost-stratificationestimatorforexpectationsofboundedfunctions g(θ)andgeometricallyergodicMarkovchains. Theorem3.1SupposethattheMarkovchainisgeometricallyergodicwithinvariantdistributionπ.If thefunctiong(θ)isboundedandnotalinearcombinationofthestrataindicatorsI j (θ),j=1,...,m, thenthereexistsarealnumberσ(g)suchthat,asn→∞,thedistributionof √ nE ˆ [g(θ)] w,ps −E[g(θ)] ´³ convergesweaklytoanormaldistributionwithmean0andvarianceσ 2 (g). Proof.TheresultfollowsfromaverificationoftheconditionsofTheorem4ofTierney’spaperandby anapplicationofthedeltamethod.SeeAppendixAforamoredetailedproof. UnderthestrongerassumptionofuniformergodicityoftheMarkovchain,anidenticalresultholds forallfunctionsg(θ)withfiniteposteriorvariance.AproofofthisresultreliesonTheorem5ofTierney’s paper,butisotherwisethesameastheoneabove. Theresultsextendbeyondthebasicpost-stratificationestimator,applyingalsotothemorecom- plexpost-stratificationestimators.Theproofofthefollowingcorollaryformodifiedpost-stratification estimatorsisalmostidenticaltothatoftheprevioustheorem.Itdiffersonlyinminordetailsspecificto modifiedpost-stratificationestimators.Non-singularityofthematrix B definedbelowisrequiredfor localcontinuityoftheestimator. Corollary3.2LetE ˆ [g(θ)] ∗ w,ps denotethemodifiedversionofthepost-stratificationestimatorbasedon 11