(Bayesian Estimation) Let $X_i\sim\text{N}(\mu,\theta^{-1})$ be iid random variables. $\mu$ and $\theta$ are unknown. The prior has a Normal-Gamma distribution with parameters $\alpha_1,\alpha_2,\beta_1,\beta_2$ as follows: $$ \pi(\mu,\theta)=c\theta^{\beta_1/2}\exp\left\{-\frac{\theta}{2}\left[ \alpha_1+\beta_2(\alpha_2-\mu)^2\right] \right\}, c\text{ is a constant }, \theta0, \mu \in \mathbb{R}$$Show that that the prior distribution is a conjugate prior.

Question

(Bayesian Estimation) Let $X_i\sim\text{N}(\mu,\theta^{-1})$ be iid random variables. $\mu$ and $\theta$ are unknown. The prior has a Normal-Gamma distribution with parameters $\alpha_1,\alpha_2,\beta_1,\beta_2$ as follows: $$ \pi(\mu,\theta)=c\theta^{\beta_1/2}\exp\left\{-\frac{\theta}{2}\left[ \alpha_1+\beta_2(\alpha_2-\mu)^2\right] \right\}, c\text{ is a constant }, \theta>0, \mu \in \mathbb{R}$$Show that that the prior distribution is a conjugate prior.

anonymous · Answer

Oh I forgot to type that we just assume that there are n iid random variables.