probability and measurements - tarantola a.

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1、ALBERTTARANTOLAALBERTTARANTOLAtobepublishedbytobepublishedby...ProbabilityandMeasurements1AlbertTarantolaUniversit´edeParis,InstitutdePhysiqueduGlobe4,placeJussieu;75005Paris;FranceE-mail:Albert.Tarantola@ipgp.jussieu.frDecember3,20011
cA.Tarantola,2001.iiiiiTothememoryofmy

2、father.Tomymotherandmywife.ivvPrefaceInthisbook,Iattempttoreachtwogoals.Thefirstispurelymathematical:toclarifysomeofthebasicconceptsofprobabilitytheory.Thesecondgoalisphysical:toclarifythemethodstobeusedwhenhandlingtheinformationbroughtbymeasurements,inordertounderstandhowac

3、curatearethepredictionswemaywishtomake.ProbabilitytheoryissolidlybasedonKolmogorovaxioms,andthereisnoproblemwhentreatingdiscreteprobabilities.ButIamveryunhappywiththeusualwayofextendingthetheorytocontinuousprobabilitydistributions.Inthistext,Iintroducethenotionof‘volumetric

4、probability’differentfromthemoreusualnotionof‘probabilitydensity’.Iclaimthatsomeofthemorebasicproblemsofthetheoryofcontinuousprobabilitydistributionscanonlynesolvedwithinthisframework,andthatmanyofthewellknown‘paradoxes’ofthetheoryarefundamentalmisunderstandings,thatItrytocl

5、arify.Istartthebookwithanintroductiontotensorcalculus,becauseIchoosetodeveloptheprobabilitytheoryconsideringmetricmanifolds.Thesecondchapterdealswiththeprobabilitytheoryperse.Itrytouseintrinsicnotionseverywhere,i.e.,Ionlyintroducedefinitionsthatmakesenseirrespectivelyofthepa

6、rticularcoordinatesbeingusedinthemanifoldunderinvestigation.Thereadershallseethatthisleadstomanydevelomentsthatareatoddswiththosefoundinusualtexts.Inphysicalapplicationsonenotonlyneedstodefineprobabilitydistributionsover(typically)large-dimensionalmanifolds.Onealsoneedstomak

7、euseofthem,andthisisachievedbysamplingtheprobabilitydistributionsusingthe‘MonteCarlo’methodsdescribedinchapter3.Thereisnomajordiscoveryexposedinthischapter,butImaketheefforttosetMonteCarlomethodsusingtheintrinsicpointofviewmentionedabove.Themetricfoundationusedhereallowstoin

8、troducetheimportantnotionof‘homogeneous’probabilitydistributions.Contrarytothe‘non