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.
self-study
tag if so and in any case tell us where you are stuck with this derivation? Note that the prior on $\sigma^2$ is not useful for that question. $\endgroup$ – Xi'an Dec 15 '15 at 8:39