ABSTRACT
Keywords and phrases Posterior morrients, prior distributions, sufiicient st,:rt,istics; trm~cat,ion p;trarneters) expone11tia.l tlistributiol~, ideritifiability, location pitritl~i(~kl', mixing distributioli, posterior rrmm
Let I'(:ylU) be the likelihood func1,ion of an intlepcrideat arid identically distributed sample y - (XI,. . ,.I:,,) from distribution p(zlH), then thc Bayes estimator ant1 posterior risk, uricler sqlmred error loss, arc posterior rnean E(Oly) and posterior variitr~ce var(OIg) respectively. For the norriial liltelihootl fimction with lirlown variance arid an arbitrary prior dist~.ibution, the explicit expressions for the posterior mean ancl vnrinnce
are delivctl Iy Pcricchi and Smith (1992). For an arl)itrn.ry locatioil parameter likelihood function and the nol.~r~itl prior distriht,ion, the exitet form of posterior mean is given by Polson (1991). Pericclli, Sansci, and Smith (1 993) also discussed the posterior c r ~ n d a n t ~ e l i ~ t i o r ~ s in Baycsim inference assurning the cspone~itial fanlily forms eitlrcr on t,hc likelillootl or on the prior. They all rrlcntioned t,lla.t imalyticitl Havcsiari cornputations wit,llo~~t the ;tss~mq)t~ior~ f normality, eitlier on 1ik~:lillootl f u r ~ c t i o ~ ~ or or1 prior distrihtion, a.rc very clifficult and ricctled i~wcitigatioir.
