Transcription of Inverse Gamma Distribution - johndcook.com
1 Inverse Gamma DistributionJohn D. CookOctober 3, 2008 AbstractThese notes write up some basic facts regarding the Inverse gammadistribution, also called the inverted Gamma Distribution . In a sensethis Distribution is unnecessary: it has the same Distribution as thereciprocal of a Gamma Distribution . However, a catalog of results forthe Inverse Gamma Distribution prevents having to repeatedly applythe transformation theorem in we derive the Distribution of the Inverse Gamma , calculateits moments, and show that it is a conjugate prior for an exponentiallikelihood ParameterizationsThere are at least a couple common parameterizations of the Gamma distri-bution.
2 For our purposes, a Gamma ( , ) Distribution has densityf(x) =1 ( ) x 1exp( x/ )forx >0. With this parameterization, a Gamma ( , ) Distribution hasmean and variance the Inverse Gamma (IG) Distribution to have the densityf(x) = ( )x 1exp( /x)forx > Relation to the Gamma distributionWith the above parameterizations, ifXhas a Gamma ( , ) distributionthenY= 1/Xhas an IG( , 1/ ) Distribution . To see this, apply thetransformation (y) =fX(1/y) ddyy 1 =1 ( ) y +1exp( 1/ y)y 2=(1/ ) ( )y 1exp( (1/ )/y)3 MomentsNext we calculate the moments ofX IG( , ). If > n,E(Xn) = ( ) 0xnx 1exp( /x)dx= ( ) 0xn 1exp( /x)dx= ( ) ( n) n= n ( n)( 1) ( n) ( n)= n( 1) ( n).
3 In particular, for >1E(X) = 1and for >2E(X2) = 2( 1)( 2)and so for >2V ar(X) =E(X2) E(X)2= 2( 1)2( 2).24 Conjugate prior for exponential likelihoodFinally, suppose than an observationX| exponential( ) anda priori IG( , ). By Bayes theorem, the posterior Distribution on given anobservationX=xis proportional to1 exp( x/ )1 +1exp( / ) =1 +2exp( ( +x)/ ).When normalized to be a probability Distribution , the result is an IG( + 1, +x) Distribution . In general, after observingx1,x2, .. ,xn, the posteriordistribution on is IG( +n, + ni=1xi).The motivation for parameterizing the Inverse Gamma Distribution theway we do is to make the posterior Distribution have the simple form document available
