Suppose that $x_{i}|\mu,\sigma^{2} \sim \mathcal{N}(\mu,\sigma^{2})$ for $i = 1,...n$. Assume that the assigned prior distributions are $\mu$ ~ $\mathcal{N}$($\mu_{0}$, $\sigma^{2}_{0}$) and $\tau \sim Gamma(ξ_{0}, ξ_{0})$ with $\tau = \frac{1}{\sigma^{2}}$

I have derived that the joint posterior distribution of $\mu$ and $\tau$, $p(\mu,\tau|\mathbf{x})$, where $\mathbf{x}$ is $x_{1},x_{2},...,x_{n}$, is $$p(\mu,\tau|\mathbf{x})\propto \tau^{\frac{n}{2}+ξ_{0}-1}exp(-\tauξ_{0})exp\{-\frac{1}{2}\tau\sum_{i=1}^{n}(x_{i}-\mu)^2-\frac{1}{2\sigma_{0}^{2}}(\mu-\mu_{0})^2\}$$

and the full conditional distributions of $\mu$ and $\tau$ , $p(\mu|\tau, \mathbf{x})$ and $p(\tau|\mu, \mathbf{x})$ as $$p(\mu|\tau, \mathbf{x}) \sim \mathcal{N}(\frac{\tau n\bar{x}\sigma_{0}^2+\mu_{0}}{n \tau\sigma_{0}^2+1} , \frac{\sigma_{0}^{2}}{n \tau\sigma_{0}^{2}+1})$$ $$p(\tau|\mu, \mathbf{x}) \sim Gamma(\frac{n}{2}+ξ_{0},\frac{\sum_{i=1}^{n}(x_{i}-\mu)^2}{2}+ξ_{0})$$ where $\bar{x} = \frac{\sum_{i=1}^{n}x_{i}}{n}.$ Now I have to derive the posterior predictive distribution $p(\tilde{x}|\mathbf{x})$, by definition $$p(\tilde{x}|\mathbf{x}) = \int\int p(\tilde{x}|\mu,\tau) p(\mu,\tau|\mathbf{x}) d\tau d\mu$$ The problem is I am not quite sure about the exact density of $p(\mu,\tau|\mathbf{x})$. I didn't think it is a Normal-Gamma distribution since it carries the term $exp\{-\frac{1}{2\sigma_{0}^{2}}(\mu-\mu_{0})^2\}$ which is without $\tau$. Hence I couldn't proceed with the integration. $$$$ Can someone show me the steps to obtain the posterior predictive distribution $p(\tilde{x}|\mathbf{x})$? Or correct any mistakes I have committed? Much appreciated!


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