91Ó°ÊÓ

Normal likelihood with conjugate prior distribution: suppose \(y\) is an independent and identically distributed sample of size \(n\) from the distribution. \(\mathrm{N}\left(\mu, \sigma^{2}\right)\), where \(\left(\mu, \sigma^{2}\right)\) have the N-Inv- \(\chi^{2}\left(\mu_{0}, \sigma_{0}^{2} / \kappa_{0} ; \nu_{0}, \sigma_{0}^{2}\right)\) prior distribution, (that is, \(\sigma^{2} \sim \operatorname{Inv}-\chi^{2}\left(\nu_{0}, \sigma_{0}^{2}\right)\) and \(\left.\mu \mid \sigma^{2} \sim \mathrm{N}\left(\mu_{0}, \sigma^{2} / \kappa_{0}\right)\right)\). The posterior distribution, \(p\left(\mu, \sigma^{2} \mid y\right)\), is also normal-inverse- \(\chi^{2}\), derive explicitly its parameters in terms of the prior parameters and the sufficient statistics of the data.