; TeX output 2004.06.02:0837 33GfK{dsrc:34MW.Note11.texC "V cmbx10StatisticsT&GeneExpressionDataAnalysisZyNote11:pMulti-NormalTheory A 33Gx!", cmsy10TDensit9yandParameters荑dsrc:65MW.Note11.texx g K`y cmr10isUUb> cmmi10p81withmeanvectorm?(p1)andvqariance(covariance)matrixV\n(p8p)ȍdsrc:66MW.Note11.texPrecisionUU(orconcentration)matrixK=VxܟpO! cmsy7 ٓR cmr71(assumenon-singular)dsrc:67MW.Note11.texDensityUUfunctionVX -p(x)=cexp( (xJ 8m2)09Kī(x 8m)=2)VXdsrc:69MW.Note11.texwithUUc=j2[ٷj^ 0er cmmi7p=2jK j^1=2dsrc:70MW.Note11.texxݷN(m R;V \l)UUorx.2N(xŷjm R;V \l)TLinearT ransforms荑dsrc:74MW.Note11.texAnyUUkw8pmatrixGQandconstantkP vectoran;y.2=a +8GCxisnormalyN(a+8GCmٱ;G OVgGqp0?)dsrc:75MW.Note11.texk<p:UUDimensionreductiondsrc:76MW.Note11.texk>p:UURankdecient(singular)distributionTKeyPropQerties:pMarginal&ConditionalDistributionssrc:80MW.Note11.texPartitionUUxoasUUx g18andUUx g2andUUconformablypartitionm?andV\nsothat&uYx~2y=^u cmex10 4x1 4x2^!0;m"=^ 4mc1 4mc2^.& Vxݫ=^ 4V1%CR 5RN2p0"lV+021K^src:84MW.Note11.texwhere$C (xş1 8;xm28)=R(andofcourseC(xş2 8;xm18)=Rp0e:)$Dimensions#areconformable{anysubsettingofx worksdsrc:87MW.Note11.texxş1 UPN(m R1ű;V \l1߫)UUandx g2N(m R2ű;V \l2߫)dsrc:88MW.Note11.texReallyfcriticaltounderstandingregressionaretheconditionaledistributions:Hereisp(xş1 8jxş2)fandthedsameUUgeneraltheorytellsyouwhatp(xş2 8jxş1)UUisdsrc:91MW.Note11.texVX (xş1 8jxş2)N(a1Ll+8B g1xN2q;W 21 )VXsrc:92MW.Note11.texwithlar1y*=m\j1 8B g1mx۟2N;Bߟ1j=RgVpٺ 12(N& W㋟1'=Vxܟ1./ 8B g1Rp0&ЍTPrecisionMatrixandDepQendencies荑dsrc:97MW.Note11.texT*akex 11`=Mx1|s;ܫtherstelementofx !|sothatx 2isalltherest. ZAnotherwayofwritingtheconditionaldistributionPabGovePisintermsoftheelementsoftheprecisionmatrixKninsteadofVSasfollows(thisisjustbasedUUonstandardlinearalgebraandrepresentationsofinversesofpartitionedmatrices).ȍdsrc:102MW.Note11.texIfUUx g1=x1|s;UUthenB1USisthe(p8 1)UUrowvectorwithj ^thelementVX b1;j Z7= K1;j =K1;1dsrc:104MW.Note11.texandUUW8ߟ1 isUUthescalarvqariance1=K1;1dsrc:105MW.Note11.texShowsUUthelinearregressionofx1ȫ(oranyotherxiTL)onallothervqariables(genes)dsrc:106MW.Note11.texNote:qZerosUUinprecisionmatricescorrespGondingto!p0J cmsl10conditionalindependenciesdsrc:107MW.Note11.texUnderliesUUthema 8jorareaofGaussiangraphicalmoGdels MikeUUW*est June2,2004 Page * 33GfK{TSingularNormal dsrc:111MW.Note11.texVisUUsingular;distributionissingular dsrc:112MW.Note11.texrankdecient:yrGankP(Vī)=kk