Linear regression with t-distribution prior for beta coefficients

Question

Having: $$y\sim N_n(X\beta, \sigma^2 I_n)$$ with prior distributions:

$$\beta\sim t_\nu(\beta_0, B_0)$$ and $$\sigma^2 \sim IG(\alpha_0/ 2, \delta_0/2)$$

What would be the conditional posterior of $\beta|\sigma^2, y, x$

I´m trying the following but don´t know if I´m in the right track:

$$\beta|\sigma^2, y, x \propto |B_0|^{N /2} \left(1+\frac{1}{\nu}(\beta-\beta_0)'B_0^{-1}(\beta- \beta_0)\right)^{-(\nu+k)/2}\times exp \left(-\frac{1}{2}\sum \sigma^{-2}(y_i-x_i'\beta)^2 \right)$$

But don't know if this would be a known distribution.

Also I´m thinking in doing: $$\beta\sim N(\beta_0, \lambda^{-1}B_0)$$ $$\lambda\sim G(\nu /2, \nu /2)$$ But I don't know if this would be ok.

Answer 1

The posterior conditional distribution:

$$\beta|y,x,\sigma^2,\lambda \sim N(\hat{\beta}, B_1)$$ with $$\hat{\beta}=B_1[\sigma^{-2}X'y+\lambda^{-1}B_0^{-1}\beta_0 ]$$ and $$B_1=(\lambda B_0^{-1}+\sigma^{-2}X'X)^{-1}$$