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1、INVERSEPROBLEMSASSTATISTICSSTEVENN.EVANSANDPHILIPB.STARKAbstract.Whatmathematicians,scientists,engineers,andstatisticiansmeanbyinverseproblem"di ers.Forastatistician,aninverseproblemisaninferenceorestimationproblem.Thedataare niteinnumberandcontainerrors,astheyd
2、oinclassicalestimationorinferenceproblems,andtheunknowntypicallyisin nite-dimensional,asitisinnonparametricregres-sion.Theadditionalcomplicationinaninverseproblemisthatthedataareonlyindirectlyrelatedtotheunknown.Canonicalabstractformulationsofstatisticalestimatio
3、nproblemssubsumethiscomplicationbyallowingprobabilitydistributionstobeindexedinmore-or-lessarbitrarywaysbyparameters,whichcanbein nite-dimensional.Standardstatisticalconcepts,questions,andconsiderationssuchasbias,variance,mean-squarederror,identi ability,con-sist
4、ency,eciency,andvariousformsofoptimality,applytoinverseproblems.Thisarticlediscussesinverseproblemsasstatisticalestimationandinferenceproblems,andpointstotheliteratureforavarietyoftechniquesandresults.1.IntroductionThispapercastsinverseproblemsasstatisticalestim
5、ationandinferenceproblems.ItwaswrittentointroducesomestandardstatisticalideasandapproachestotheInverseProblemscommunity.Itismostlyexpository,butSection2.4containsnewresultsconcerningconsistentestimationinlinearinverseproblems.Inforwardproblemsinstatistics,onehasa
6、classofpossibledescriptionsoftheworld,andaforwardoperatorthatmapseachdescriptionintoaprobabilitymeasurefortheobservables.Thedataconsistofarealizationof(i.e.,asamplefrom)theprobabilitymeasure.Theprob-abilitymeasuretellsthewholestory:itcapturesanystochasticvariabil
7、ityinthetruth,"contaminationbymeasurementerror,systematicerror,etc.Werefertoeachpossibledescrip-tionoftheworldasamodel.AppliedmathematiciansgenerallywriteforwardproblemsasaDate:31August2001;updated18September2001.TechnicalReport609,DepartmentofStatistics,U.C.Ber
8、keley.Keywordsandphrases.Inverseproblems,statistics,decisiontheory,ill-posedproblem,regularization,infer-ence,constraints.12STEVENN.EVANSANDPHILIPB.STARKcompos
