hyperparameters it is typical to refer to any parameters of the prior density as hyperparameters of the bayesian model instead of giving directly prior for (or )it turns out to be convenient to give the following prior distribution to / gamma(abs inverse gamma ( the smaller and arethe less informative is the prior under this priors is said to have an inverse gamma distribution if / gamma(ab)then the pdf of is proportional to exp (- / /za+ (exercise the bayesian posterior is then given byg(us tg(ux ( ( us exp - / exp np ( ) /( ) ( nx exp ph ( ) + ( ) / ( xn ) ( ) - / - - ( exp ph / where : xi (xi xn ) is the (scaledsample variance all inference about (us is then represented by the posterior pdf to facilitate computations it is helpful to find out if the posterior belongs to recognizable family of distributions for examplethe conditional pdf of given and is ( ) ( xn ) ( texp ph / which after simplification can be recognized as the pdf of ( tn gn xn ( gn )ngn / ( where we have defined the weight parametergn :sn ph sn we can then see that the posterior mean [ tgn xn ( gn ) is weighted linear combination of the prior mean and the sample average xn furtheras the weight gn and thus the posterior mean approaches the maximum likelihood estimate xn improper prior it is sometimes possible to use prior (ththat is not bona fide probability densityin the sense that (thdth as long as the resulting posterior (th tg( th) (this proper pdf such prior is called an improper prior example (normal model (cont )an example of an improper prior is obtained from ( when we let ph (the larger ph isthe more uninformative is the priortheng( is flat priorbut (udu making it an improper prior neverthelessthe posterior is proper densityand in particular the conditional posterior of ( tsimplifies to ( tn xn / reciprocal gamma distribution would have been better name