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As a complete beginner in the world of Bayesian statistics, I unfortunately have no idea of how to start this problem.

I am given that our data is distributed in: $x_n|(\mu,\sigma^2)$ ~ $N(\mu,\sigma^2)$. Using this, we are also given that the priors are respectively $\mu$ ~ $N(\mu_0,\sigma^2_0)$ and $\sigma^2$ ~ $Gamma^{-1}(a,b)$. I assume that the prior for sigma squared is simply the inverse gamma function. We are also given all values for $\mu_0,\sigma^2_0,a,b$ with them being $2,1,1,$ and $1$ respectively. That means they are all fixed and known which should be very helpful in getting the full posterior and full conditionals.

I have started this problem by attempting to build a DAG model in which the priors both point to the data distribution which is also normal. Is it as simple as just multiplying the two priors (pdfs) to get the posterior? In which case, the distribution for $\mu$ is simple since it is just normal and $\sigma^2$ would just be $~Gamma(a,\frac{1}{b})$. But then, how would I obtain the conditionals since all parameters are specified and fixed?

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I assume that your goal is to obtain the posterior $\mu, \sigma^2 \vert x_n$. Based on Bayes' theorem, the posterior is given by $$ \begin{aligned} p(\mu, \sigma^2 \vert x_n) \propto p(x_n \vert \mu, \sigma^2) \ \times \ p(\mu, \sigma^2). \end{aligned} $$

The likelihood $p(x_n \vert \mu, \sigma^2)$ is a Gaussian $\mathcal{N}(\mu, \sigma^2)$ and the prior for $\mu$ and $\sigma$ are independent as $\mathcal{N}(\mu_0, \sigma_0^2)$ and ${\rm Gamma}^{-1}(a, b)$, respectively. Therefore, with the parameters $\mu_0, \sigma^2_0, a, b$ and the data $x_n$ specified, the posterior is given by $$ p(\mu, \sigma^2 \vert x_n) \propto \mathcal{N}(\mu, \sigma^2)(x_n) \ \times \ \mathcal{N}(\mu_0, \sigma_0^2)(\mu) \ \times \ {\rm Gamma}^{-1}(a, b)(\sigma^2). $$

With the above equation, you can either do a Monte Carlo sampling or work out the analytical expression for the posterior.

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