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AGeneralBayesianApproachToBlindSourceSeparationWithCorrelationDanielB.RoweMarch22,1999AbstractThispaperpresentsaBayesianstatisticalapproachtoblindsepa-rationofcorrelatedsourceswhichisageneralizationofthemethodsinRowe(1999).TheBayesianblindseparationofsourcesmodelisex-tendedtothecasewhereboththeobservedmixedsignalsvectorsandtheunobservedsourcesignalvectorscouldbecorrelated.Further,thelinearsynthesismodelisextendedtoaccomodatethiscorrelationandallowthemixingmatrixtochange.Forsimplicity,itisassumedthatthemixingmatrixisconstant,andcorrelationsimpli cationsareexploredfortheobservedmixedsignalsandtheunobservedsources.1CorrelatedModel1.1IntroductionTheproblemaddressedbyblindsourceseparationisthatofseparatingun-observablesourcesignalswhenmixedsignalsareobserved.AlinearsynthesismodelisadoptedwheretheobservationsarelinearcombinationsofthesourcesandaBayesianstatisticalapproachistaken.Tomotivatetheblindseparationofsourcesmodel,thecontextofthe\cock-tailpartyproblemisadopted.Atacocktailparty,therearepmicrophonesthatrecordorobservempartygoersorspeakersatntimeincrements.Theobservedconversationsconsistofmixturesoftrueunobservableconversations.Thep-dimensionalmixedsignalvectorsxi=(xi1;:::;xip)0areobservedandtheobjectiveistoseparatetheseobservedsignalvectorsintom-dimensionaltrueunderlyingsourcesignalvectors,si=(xi1;:::;xim)0wherei=1;:::;n.Theobservationvectorsarestackedintoasinglevectorwhichisnp 1,andthemodeliswritenas(xj ;m;T;s)= +Ts+ ;(np 1)(np 1)(np nm)(nm 1)(np 1)(1.1)1wherex=annp-dimensionalvectorofobservedsignalsx=(x01;:::;x0n)0, =annpdimensionalgeneralmeanvector, =( 01;:::; 0n)0,T=anp nmmatrixofmixingconstants,s=annm-dimensionalunobservedtruesourcesignalvector,s=(s01;:::;s0n)0, =annp-dimensionalobservationerrorvector,and =( 01;:::; 0n)0.1.2LikelihoodTheerrorsoftheobservationsareassumedtobenormallydistributedwithmeanzeroandcovariancematrix ,ornotationally( j ) N(0; ):(1.1)Fromthisassumption,thedistributionoftheobservationsis(xj ; ;m;T;s) N( +Ts; );(1.2)andthelikelihoodfortheobservationsisp(xj ; ;m;T;s)=(2 ) np2j j 12e 12(x Ts)0 1(x Ts):(1.3)Thevector istheunobservedmeanbackgroundsignal.Forsimplicity,itisassumedthatthebackgroundsignalisthesameovertimethus i= foralliand =e ,whereeisann 1vectorofones.Itcanbeshownthatthemaximumlikelihoodestimatorforthemean is x,sowithoutlossofgeneralityandforsimplicity,theobservationvectorsareassumedtohavebeencenteredaboutthesamplemean.Themodelandlikelihoodnowbecomes(xjm;T;s)=Ts+ ;(np 1)(np nm)(nm 1)(np 1)(1.4)andp(xj ;m;T;s)=(2 ) np2j j 12e 12(x Ts)0 1(x Ts):(1.5)Followingtheassumptionthatthemixingmatrixisconstantovertime,itisassumedthatT=In andthustheusuallinearsynthesismodel2(xjm;T;s)=(In )s+ ;(np 1)(np nm)(nm 1)(np 1)(1.6)isobtainedwithlikelihoodp(xj ;m;T;s)=(2 ) np2j j 12e 12[x (In )s]0 1[x (In )s]:(1.7)Thegoalistorecovertheoriginalsourcesorunmixthesourcessbycom-putingestimatesoftheirvaluesfromprobabilitydistributions.Knowledgeastothemixingprocessisalsodesiredandisobtainedbyestimatingthemixingmatrix andtheerrormatrix .1.3PriorsUncertaintyabouttheparametersisrepresentedbyusingnaturalconjugatepriordistributions.RecallasinRowe(1999)thatthesourcesarerandomvariableswithanassociateddistribution.Thisdistributionisincludedwiththepriorsforthemixingmatrixandtheobservationerrorcovariancematrix.Itisassumedthatthejointpriordistributionfortheunknownparametersisgivenbyp( ;s; ; )=p( )p(sjm; )p( jm)p( jm);(1.8)wherep( )=c(np; )j j 2e 12tr 1A; 0; 2np;(1.9)p(sjm; )=(2 ) nm2j j 12e 12s0 1s; 0(1.10)p( jm)=(2 ) pm2j j 12e 12( 0)0 1( 0); 0;(1.11)andc(np; )isaconstantdependingonlyonnpand .Thepriorfor willbediscussedlater.Itisassumedapriori,thattheerrordisturbancecovariancematrixisinvertedWishart.Further,itisassumedthatthesourcesandthemixingmatrixareindependentlynormallydistributedgiventhenumberoffactors.Withoutlossofgenerality,itisassumedthatthevarianceforthesourcesisunitysothat isacorrelationmatrix(seePress1982p.331).Notethat 0 ( 1;:::; p),andthemixingconstantshavebeenwrittenas vec( 0)=( 01;:::; 0p)0.(Vectorswillbedenotedaslowercaseandmatricesas3uppercaseletters.)Alsonotethatthefollowingmodelassumptionregardingthedistributionoftheunobservedsourceshasbeenmade.Itisassumedthat(II)(sjm; ) N(0; ),thisisanalogoustoassumption(b)intheRowe(1999)model.Itdi ersinthathere,thesourcesignalvectorsareallowedtobecorrelated.Itisalsoassumedthat(III)(sjm; )and( j )areindependentrandomvectors.Thisistheidenticaltoassumption(c)intheRowe(1999)model.Thisas-sumptionisevidentfromthelikelihoodandthemodelpriordistributionforthesources.1.4PosteriorByBayes’rule,thejointposteriordistributionfortheunknownparametersofinterestisgivenbyp( ;s; ; jx;m)/p( ;s; ; jm)p(xj ;s; ;m)/p( )p(sjm)p( jm)p( jm)p(xj ;m;s; )/j j 2e 12tr 1Aj j 12e 12s0 1s j j 12e 12( 0)0 1( 0) j j 12e 12[x (In )s]0 1[x (In )s]p( jm)/j j ( +1)2e 12tr 1A j j 12e 12s0 1sj j 12e 12( 0)0 1( 0) e 12[x (In )s]0 1[x (In )s]p( jm)(1.12)notethatapriordistributionmuststillbeassessesfor ,whichmusthavethepropertythatthediagonalelementsareunity.