(efinallyshow that in the -step of the em algorithm th(tis updated from th( - as followspn ( - xi = (tu pn ( - = wi ( - (tw (xi ( )(xi ( ) = and (tis defined implicitly through the solution of the nonlinear equationpn ( - ( -   ( (tln( + + = ln -ps +ps ln + generalization of both the gamma and inverse-gamma distribution is the generalized inverse-gamma distributionwhich has density ( ( /bp/ - (as+ /ss ababs ( where is the modified bessel function of the second kindwhich can be defined as the integral ( - cosh(tcosh(ptdtx ( we write gig(abpto denote that has pdf of the form ( the function has many interesting properties special cases include - / ( - / (xe - / (xe more generallyk satisfies the recursion + (xk - ( (xx ( (ausing the change of variables ez /bshow that - (as+ /sds ab)( /ap/ (blet gig(abpshow that and + abes ab( ab + es - ab( generalized inverse-gamma distribution modified bessel function of the second kind