Bayesian Experimental Design - Mathematical Sciences Home Pages
Bayesian Experimental Design - Mathematical Sciences Home Pages
Bayesian Experimental Design - Mathematical Sciences Home Pages
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
thepriordistribution. secondderivativesofthelogarithmofthepriordensityfunction,ortheprecisionmatrixof alizedmaximumlikelihoodestimateofasinBerger,1985,p.133),andRisthematrixof<br />
theexpectedinformationmatrix,almostalwaysgivesabetternormalapproximationtothe theremaybereasonstopreferoneapproximationtoanother,andtheobserved,ratherthan andvarianceasthemeanandvarianceoftheapproximatingnormaldistribution,orusing theobservedratherthanexpectedFisherinformationmatrix.Althoughinspecicproblems Severalotherapproximationsarepossible,forexampleusingtheexactposteriormean<br />
theexpectedFisherinformationmatrixisusuallyalgebraicallymuchmoretractable.Using approximationsotherthan(12)and(13)isanareaforfutureresearch. posteriordistribution,ingeneralthereisnoobviouslybestonetouse.Fordesignpurposes U1()isgivenbyequation(4),asinthelinearmodel.U1()istheexactexpectedutility, model whichinvolvesp(yj),themarginaldistributionofthedataforadesign.Asinthelinear If,forillustration,Shannoninformationisthechoiceofutilitythentheexpectedutility<br />
utilityonlydependsonythroughsomeconsistentestimate^afurtherapproximation,of thesameorderas(12)and(13),istotakethepredictivedistributionof^tobetheprior Inmostcasesthismarginaldistributionofymustalsobeapproximated.Whentheposterior p(yj)=Zp(yj;)p()d:<br />
distribution.Usingthisapproximationtogetherwith(12)givesanapproximatevalueof U1(): AsinearliersectionsU()willbeusedtodenoteexactexpectedutilityand()adesign criterion.Theconstanttermsandmultiplierin(14)canbedroppedtogive ?k2log(2)?k2+12ZlogdetfnI(;)gp()d: 1()=ZlogdetfnI(;)gp()d (14) (15)<br />
28
Change language