If I have the following model:
$$y\sim N_n(X\beta, \sigma^2 I_n)$$ with prior distributions:
$$\beta\sim N_n(\beta_0, B_0)$$ and $$\sigma^2 \sim IG(\alpha_0/ 2, \delta_0/2)$$
What would be the posterior of $\theta=(\beta,\sigma^2)$?
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1I didn't locate the duplicate I was after, but here's a pointer to some related answers ... one of the main parts of the calculation is discussed in the answers Bayesian regression full conditional distribution and Posterior Distribution for Bayesian Linear Regression – Glen_b Dec 14 '15 at 03:38
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The joint posterior is a Normal-Inverse Gamma, with parameters $(\mu_n, B_n, \alpha_n, \beta_n)$ where:
$$B_n = \frac{1}{B_{0}^{-1} + n}$$
$$\mu_n = B_n(B_0^{-1}\mu_0 + n\bar{x})$$
$$\alpha_n = \alpha_0 + \frac{n}{2}$$
$$\beta_n = \beta_0 + \frac{1}{2}[\mu_0^{2}B_0^{-1} + \sum_{i}x_{i}^{2} - \mu^{2}_nB_n^{-1}]$$
Deriving these parameters can be beastly (see here for more details). Usually people solve for the full conditionals of $\beta$ and $\sigma^{2}$ (they each take a more manageable closed form distribution) and use Gibbs sampling to find the posterior for each parameter.
ilanman
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Thanks for your answer. What about if I change the prior of $\beta$ to be a multivariate t distribution $t_nu(\beta_0, B_0)$ – user2246905 Dec 15 '15 at 04:05
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I think that‘s not correct. The above question does not assume that $\beta$ depends on $\sigma$. Your answer only works for $\beta\sim N_n(\beta_0, \sigma B_0)$. – Frederik Ziebell Jun 25 '20 at 21:10