probability and measurements - tarantola a.

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1、ALBERTTARANTOLAALBERTTARANTOLAtobepublishedbytobepublishedby...ProbabilityandMeasurements1AlbertTarantolaUniversit´edeParis,InstitutdePhysiqueduGlobe4,placeJussieu;75005Paris;FranceE-mail:Albert.Tarantola@ipgp.jussieu.frDecember3,20011
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2、1.iiiiiTothememoryofmyfather.Tomymotherandmywife.ivvPrefaceInthisbook,Iattempttoreachtwogoals.Thefirstispurelymathematical:toclarifysomeofthebasicconceptsofprobabilitytheory.Thesecondgoalisphysical:toclarifythemethodstobeusedwhenhandlingtheinformationbr

3、oughtbymeasurements,inordertounderstandhowaccuratearethepredictionswemaywishtomake.ProbabilitytheoryissolidlybasedonKolmogorovaxioms,andthereisnoproblemwhentreatingdiscreteprobabilities.ButIamveryunhappywiththeusualwayofextendingthetheorytocontinuouspr

4、obabilitydistributions.Inthistext,Iintroducethenotionof‘volumetricprobability’differentfromthemoreusualnotionof‘probabilitydensity’.Iclaimthatsomeofthemorebasicproblemsofthetheoryofcontinuousprobabilitydistributionscanonlynesolvedwithinthisframework,and

5、thatmanyofthewellknown‘paradoxes’ofthetheoryarefundamentalmisunderstandings,thatItrytoclarify.Istartthebookwithanintroductiontotensorcalculus,becauseIchoosetodeveloptheprobabilitytheoryconsideringmetricmanifolds.Thesecondchapterdealswiththeprobabilityt

6、heoryperse.Itrytouseintrinsicnotionseverywhere,i.e.,Ionlyintroducedefinitionsthatmakesenseirrespectivelyoftheparticularcoordinatesbeingusedinthemanifoldunderinvestigation.Thereadershallseethatthisleadstomanydevelomentsthatareatoddswiththosefoundinusualt

7、exts.Inphysicalapplicationsonenotonlyneedstodefineprobabilitydistributionsover(typically)large-dimensionalmanifolds.Onealsoneedstomakeuseofthem,andthisisachievedbysamplingtheprobabilitydistributionsusingthe‘MonteCarlo’methodsdescribedinchapter3.Thereisn

8、omajordiscoveryexposedinthischapter,butImaketheefforttosetMonteCarlomethodsusingtheintrinsicpointofviewmentionedabove.Themetricfoundationusedhereallowstointroducetheimportantnotionof‘homogeneous’probabilitydistributions.Contrarytothe‘non