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1、GaussianmodelLetusstartwithaclassicalexampleinwhichtheresponsesignalfromadetectorisdescribedbyaGaussianerrorfunctionaroundthetruevaluewithastandarddeviation,whichisassumedtobeexactlyknown.Thismodelisthebest-knownamongphysicistsand,indeed,theGaussianpdfisalsoknownasnormalbecauseit
2、isoftenassumedthaterrorsare'normally'distributedaccordingtothisfunction.ApplyingBayes'theoremforcontinuousvariables(seeTab. 1),fromthelikelihood(25)wegetfor(26)Consideringallvaluesofequallylikelyoveraverylargeinterval,wecanmodelthepriorwithaconstant,whichsimplifiesinEq. (26),yiel
3、ding(27)Expectationandstandarddeviationoftheposteriordistributionareand,respectively.Thisparticularresultcorrespondstowhatisoftendoneintuitivelyinpractice.Butonehastopayattentiontotheassumedconditionsunderwhichtheresultislogicallyvalid:Gaussianlikelihoodanduniformprior.Moreover,w
4、ecanspeakabouttheprobabilityoftruevaluesonlyinthesubjectivesense.Itisrecognizedthatphysicists,andscientistsingeneral,arehighlyconfusedaboutthispoint(D'Agostini1999a).Anoteworthycaseofapriorforwhichthenaiveinversiongivesparadoxicalresultsiswhenthevalueofaquantityisconstrainedtobei
5、nthe`physicalregion,'forexample,whilefallsoutsideit(oritisatitsedge).Thesimplestpriorthatcurestheproblemisastepfunction,andtheresultisequivalenttosimplyrenormalizingthepdfinthephysicalregion(thisresultcorrespondstoa`prescription'sometimesusedbypractitionerswithafrequentistbackgro
6、undwhentheyencounterthiskindofproblem).AnotherinterestingcaseiswhenthepriorknowledgecanbemodeledwithaGaussianfunction,forexample,describingourknowledgefromapreviousinference(28)InsertingEq. (28)intoEq. (26),weget(29)where(30) (31)(32)Wecanthenseethatthecasecorrespondstothelimitof
7、aGaussianpriorwithverylargeandfinite.TheformulafortheexpectedvaluecombiningpreviousknowledgeandpresentexperimentalinformationhasbeenwritteninseveralwaysinEq.(31).AnotherenlightingwayofwritingEq.(30)isconsideringandtheestimatesofattimesand,respectivelybeforeandaftertheobservationh
8、appenedattime.Indicatingtheestimatesatdi
